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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

The Voronoi diagrams technique, introduced by Cabello [SODA'17] to compute the diameter of planar graphs in subquadratic time, has revolutionized the field of distance computations in planar graphs. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted planar graphs, as well as some new limitations.
- In the static case, we give n^{3+o(1)}/D² and Õ(n⋅D²) time algorithms for computing the diameter of a planar graph G with diameter D. These are faster than the state of the art Õ(n^{5/3}) [SODA'18] when D < n^{1/3} or D > n^{2/3}.
- In the fault-tolerant setting, we give an n^{7/3+o(1)} time algorithm for computing the diameter of G⧵ {e} for every edge e in G (the replacement diameter problem). This should be compared with the naive Õ(n^{8/3}) time algorithm that runs the static algorithm for every edge.
- In the incremental setting, where we wish to maintain the diameter while adding edges, we present an algorithm with total running time n^{7/3+o(1)}. This should be compared with the naive Õ(n^{8/3}) time algorithm that runs the static algorithm after every update.
- We give a lower bound (conditioned on the SETH) ruling out an amortized O(n^{1-ε}) update time for maintaining the diameter in weighted planar graph. The lower bound holds even for incremental or decremental updates.
Our upper bounds are obtained by novel uses and manipulations of Voronoi diagrams. These include maintaining the Voronoi diagram when edges of the graph are deleted, allowing the sites of the Voronoi diagram to lie on a BFS tree level (rather than on boundaries of r-division), and a new reduction from incremental diameter to incremental distance oracles that could be of interest beyond planar graphs. Our lower bound is the first lower bound for a dynamic planar graph problem that is conditioned on the SETH.

Amir Abboud, Shay Mozes, and Oren Weimann. What Else Can Voronoi Diagrams Do for Diameter in Planar Graphs?. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 4:1-4:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abboud_et_al:LIPIcs.ESA.2023.4, author = {Abboud, Amir and Mozes, Shay and Weimann, Oren}, title = {{What Else Can Voronoi Diagrams Do for Diameter in Planar Graphs?}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {4:1--4:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.4}, URN = {urn:nbn:de:0030-drops-186575}, doi = {10.4230/LIPIcs.ESA.2023.4}, annote = {Keywords: Planar graphs, diameter, dynamic graphs, fault tolerance} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Let G = (V,E) be an undirected unweighted planar graph. Let S = {s_1,…,s_k} be the vertices of some face in G and let T ⊆ V be an arbitrary set of vertices. The Okamura-Seymour metric compression problem asks to compactly encode the S-to-T distances.
Consider a vector storing the distances from an arbitrary vertex v to all vertices S = {s_1,…,s_k} in their cyclic order. The pattern of v is obtained by taking the difference between every pair of consecutive values of this vector. In STOC'19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted p_#, is only O(k³). This resulted in a simple Õ(min{k⁴+|T|, k⋅|T|}) space compression of the Okamura-Seymour metric.
We give an alternative proof of the p_# = O(k³) bound that exploits planarity beyond the VC-dimension argument. Namely, our proof relies on cut-cycle duality, as well as on the fact that distances among vertices of S are bounded by k. Our method implies the following:
(1) An Õ(p_#+k+|T|) space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to Õ(min{k³+|T|, k⋅|T|}).
(2) An optimal Õ(k+|T|) space compression of the Okamura-Seymour metric, in the case where the vertices of T induce a connected component in G.
(3) A tight bound of p_# = Θ(k²) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing p_# = O(k³).

Shay Mozes, Nathan Wallheimer, and Oren Weimann. Improved Compression of the Okamura-Seymour Metric. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 27:1-27:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mozes_et_al:LIPIcs.ISAAC.2022.27, author = {Mozes, Shay and Wallheimer, Nathan and Weimann, Oren}, title = {{Improved Compression of the Okamura-Seymour Metric}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {27:1--27:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.27}, URN = {urn:nbn:de:0030-drops-173123}, doi = {10.4230/LIPIcs.ISAAC.2022.27}, annote = {Keywords: Shortest paths, planar graphs, metric compression, distance oracles} }

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**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Given two strings S and P, the Episode Matching problem is to find the shortest substring of S that contains P as a subsequence. The best known upper bound for this problem is Õ(nm) by Das et al. (1997), where n,m are the lengths of S and P, respectively. Although the problem is well studied and has many applications in data mining, this bound has never been improved. In this paper we show why this is the case by proving that no O((nm)^{1-ε}) algorithm (even for binary strings) exists, unless the Strong Exponential Time Hypothesis (SETH) is false.
We then consider the indexing version of the problem, where S is preprocessed into a data structure for answering episode matching queries P. We show that for any τ, there is a data structure using O(n+(n/(τ)) ^k) space that answers episode matching queries for any P of length k in O(k⋅ τ ⋅ log log n) time. We complement this upper bound with an almost matching lower bound, showing that any data structure that answers episode matching queries for patterns of length k in time O(n^δ), must use Ω(n^{k-kδ-o(1)}) space, unless the Strong k-Set Disjointness Conjecture is false. Finally, for the special case of k = 2, we present a faster construction of the data structure using fast min-plus multiplication of bounded integer matrices.

Philip Bille, Inge Li Gørtz, Shay Mozes, Teresa Anna Steiner, and Oren Weimann. The Fine-Grained Complexity of Episode Matching. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 4:1-4:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bille_et_al:LIPIcs.CPM.2022.4, author = {Bille, Philip and G{\o}rtz, Inge Li and Mozes, Shay and Steiner, Teresa Anna and Weimann, Oren}, title = {{The Fine-Grained Complexity of Episode Matching}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {4:1--4:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.4}, URN = {urn:nbn:de:0030-drops-161312}, doi = {10.4230/LIPIcs.CPM.2022.4}, annote = {Keywords: Pattern matching, fine-grained complexity, longest common subsequence} }

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

We present a truly subquadratic size distance oracle for reporting, in constant time, the exact shortest-path distance between any pair of vertices of an undirected, unweighted planar graph G. For any ε > 0, our distance oracle requires O(n^{5/3+ε}) space and is capable of answering shortest-path distance queries exactly for any pair of vertices of G in worst-case time O(log (1/ε)). Previously no truly sub-quadratic size distance oracles with constant query time for answering exact shortest paths distance queries existed.

Viktor Fredslund-Hansen, Shay Mozes, and Christian Wulff-Nilsen. Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 25:1-25:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fredslundhansen_et_al:LIPIcs.ISAAC.2021.25, author = {Fredslund-Hansen, Viktor and Mozes, Shay and Wulff-Nilsen, Christian}, title = {{Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {25:1--25:12}, 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.2021.25}, URN = {urn:nbn:de:0030-drops-154586}, doi = {10.4230/LIPIcs.ISAAC.2021.25}, annote = {Keywords: distance oracle, planar graph, shortest paths, subquadratic} }

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

We give an O(k³ Δ n log n min(k, log² n) log²(nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks. This improves by a factor of max(k²,k log² n) over the fastest algorithm previously known for this problem [Wang, SODA 2019].
The speedup is obtained by two independent ideas. First we replace an iterative procedure of Wang that uses O(k) invocations of an O(k³ n log³ n)-time algorithm for maximum flow algorithm in a planar graph with k apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al. Second, we show two alternatives for computing flows in the k-apex graphs that arise in our modification of Wang’s procedure faster than the algorithm of Borradaile et al. In doing so, we introduce and analyze a sequential implementation of the parallel highest-distance push-relabel algorithm of Goldberg and Tarjan [JACM 1988].

Julian Enoch, Kyle Fox, Dor Mesica, and Shay Mozes. A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 72:1-72:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{enoch_et_al:LIPIcs.ISAAC.2021.72, author = {Enoch, Julian and Fox, Kyle and Mesica, Dor and Mozes, Shay}, title = {{A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {72:1--72:16}, 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.2021.72}, URN = {urn:nbn:de:0030-drops-155057}, doi = {10.4230/LIPIcs.ISAAC.2021.72}, annote = {Keywords: flow, planar graphs, vertex capacities} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We consider the problem of preprocessing two strings S and T, of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i,j in S and positions a,b in T, return the optimal alignment score of S[i..j] and T[a..b]. Let N = mn. We present an oracle with preprocessing time N^{1+o(1)} and space N^{1+o(1)} that answers queries in log^{2+o(1)}N time. In other words, we show that we can efficiently query for the alignment score of every pair of substrings after preprocessing the input for almost the same time it takes to compute just the alignment of S and T. Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS).
The best previously known oracle with construction time and size 𝒪(N) has slow Ω(√N) query time [Sakai, TCS 2019], and the one with size N^{1+o(1)} and query time log^{2+o(1)}N (using a planar graph distance oracle) has slow Ω(N^{3/2}) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a √ N factor.

Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, and Oren Weimann. An Almost Optimal Edit Distance Oracle. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 48:1-48:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{charalampopoulos_et_al:LIPIcs.ICALP.2021.48, author = {Charalampopoulos, Panagiotis and Gawrychowski, Pawe{\l} and Mozes, Shay and Weimann, Oren}, title = {{An Almost Optimal Edit Distance Oracle}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {48:1--48:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.48}, URN = {urn:nbn:de:0030-drops-141175}, doi = {10.4230/LIPIcs.ICALP.2021.48}, annote = {Keywords: longest common subsequence, edit distance, planar graphs, Voronoi diagrams} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We give a randomized algorithm that finds a minimum cut in an undirected weighted m-edge n-vertex graph G with high probability in O(m log² n) time. This is the first improvement to Karger’s celebrated O(m log³ n) time algorithm from 1996. Our main technical contribution is a deterministic O(m log n) time algorithm that, given a spanning tree T of G, finds a minimum cut of G that 2-respects (cuts two edges of) T.

Paweł Gawrychowski, Shay Mozes, and Oren Weimann. Minimum Cut in O(m log² n) Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 57:1-57:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gawrychowski_et_al:LIPIcs.ICALP.2020.57, author = {Gawrychowski, Pawe{\l} and Mozes, Shay and Weimann, Oren}, title = {{Minimum Cut in O(m log² n) Time}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.57}, URN = {urn:nbn:de:0030-drops-124646}, doi = {10.4230/LIPIcs.ICALP.2020.57}, annote = {Keywords: Minimum cut, Minimum 2-respecting cut} }

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**Published in:** LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

We consider the problem of dynamically maintaining an optimal alignment of two strings, each of length at most n, as they undergo insertions, deletions, and substitutions of letters. The string alignment problem generalizes the longest common subsequence (LCS) problem and the edit distance problem (also with non-unit costs, as long as insertions and deletions cost the same). The conditional lower bound of Backurs and Indyk [J. Comput. 2018] for computing the LCS in the static case implies that strongly sublinear update time for the dynamic string alignment problem would refute the Strong Exponential Time Hypothesis. We essentially match this lower bound when the alignment weights are constants, by showing how to process each update in 𝒪̃(n) time. When the weights are integers bounded in absolute value by some w=n^{𝒪(1)}, we can maintain the alignment in 𝒪̃(n ⋅ min {√ n,w}) time per update. For the 𝒪̃(nw)-time algorithm, we heavily rely on Tiskin’s work on semi-local LCS, and in particular, in an implicit way, on his algorithm for computing the (min,+)-product of two simple unit-Monge matrices [Algorithmica 2015]. As for the 𝒪̃(n√n)-time algorithm, we employ efficient data structures for computing distances in planar graphs.

Panagiotis Charalampopoulos, Tomasz Kociumaka, and Shay Mozes. Dynamic String Alignment. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 9:1-9:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{charalampopoulos_et_al:LIPIcs.CPM.2020.9, author = {Charalampopoulos, Panagiotis and Kociumaka, Tomasz and Mozes, Shay}, title = {{Dynamic String Alignment}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {9:1--9:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.9}, URN = {urn:nbn:de:0030-drops-121344}, doi = {10.4230/LIPIcs.CPM.2020.9}, annote = {Keywords: string alignment, edit distance, longest common subsequence, (unit-)Monge matrices, (min,+)-product} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator.
We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in O~(n) time a distance emulator of size O~(min(k^2,sqrt{k * n})). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3}).

Hsien-Chih Chang, Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Near-Optimal Distance Emulator for Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chang_et_al:LIPIcs.ESA.2018.16, author = {Chang, Hsien-Chih and Gawrychowski, Pawel and Mozes, Shay and Weimann, Oren}, title = {{Near-Optimal Distance Emulator for Planar Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {16:1--16:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.16}, URN = {urn:nbn:de:0030-drops-94796}, doi = {10.4230/LIPIcs.ESA.2018.16}, annote = {Keywords: planar graphs, shortest paths, metric compression, distance preservers, distance emulators, distance oracles} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

In the k-dispersion problem, we need to select k nodes of a given graph so as to maximize the minimum distance between any two chosen nodes. This can be seen as a generalization of the independent set problem, where the goal is to select nodes so that the minimum distance is larger than 1. We design an optimal O(n) time algorithm for the dispersion problem on trees consisting of n nodes, thus improving the previous O(n log n) time solution from 1997.
We also consider the weighted case, where the goal is to choose a set of nodes of total weight at least W. We present an O(n log^2n) algorithm improving the previous O(n log^4 n) solution. Our solution builds on the search version (where we know the minimum distance lambda between the chosen nodes) for which we present tight Theta(n log n) upper and lower bounds.

Pawel Gawrychowski, Nadav Krasnopolsky, Shay Mozes, and Oren Weimann. Dispersion on Trees. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 40:1-40:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gawrychowski_et_al:LIPIcs.ESA.2017.40, author = {Gawrychowski, Pawel and Krasnopolsky, Nadav and Mozes, Shay and Weimann, Oren}, title = {{Dispersion on Trees}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {40:1--40:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.40}, URN = {urn:nbn:de:0030-drops-78438}, doi = {10.4230/LIPIcs.ESA.2017.40}, annote = {Keywords: parametric search, dispersion, k-center, dynamic programming} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v,c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log log n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log n) time. We complement this by showing that O(polylog n) update time implies Omega(log n \ log log n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor/descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.

Pawel Gawrychowski, Gad M. Landau, Shay Mozes, and Oren Weimann. The Nearest Colored Node in a Tree. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 25:1-25:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gawrychowski_et_al:LIPIcs.CPM.2016.25, author = {Gawrychowski, Pawel and Landau, Gad M. and Mozes, Shay and Weimann, Oren}, title = {{The Nearest Colored Node in a Tree}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {25:1--25:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.25}, URN = {urn:nbn:de:0030-drops-60674}, doi = {10.4230/LIPIcs.CPM.2016.25}, annote = {Keywords: Marked ancestor, Vertex-label distance oracles, Nearest colored descend- ant, Top-trees} }

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