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**Published in:** LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

We study the Solo-Chess problem which has been introduced in [Aravind et al., FUN 2022]. This is a single-player variant of chess in which the player must clear all but one piece from the board via a sequence captures while ensuring that the number of captures performed by each piece does not exceed the piece’s budget. The time complexity of finding a winning sequence of captures has already been pinpointed for several combination of piece types and initial budgets. We contribute to a better understanding of the computational landscape of Solo-Chess by closing two problems left open in [Aravind et al., FUN 2022]. Namely, we show that Solo-Chess is hard even when all pieces are restricted to be only rooks with budget exactly 2, or only knights with budget exactly 11.

Davide Bilò, Luca Di Donato, Luciano Gualà, and Stefano Leucci. Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2024.4, author = {Bil\`{o}, Davide and Di Donato, Luca and Gual\`{a}, Luciano and Leucci, Stefano}, title = {{Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard}}, booktitle = {12th International Conference on Fun with Algorithms (FUN 2024)}, pages = {4:1--4:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-314-0}, ISSN = {1868-8969}, year = {2024}, volume = {291}, editor = {Broder, Andrei Z. and Tamir, Tami}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.4}, URN = {urn:nbn:de:0030-drops-199121}, doi = {10.4230/LIPIcs.FUN.2024.4}, annote = {Keywords: solo chess, puzzle games, board games, NP-completeness} }

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**Published in:** LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

There was a mix-up in Escher’s bar and n customers sitting at the same table have each received a beer ordered by somebody else in the party. The drinks can be rearranged by swapping them in pairs, but the eccentric table shape only allows drinks to be exchanged between people sitting on opposite sides of the table. We study the problem of finding the minimum number of swaps needed so that each customer receives its desired beer before it gets warm.
Formally, we consider the Colored Token Swapping problem on complete bipartite graphs. This problem is known to be solvable in polynomial time when all ordered drinks are different [Yamanaka et al., FUN 2014], but no results are known for the more general case in which multiple people in the party can order the same beer. We prove that Colored Token Swapping on complete bipartite graphs is NP-hard and that it is fixed-parameter tractable when parameterized by the number of distinct types of beer served by the bar.

Davide Bilò, Maurizio Fiusco, Luciano Gualà, and Stefano Leucci. Swapping Mixed-Up Beers to Keep Them Cool. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2024.5, author = {Bil\`{o}, Davide and Fiusco, Maurizio and Gual\`{a}, Luciano and Leucci, Stefano}, title = {{Swapping Mixed-Up Beers to Keep Them Cool}}, booktitle = {12th International Conference on Fun with Algorithms (FUN 2024)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-314-0}, ISSN = {1868-8969}, year = {2024}, volume = {291}, editor = {Broder, Andrei Z. and Tamir, Tami}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.5}, URN = {urn:nbn:de:0030-drops-199132}, doi = {10.4230/LIPIcs.FUN.2024.5}, annote = {Keywords: Colored Token Swapping, Complete Bipartite Graphs, Labeled Token Swapping, FPT Algorithms, NP-Hardness} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

A temporal graph is an undirected graph G = (V,E) along with a function λ : E → ℕ^+ that assigns a time-label to each edge in E. A path in G such that the traversed time-labels are non-decreasing is called a temporal path. Accordingly, the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal α-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of α. The size of H is measured as the number of its edges.
In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2k-1)-spanner with Õ(kn^{1+1/k}) edges, where k > 1 is an integer parameter of choice. Choosing k = ⌊log n⌋, we obtain a temporal O(log n)-spanner with Õ(n) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity.
We then turn our attention to general temporal graphs. Since Ω(n²) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that Õ(n/log(1+ε)) edges suffice to obtain a stretch of (1+ε), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω(n²) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of Õ(n² / β) on the size of any temporal β-additive spanner, which we show to be tight up to polylogarithmic factors.
Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Sparse Temporal Spanners with Low Stretch. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2022.19, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Rossi, Mirko}, title = {{Sparse Temporal Spanners with Low Stretch}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.19}, URN = {urn:nbn:de:0030-drops-169575}, doi = {10.4230/LIPIcs.ESA.2022.19}, annote = {Keywords: temporal spanners, temporal graphs, graph sparsification, approximate distances} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem.
We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {12:1--12:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.12}, URN = {urn:nbn:de:0030-drops-158221}, doi = {10.4230/LIPIcs.STACS.2022.12}, annote = {Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We study the problem of designing a resilient data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC'04] in which up to δ memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports resilient (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in O(δ) worst-case time per operation.

Luciano Gualà, Stefano Leucci, and Isabella Ziccardi. Resilient Level Ancestor, Bottleneck, and Lowest Common Ancestor Queries in Dynamic Trees. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 66:1-66:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{guala_et_al:LIPIcs.ISAAC.2021.66, author = {Gual\`{a}, Luciano and Leucci, Stefano and Ziccardi, Isabella}, title = {{Resilient Level Ancestor, Bottleneck, and Lowest Common Ancestor Queries in Dynamic Trees}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {66:1--66:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.66}, URN = {urn:nbn:de:0030-drops-154998}, doi = {10.4230/LIPIcs.ISAAC.2021.66}, annote = {Keywords: level ancestor queries, lowest common ancestor queries, bottleneck vertex queries, resilient data structures, faulty-RAM model, dynamic trees} }

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**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house.
The garden consists of n bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the Bamboo Garden Trimming (BGT) problem, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan, i.e., the maximum height ever reached by a bamboo.
Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than x. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of O(log n) and 4 for the best choice of x = 2, respectively. We prove the first constant upper bound of 9 for Reduce-Max and improve the one for Reduce-Fastest(x) to (3+√5)/2 < 2.62 for x = 1+1/√5.
Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x).

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Giacomo Scornavacca. Cutting Bamboo down to Size. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2021.5, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Scornavacca, Giacomo}, title = {{Cutting Bamboo down to Size}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.5}, URN = {urn:nbn:de:0030-drops-127663}, doi = {10.4230/LIPIcs.FUN.2021.5}, annote = {Keywords: bamboo garden trimming, trimming oracles, approximation algorithms, pinwheel scheduling} }

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Extended Abstract

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast?
In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent.
We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise).
We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

Andrea Clementi, Luciano Gualà, Emanuele Natale, Francesco Pasquale, Giacomo Scornavacca, and Luca Trevisan. Consensus vs Broadcast, with and Without Noise (Extended Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{clementi_et_al:LIPIcs.ITCS.2020.42, author = {Clementi, Andrea and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo and Trevisan, Luca}, title = {{Consensus vs Broadcast, with and Without Noise}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {42:1--42:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.42}, URN = {urn:nbn:de:0030-drops-117277}, doi = {10.4230/LIPIcs.ITCS.2020.42}, annote = {Keywords: Distributed Computing, Consensus, Broadcast, Gossip Models, Noisy Communication Channels} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The Undecided-State Dynamics is a well-known protocol for distributed consensus. We analyze it in the parallel PULL communication model on the complete graph with n nodes for the binary case (every node can either support one of two possible colors, or be in the undecided state).
An interesting open question is whether this dynamics is an efficient Self-Stabilizing protocol, namely, starting from an arbitrary initial configuration, it reaches consensus quickly (i.e., within a polylogarithmic number of rounds). Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Theta(n)) towards the majority color.
In this paper we present an unconditional analysis of the Undecided-State Dynamics that answers to the above question in the affirmative. We prove that, starting from any initial configuration, the process reaches a monochromatic configuration within O(log n) rounds, with high probability. This bound turns out to be tight. Our analysis also shows that, if the initial configuration has bias Omega(sqrt(n log n)), then the dynamics converges toward the initial majority color, with high probability.

Andrea Clementi, Mohsen Ghaffari, Luciano Gualà, Emanuele Natale, Francesco Pasquale, and Giacomo Scornavacca. A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{clementi_et_al:LIPIcs.MFCS.2018.28, author = {Clementi, Andrea and Ghaffari, Mohsen and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo}, title = {{A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.28}, URN = {urn:nbn:de:0030-drops-96107}, doi = {10.4230/LIPIcs.MFCS.2018.28}, annote = {Keywords: Distributed Consensus, Self-Stabilization, PULL Model, Markov Chains} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Neeldhara Misra. On the Complexity of Two Dots for Narrow Boards and Few Colors. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.7, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Misra, Neeldhara}, title = {{On the Complexity of Two Dots for Narrow Boards and Few Colors}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.7}, URN = {urn:nbn:de:0030-drops-87988}, doi = {10.4230/LIPIcs.FUN.2018.7}, annote = {Keywords: puzzle, NP-complete, perfect information, combinatorial game theory} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors.

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.8, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.8}, URN = {urn:nbn:de:0030-drops-87994}, doi = {10.4230/LIPIcs.FUN.2018.8}, annote = {Keywords: peg duotaire, pspace-completeness, peg solitaire, two-player games} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \in S x T, where S, T \subseteq V(G), and every e \in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)).
More precisely, we provide the following results:
(1) DSO:
For every S,T \subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\sqrt{|S||T|}) and query time of
O(\sqrt{|S||T|}). At the expense of having quasi-polynomial query time,
the size of the oracle can be improved to tilde{O}(n|S|+|T|\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings.
(2) FTP:
We show the construction of a path-reporting DSO of size tilde{O}(n^{4/3}(|S||T|)^{1/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors).
(3) Routing and Labeling Schemes:
For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.

Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.13, author = {Bil\`{o}, Davide and Choudhary, Keerti and Gual\`{a}, Luciano and Leucci, Stefano and Parter, Merav and Proietti, Guido}, title = {{Efficient Oracles and Routing Schemes for Replacement Paths}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.13}, URN = {urn:nbn:de:0030-drops-85249}, doi = {10.4230/LIPIcs.STACS.2018.13}, annote = {Keywords: Fault tolerant, Shortest path, Oracle, Routing} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

A tree sigma-spanner of a positively real-weighted n-vertex and m-edge undirected graph G is a spanning tree T of G which approximately preserves (i.e., up to a multiplicative stretch factor sigma) distances in G.
Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in T, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge -- a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in G, but also in all the scenarios in which an edge of T has failed. For this problem we provide a very efficient solution, running in O(n^2 log^4 n) time, which drastically improves (almost by a quadratic factor in n in dense graphs!) on the previous known best result.

Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2017.14, author = {Bil\`{o}, Davide and Colella, Feliciano and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.14}, URN = {urn:nbn:de:0030-drops-82663}, doi = {10.4230/LIPIcs.ISAAC.2017.14}, annote = {Keywords: Transient edge failure, Swap algorithm, Tree spanner} }

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Brief Announcement

**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

The Undecided-State Dynamics is a well-known protocol that achieves Consensus in distributed systems formed by a set of n anonymous nodes interacting via a communication network. We consider this dynamics in the parallel PULL communication model on the complete graph for the binary case, i.e., when every node can either support one of two possible colors or stay in the undecided state. Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Theta(n)) towards the majority color. A interesting open question here is whether this dynamics reaches consensus quickly, i.e. within a polylogarithmic number of rounds. In this paper we present an unconditional analysis of the Undecided-State Dynamics which answers to the above question in the affirmative. Our analysis shows that, starting from any initial configuration, the Undecided-State Dynamics reaches a monochromatic configuration within O(log^2 n) rounds, with high probability (w.h.p.). Moreover, we prove that if the initial configuration has bias Omega(sqrt(n log n)), then the dynamics converges toward the initial majority color within O(log n) round, w.h.p. At the heart of our approach there is a new analysis of the symmetry-breaking phase that the process must perform in order to escape from (almost-)unbiased configurations. Previous symmetry-breaking analysis of consensus dynamics essentially concern sequential communication models (such as Population Protocols) and/or symmetric updated rules (such as majority rules).

Andrea Clementi, Luciano Gualà, Francesco Pasquale, and Giacomo Scornavacca. Brief Announcement: On the Parallel Undecided-State Dynamics with Two Colors. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 47:1-47:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{clementi_et_al:LIPIcs.DISC.2017.47, author = {Clementi, Andrea and Gual\`{a}, Luciano and Pasquale, Francesco and Scornavacca, Giacomo}, title = {{Brief Announcement: On the Parallel Undecided-State Dynamics with Two Colors}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {47:1--47:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.47}, URN = {urn:nbn:de:0030-drops-79724}, doi = {10.4230/LIPIcs.DISC.2017.47}, annote = {Keywords: Distributed Consensus, Dynamics, Gossip model, Markov chains} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0<epsilon<1, another sensitivity oracle having size O(n*1/epsilon*log(1/epsilon)), and is able to report a (1+epsilon)-approximate distance from s to any vertex of G in O(log(n)*1/epsilon*log(1/epsilon)) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.

Davide Bilo, Luciano Guala, Stefano Leucci, and Guido Proietti. Compact and Fast Sensitivity Oracles for Single-Source Distances. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2016.13, author = {Bilo, Davide and Guala, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Compact and Fast Sensitivity Oracles for Single-Source Distances}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.13}, URN = {urn:nbn:de:0030-drops-63640}, doi = {10.4230/LIPIcs.ESA.2016.13}, annote = {Keywords: fault-tolerant shortest-path tree, approximate distance, distance sensitivity oracle} }

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**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csakany and Juhasz (Mathematics Magazine, 2000).

Luciano Gualà, Stefano Leucci, Emanuele Natale, and Roberto Tauraso. Large Peg-Army Maneuvers. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{guala_et_al:LIPIcs.FUN.2016.18, author = {Gual\`{a}, Luciano and Leucci, Stefano and Natale, Emanuele and Tauraso, Roberto}, title = {{Large Peg-Army Maneuvers}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.18}, URN = {urn:nbn:de:0030-drops-58709}, doi = {10.4230/LIPIcs.FUN.2016.18}, annote = {Keywords: Complexity of Games, Solitaire Army} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer f >= 1, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) sigma-approximate single-source shortest-path tree (sigma-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched at most by a factor of sigma. To this respect, we provide an algorithm that efficiently computes an f-EFT (2|F|+1)-ASPT of size O(f n). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1), plus an additive term of (f+1)*log(n).
Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle (SSDO), that can be built in ~{O}(f m) time, has size O(fn *log^2(n)), and is able to report, after the failure of the edge set F, in O(|F|^2 * log^2(n)) time a (2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G after that a batch of k simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(m * log^3(n)) time a sensitivity oracle of size O(m * log^2(n)), that reports in O(k^2 * log^2(n)) time the (at most 2k) edges either exiting from or entering into the MSF. As a result of independent interest, it is worth noticing that our MSF oracle can be employed to handle arbitrary sequences of o(sqrt[4]{n}/log(n)) (non-simultaneous) updates with a worst-case time per update of o(sqrt{n}). Thus, for relatively short sequences of updates, our oracle should be preferred w.r.t. the best-known (in a worst-case sense) MSF fully-dynamic algorithm, requiring O(sqrt{n}) time per update.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2016.18, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.18}, URN = {urn:nbn:de:0030-drops-57196}, doi = {10.4230/LIPIcs.STACS.2016.18}, annote = {Keywords: fault-tolerant shortest-path tree, distance oracle, minimum spanning tree} }

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**Published in:** LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

We study the Solo-Chess problem which has been introduced in [Aravind et al., FUN 2022]. This is a single-player variant of chess in which the player must clear all but one piece from the board via a sequence captures while ensuring that the number of captures performed by each piece does not exceed the piece’s budget. The time complexity of finding a winning sequence of captures has already been pinpointed for several combination of piece types and initial budgets. We contribute to a better understanding of the computational landscape of Solo-Chess by closing two problems left open in [Aravind et al., FUN 2022]. Namely, we show that Solo-Chess is hard even when all pieces are restricted to be only rooks with budget exactly 2, or only knights with budget exactly 11.

Davide Bilò, Luca Di Donato, Luciano Gualà, and Stefano Leucci. Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2024.4, author = {Bil\`{o}, Davide and Di Donato, Luca and Gual\`{a}, Luciano and Leucci, Stefano}, title = {{Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard}}, booktitle = {12th International Conference on Fun with Algorithms (FUN 2024)}, pages = {4:1--4:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-314-0}, ISSN = {1868-8969}, year = {2024}, volume = {291}, editor = {Broder, Andrei Z. and Tamir, Tami}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.4}, URN = {urn:nbn:de:0030-drops-199121}, doi = {10.4230/LIPIcs.FUN.2024.4}, annote = {Keywords: solo chess, puzzle games, board games, NP-completeness} }

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**Published in:** LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

There was a mix-up in Escher’s bar and n customers sitting at the same table have each received a beer ordered by somebody else in the party. The drinks can be rearranged by swapping them in pairs, but the eccentric table shape only allows drinks to be exchanged between people sitting on opposite sides of the table. We study the problem of finding the minimum number of swaps needed so that each customer receives its desired beer before it gets warm.
Formally, we consider the Colored Token Swapping problem on complete bipartite graphs. This problem is known to be solvable in polynomial time when all ordered drinks are different [Yamanaka et al., FUN 2014], but no results are known for the more general case in which multiple people in the party can order the same beer. We prove that Colored Token Swapping on complete bipartite graphs is NP-hard and that it is fixed-parameter tractable when parameterized by the number of distinct types of beer served by the bar.

Davide Bilò, Maurizio Fiusco, Luciano Gualà, and Stefano Leucci. Swapping Mixed-Up Beers to Keep Them Cool. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2024.5, author = {Bil\`{o}, Davide and Fiusco, Maurizio and Gual\`{a}, Luciano and Leucci, Stefano}, title = {{Swapping Mixed-Up Beers to Keep Them Cool}}, booktitle = {12th International Conference on Fun with Algorithms (FUN 2024)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-314-0}, ISSN = {1868-8969}, year = {2024}, volume = {291}, editor = {Broder, Andrei Z. and Tamir, Tami}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.5}, URN = {urn:nbn:de:0030-drops-199132}, doi = {10.4230/LIPIcs.FUN.2024.5}, annote = {Keywords: Colored Token Swapping, Complete Bipartite Graphs, Labeled Token Swapping, FPT Algorithms, NP-Hardness} }

Document

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

A temporal graph is an undirected graph G = (V,E) along with a function λ : E → ℕ^+ that assigns a time-label to each edge in E. A path in G such that the traversed time-labels are non-decreasing is called a temporal path. Accordingly, the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal α-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of α. The size of H is measured as the number of its edges.
In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2k-1)-spanner with Õ(kn^{1+1/k}) edges, where k > 1 is an integer parameter of choice. Choosing k = ⌊log n⌋, we obtain a temporal O(log n)-spanner with Õ(n) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity.
We then turn our attention to general temporal graphs. Since Ω(n²) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that Õ(n/log(1+ε)) edges suffice to obtain a stretch of (1+ε), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω(n²) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of Õ(n² / β) on the size of any temporal β-additive spanner, which we show to be tight up to polylogarithmic factors.
Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Sparse Temporal Spanners with Low Stretch. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2022.19, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Rossi, Mirko}, title = {{Sparse Temporal Spanners with Low Stretch}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.19}, URN = {urn:nbn:de:0030-drops-169575}, doi = {10.4230/LIPIcs.ESA.2022.19}, annote = {Keywords: temporal spanners, temporal graphs, graph sparsification, approximate distances} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem.
We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {12:1--12:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.12}, URN = {urn:nbn:de:0030-drops-158221}, doi = {10.4230/LIPIcs.STACS.2022.12}, annote = {Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow} }

Document

**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We study the problem of designing a resilient data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC'04] in which up to δ memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports resilient (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in O(δ) worst-case time per operation.

Luciano Gualà, Stefano Leucci, and Isabella Ziccardi. Resilient Level Ancestor, Bottleneck, and Lowest Common Ancestor Queries in Dynamic Trees. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 66:1-66:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{guala_et_al:LIPIcs.ISAAC.2021.66, author = {Gual\`{a}, Luciano and Leucci, Stefano and Ziccardi, Isabella}, title = {{Resilient Level Ancestor, Bottleneck, and Lowest Common Ancestor Queries in Dynamic Trees}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {66:1--66:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.66}, URN = {urn:nbn:de:0030-drops-154998}, doi = {10.4230/LIPIcs.ISAAC.2021.66}, annote = {Keywords: level ancestor queries, lowest common ancestor queries, bottleneck vertex queries, resilient data structures, faulty-RAM model, dynamic trees} }

Document

**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house.
The garden consists of n bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the Bamboo Garden Trimming (BGT) problem, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan, i.e., the maximum height ever reached by a bamboo.
Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than x. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of O(log n) and 4 for the best choice of x = 2, respectively. We prove the first constant upper bound of 9 for Reduce-Max and improve the one for Reduce-Fastest(x) to (3+√5)/2 < 2.62 for x = 1+1/√5.
Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x).

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Giacomo Scornavacca. Cutting Bamboo down to Size. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2021.5, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Scornavacca, Giacomo}, title = {{Cutting Bamboo down to Size}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.5}, URN = {urn:nbn:de:0030-drops-127663}, doi = {10.4230/LIPIcs.FUN.2021.5}, annote = {Keywords: bamboo garden trimming, trimming oracles, approximation algorithms, pinwheel scheduling} }

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Extended Abstract

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast?
In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent.
We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise).
We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

Andrea Clementi, Luciano Gualà, Emanuele Natale, Francesco Pasquale, Giacomo Scornavacca, and Luca Trevisan. Consensus vs Broadcast, with and Without Noise (Extended Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{clementi_et_al:LIPIcs.ITCS.2020.42, author = {Clementi, Andrea and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo and Trevisan, Luca}, title = {{Consensus vs Broadcast, with and Without Noise}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {42:1--42:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.42}, URN = {urn:nbn:de:0030-drops-117277}, doi = {10.4230/LIPIcs.ITCS.2020.42}, annote = {Keywords: Distributed Computing, Consensus, Broadcast, Gossip Models, Noisy Communication Channels} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The Undecided-State Dynamics is a well-known protocol for distributed consensus. We analyze it in the parallel PULL communication model on the complete graph with n nodes for the binary case (every node can either support one of two possible colors, or be in the undecided state).
An interesting open question is whether this dynamics is an efficient Self-Stabilizing protocol, namely, starting from an arbitrary initial configuration, it reaches consensus quickly (i.e., within a polylogarithmic number of rounds). Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Theta(n)) towards the majority color.
In this paper we present an unconditional analysis of the Undecided-State Dynamics that answers to the above question in the affirmative. We prove that, starting from any initial configuration, the process reaches a monochromatic configuration within O(log n) rounds, with high probability. This bound turns out to be tight. Our analysis also shows that, if the initial configuration has bias Omega(sqrt(n log n)), then the dynamics converges toward the initial majority color, with high probability.

Andrea Clementi, Mohsen Ghaffari, Luciano Gualà, Emanuele Natale, Francesco Pasquale, and Giacomo Scornavacca. A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{clementi_et_al:LIPIcs.MFCS.2018.28, author = {Clementi, Andrea and Ghaffari, Mohsen and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo}, title = {{A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.28}, URN = {urn:nbn:de:0030-drops-96107}, doi = {10.4230/LIPIcs.MFCS.2018.28}, annote = {Keywords: Distributed Consensus, Self-Stabilization, PULL Model, Markov Chains} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Neeldhara Misra. On the Complexity of Two Dots for Narrow Boards and Few Colors. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.7, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Misra, Neeldhara}, title = {{On the Complexity of Two Dots for Narrow Boards and Few Colors}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.7}, URN = {urn:nbn:de:0030-drops-87988}, doi = {10.4230/LIPIcs.FUN.2018.7}, annote = {Keywords: puzzle, NP-complete, perfect information, combinatorial game theory} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors.

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.8, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.8}, URN = {urn:nbn:de:0030-drops-87994}, doi = {10.4230/LIPIcs.FUN.2018.8}, annote = {Keywords: peg duotaire, pspace-completeness, peg solitaire, two-player games} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \in S x T, where S, T \subseteq V(G), and every e \in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)).
More precisely, we provide the following results:
(1) DSO:
For every S,T \subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\sqrt{|S||T|}) and query time of
O(\sqrt{|S||T|}). At the expense of having quasi-polynomial query time,
the size of the oracle can be improved to tilde{O}(n|S|+|T|\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings.
(2) FTP:
We show the construction of a path-reporting DSO of size tilde{O}(n^{4/3}(|S||T|)^{1/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors).
(3) Routing and Labeling Schemes:
For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.

Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.13, author = {Bil\`{o}, Davide and Choudhary, Keerti and Gual\`{a}, Luciano and Leucci, Stefano and Parter, Merav and Proietti, Guido}, title = {{Efficient Oracles and Routing Schemes for Replacement Paths}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.13}, URN = {urn:nbn:de:0030-drops-85249}, doi = {10.4230/LIPIcs.STACS.2018.13}, annote = {Keywords: Fault tolerant, Shortest path, Oracle, Routing} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

A tree sigma-spanner of a positively real-weighted n-vertex and m-edge undirected graph G is a spanning tree T of G which approximately preserves (i.e., up to a multiplicative stretch factor sigma) distances in G.
Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in T, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge -- a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in G, but also in all the scenarios in which an edge of T has failed. For this problem we provide a very efficient solution, running in O(n^2 log^4 n) time, which drastically improves (almost by a quadratic factor in n in dense graphs!) on the previous known best result.

Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2017.14, author = {Bil\`{o}, Davide and Colella, Feliciano and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.14}, URN = {urn:nbn:de:0030-drops-82663}, doi = {10.4230/LIPIcs.ISAAC.2017.14}, annote = {Keywords: Transient edge failure, Swap algorithm, Tree spanner} }

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Brief Announcement

**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

The Undecided-State Dynamics is a well-known protocol that achieves Consensus in distributed systems formed by a set of n anonymous nodes interacting via a communication network. We consider this dynamics in the parallel PULL communication model on the complete graph for the binary case, i.e., when every node can either support one of two possible colors or stay in the undecided state. Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Theta(n)) towards the majority color. A interesting open question here is whether this dynamics reaches consensus quickly, i.e. within a polylogarithmic number of rounds. In this paper we present an unconditional analysis of the Undecided-State Dynamics which answers to the above question in the affirmative. Our analysis shows that, starting from any initial configuration, the Undecided-State Dynamics reaches a monochromatic configuration within O(log^2 n) rounds, with high probability (w.h.p.). Moreover, we prove that if the initial configuration has bias Omega(sqrt(n log n)), then the dynamics converges toward the initial majority color within O(log n) round, w.h.p. At the heart of our approach there is a new analysis of the symmetry-breaking phase that the process must perform in order to escape from (almost-)unbiased configurations. Previous symmetry-breaking analysis of consensus dynamics essentially concern sequential communication models (such as Population Protocols) and/or symmetric updated rules (such as majority rules).

Andrea Clementi, Luciano Gualà, Francesco Pasquale, and Giacomo Scornavacca. Brief Announcement: On the Parallel Undecided-State Dynamics with Two Colors. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 47:1-47:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{clementi_et_al:LIPIcs.DISC.2017.47, author = {Clementi, Andrea and Gual\`{a}, Luciano and Pasquale, Francesco and Scornavacca, Giacomo}, title = {{Brief Announcement: On the Parallel Undecided-State Dynamics with Two Colors}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {47:1--47:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.47}, URN = {urn:nbn:de:0030-drops-79724}, doi = {10.4230/LIPIcs.DISC.2017.47}, annote = {Keywords: Distributed Consensus, Dynamics, Gossip model, Markov chains} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0<epsilon<1, another sensitivity oracle having size O(n*1/epsilon*log(1/epsilon)), and is able to report a (1+epsilon)-approximate distance from s to any vertex of G in O(log(n)*1/epsilon*log(1/epsilon)) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.

Davide Bilo, Luciano Guala, Stefano Leucci, and Guido Proietti. Compact and Fast Sensitivity Oracles for Single-Source Distances. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2016.13, author = {Bilo, Davide and Guala, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Compact and Fast Sensitivity Oracles for Single-Source Distances}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.13}, URN = {urn:nbn:de:0030-drops-63640}, doi = {10.4230/LIPIcs.ESA.2016.13}, annote = {Keywords: fault-tolerant shortest-path tree, approximate distance, distance sensitivity oracle} }

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**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csakany and Juhasz (Mathematics Magazine, 2000).

Luciano Gualà, Stefano Leucci, Emanuele Natale, and Roberto Tauraso. Large Peg-Army Maneuvers. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{guala_et_al:LIPIcs.FUN.2016.18, author = {Gual\`{a}, Luciano and Leucci, Stefano and Natale, Emanuele and Tauraso, Roberto}, title = {{Large Peg-Army Maneuvers}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.18}, URN = {urn:nbn:de:0030-drops-58709}, doi = {10.4230/LIPIcs.FUN.2016.18}, annote = {Keywords: Complexity of Games, Solitaire Army} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer f >= 1, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) sigma-approximate single-source shortest-path tree (sigma-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched at most by a factor of sigma. To this respect, we provide an algorithm that efficiently computes an f-EFT (2|F|+1)-ASPT of size O(f n). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1), plus an additive term of (f+1)*log(n).
Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle (SSDO), that can be built in ~{O}(f m) time, has size O(fn *log^2(n)), and is able to report, after the failure of the edge set F, in O(|F|^2 * log^2(n)) time a (2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G after that a batch of k simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(m * log^3(n)) time a sensitivity oracle of size O(m * log^2(n)), that reports in O(k^2 * log^2(n)) time the (at most 2k) edges either exiting from or entering into the MSF. As a result of independent interest, it is worth noticing that our MSF oracle can be employed to handle arbitrary sequences of o(sqrt[4]{n}/log(n)) (non-simultaneous) updates with a worst-case time per update of o(sqrt{n}). Thus, for relatively short sequences of updates, our oracle should be preferred w.r.t. the best-known (in a worst-case sense) MSF fully-dynamic algorithm, requiring O(sqrt{n}) time per update.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2016.18, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.18}, URN = {urn:nbn:de:0030-drops-57196}, doi = {10.4230/LIPIcs.STACS.2016.18}, annote = {Keywords: fault-tolerant shortest-path tree, distance oracle, minimum spanning tree} }

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