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

A weighted directed graph G = (V,A,c), where A ⊆ V× V and c:A → ℝ, naturally describes a road network in which an electric car, or vehicle (EV), can roam. An arc uv ∈ A models a road segment connecting the two vertices (junctions) u and v. The cost c(uv) of the arc uv is the amount of energy the car needs to travel from u to v. This amount can be positive, zero or negative. We consider both the more realistic scenario where there are no negative cycles in the graph, as well as the more challenging scenario, which can also be motivated, where negative cycles may be present.
The electric car has a battery that can store up to B units of energy. The car can traverse an arc uv ∈ A only if it is at u and the charge b in its battery satisfies b ≥ c(uv). If the car traverses the arc uv then it reaches v with a charge of min{b-c(uv),B} in its battery. Arcs with a positive cost deplete the battery while arcs with negative costs may charge the battery, but not above its capacity of B. If the car is at a vertex u and cannot traverse any outgoing arcs of u, then it is stuck and cannot continue traveling.
We consider the following natural problem: Given two vertices s,t ∈ V, can the car travel from s to t, starting at s with an initial charge b, where 0 ≤ b ≤ B? If so, what is the maximum charge with which the car can reach t? Equivalently, what is the smallest depletion δ_{B,b}(s,t) such that the car can reach t with a charge of b-δ_{B,b}(s,t) in its battery, and which path should the car follow to achieve this? We also refer to δ_{B,b}(s,t) as the energetic cost of traveling from s to t. We let δ_{B,b}(s,t) = ∞ if the car cannot travel from s to t starting with an initial charge of b. The problem of computing energetic costs is a strict generalization of the standard shortest paths problem.
When there are no negative cycles, the single-source version of the problem can be solved using simple adaptations of the classical Bellman-Ford and Dijkstra algorithms. More involved algorithms are required when the graph may contain negative cycles.

Dani Dorfman, Haim Kaplan, Robert E. Tarjan, and Uri Zwick. Optimal Energetic Paths for Electric Cars. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dorfman_et_al:LIPIcs.ESA.2023.42, author = {Dorfman, Dani and Kaplan, Haim and Tarjan, Robert E. and Zwick, Uri}, title = {{Optimal Energetic Paths for Electric Cars}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {42:1--42:17}, 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.2023.42}, URN = {urn:nbn:de:0030-drops-186955}, doi = {10.4230/LIPIcs.ESA.2023.42}, annote = {Keywords: Electric cars, Optimal Paths, Battery depletion} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider word RAM data structures for maintaining ordered sets of integers whose select and rank operations are allowed to return approximate results, i.e., ranks, or items whose rank, differ by less than Delta from the exact answer, where Delta=Delta(n) is an error parameter. Related to approximate select and rank is approximate (one-dimensional) nearest-neighbor. A special case of approximate select queries are approximate min queries. Data structures that support approximate min operations are known as approximate heaps (priority queues). Related to approximate heaps are soft heaps, which are approximate heaps with a different notion of approximation.
We prove the optimality of all the data structures presented, either through matching cell-probe lower bounds, or through equivalences to well studied static problems. For approximate select, rank, and nearest-neighbor operations we get matching cell-probe lower bounds. We prove an equivalence between approximate min operations, i.e., approximate heaps, and the static partitioning problem. Finally, we prove an equivalence between soft heaps and the classical sorting problem, on a smaller number of items.
Our results have many interesting and unexpected consequences. It turns out that approximation greatly speeds up some of these operations, while others are almost unaffected. In particular, while select and rank have identical operation times, both in comparison-based and word RAM implementations, an interesting separation emerges between the approximate versions of these operations in the word RAM model. Approximate select is much faster than approximate rank. It also turns out that approximate min is exponentially faster than the more general approximate select. Next, we show that implementing soft heaps is harder than implementing approximate heaps. The relation between them corresponds to the relation between sorting and partitioning.
Finally, as an interesting byproduct, we observe that a combination of known techniques yields a deterministic word RAM algorithm for (exactly) sorting n items in O(n log log_w n) time, where w is the word length. Even for the easier problem of finding duplicates, the best previous deterministic bound was O(min{n log log n,n log_w n}). Our new unifying bound is an improvement when w is sufficiently large compared with n.

Mikkel Thorup, Or Zamir, and Uri Zwick. Dynamic Ordered Sets with Approximate Queries, Approximate Heaps and Soft Heaps. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 95:1-95:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{thorup_et_al:LIPIcs.ICALP.2019.95, author = {Thorup, Mikkel and Zamir, Or and Zwick, Uri}, title = {{Dynamic Ordered Sets with Approximate Queries, Approximate Heaps and Soft Heaps}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {95:1--95:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.95}, URN = {urn:nbn:de:0030-drops-106712}, doi = {10.4230/LIPIcs.ICALP.2019.95}, annote = {Keywords: Order queries, word RAM, lower bounds} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We present an improved exponential time algorithm for Energy Games, and hence also for Mean Payoff Games. The running time of the new algorithm is O (min(m n W, m n 2^{n/2} log W)), where n is the number of vertices, m is the number of edges, and when the edge weights are integers of absolute value at most W. For small values of W, the algorithm matches the performance of the pseudopolynomial time algorithm of Brim et al. on which it is based. For W >= n2^{n/2}, the new algorithm is faster than the algorithm of Brim et al. and is currently the fastest deterministic algorithm for Energy Games and Mean Payoff Games. The new algorithm is obtained by introducing a technique of forecasting repetitive actions performed by the algorithm of Brim et al., along with the use of an edge-weight scaling technique.

Dani Dorfman, Haim Kaplan, and Uri Zwick. A Faster Deterministic Exponential Time Algorithm for Energy Games and Mean Payoff Games (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 114:1-114:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dorfman_et_al:LIPIcs.ICALP.2019.114, author = {Dorfman, Dani and Kaplan, Haim and Zwick, Uri}, title = {{A Faster Deterministic Exponential Time Algorithm for Energy Games and Mean Payoff Games}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {114:1--114:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.114}, URN = {urn:nbn:de:0030-drops-106909}, doi = {10.4230/LIPIcs.ICALP.2019.114}, annote = {Keywords: Energy Games, Mean Payoff Games, Scaling} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

We use soft heaps to obtain simpler optimal algorithms for selecting the k-th smallest item, and the set of k smallest items, from a heap-ordered tree, from a collection of sorted lists, and from X+Y, where X and Y are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the k-th smallest item, or the set of k smallest items, from a collection of m sorted lists we obtain a new optimal "output-sensitive" algorithm that performs only O(m + sum_{i=1}^m log(k_i+1)) comparisons, where k_i is the number of items of the i-th list that belong to the overall set of k smallest items.

Haim Kaplan, László Kozma, Or Zamir, and Uri Zwick. Selection from Heaps, Row-Sorted Matrices, and X+Y Using Soft Heaps. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kaplan_et_al:OASIcs.SOSA.2019.5, author = {Kaplan, Haim and Kozma, L\'{a}szl\'{o} and Zamir, Or and Zwick, Uri}, title = {{Selection from Heaps, Row-Sorted Matrices, and X+Y Using Soft Heaps}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {5:1--5:21}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.5}, URN = {urn:nbn:de:0030-drops-100315}, doi = {10.4230/OASIcs.SOSA.2019.5}, annote = {Keywords: selection, soft heap} }

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

The pairing heap is a classical heap data structure introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. It is remarkable both for its simplicity and for its excellent performance in practice. The "magic" of pairing heaps lies in the restructuring that happens after the deletion of the smallest item. The resulting collection of trees is consolidated in two rounds: a left-to-right pairing round, followed by a right-to-left accumulation round. Fredman et al. showed, via an elegant correspondence to splay trees, that in a pairing heap of size n all heap operations take O(log n) amortized time. They also proposed an arguably more natural variant, where both pairing and accumulation are performed in a combined left-to-right round (called the forward variant of pairing heaps). The analogy to splaying breaks down in this case, and the analysis of the forward variant was left open.
In this paper we show that inserting an item and deleting the minimum in a forward-variant pairing heap both take amortized time O(log(n) * 4^(sqrt(log n))). This is the first improvement over the O(sqrt(n)) bound showed by Fredman et al. three decades ago. Our analysis relies on a new potential function that tracks parent-child rank-differences in the heap.

Dani Dorfman, Haim Kaplan, László Kozma, and Uri Zwick. Pairing heaps: the forward variant. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dorfman_et_al:LIPIcs.MFCS.2018.13, author = {Dorfman, Dani and Kaplan, Haim and Kozma, L\'{a}szl\'{o} and Zwick, Uri}, title = {{Pairing heaps: the forward variant}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {13:1--13:14}, 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.2018.13}, URN = {urn:nbn:de:0030-drops-95956}, doi = {10.4230/LIPIcs.MFCS.2018.13}, annote = {Keywords: data structure, priority queue, pairing heap} }

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

We revisit multipass pairing heaps and path-balanced binary search trees (BSTs), two classical algorithms for data structure maintenance. The pairing heap is a simple and efficient "self-adjusting" heap, introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. In the multipass variant (one of the original pairing heap variants described by Fredman et al.) the minimum item is extracted via repeated pairing rounds in which neighboring siblings are linked.
Path-balanced BSTs, proposed by Sleator (cf. Subramanian, 1996), are a natural alternative to Splay trees (Sleator and Tarjan, 1983). In a path-balanced BST, whenever an item is accessed, the search path leading to that item is re-arranged into a balanced tree.
Despite their simplicity, both algorithms turned out to be difficult to analyse. Fredman et al. showed that operations in multipass pairing heaps take amortized O(log n * log log n / log log log n) time. For searching in path-balanced BSTs, Balasubramanian and Raman showed in 1995 the same amortized time bound of O(log n * log log n / log log log n), using a different argument.
In this paper we show an explicit connection between the two algorithms and improve both bounds to O(log n * 2^{log^* n} * log^* n), respectively O(log n * 2^{log^* n} * (log^* n)^2), where log^* denotes the slowly growing iterated logarithm function. These are the first improvements in more than three, resp. two decades, approaching the information-theoretic lower bound of Omega(log n).

Dani Dorfman, Haim Kaplan, László Kozma, Seth Pettie, and Uri Zwick. Improved Bounds for Multipass Pairing Heaps and Path-Balanced Binary Search Trees. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dorfman_et_al:LIPIcs.ESA.2018.24, author = {Dorfman, Dani and Kaplan, Haim and Kozma, L\'{a}szl\'{o} and Pettie, Seth and Zwick, Uri}, title = {{Improved Bounds for Multipass Pairing Heaps and Path-Balanced Binary Search Trees}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {24:1--24:13}, 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.2018.24}, URN = {urn:nbn:de:0030-drops-94879}, doi = {10.4230/LIPIcs.ESA.2018.24}, annote = {Keywords: data structure, priority queue, pairing heap, binary search tree} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2^{Omega(sqrt(n*log(n)))} steps. This improves on a 2^{Omega(sqrt^3(n))} lower bound of Matoušek and Szabó. As Random-Facet performs an expected number of 2^{O(sqrt(n)} steps on any n-dimensional AUSO, this established our result. Improving our 2^{Omega(sqrt(n*log(n)))} lower bound seems to require radically new techniques.

Thomas Dueholm Hansen and Uri Zwick. Random-Edge Is Slower Than Random-Facet on Abstract Cubes. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hansen_et_al:LIPIcs.ICALP.2016.51, author = {Hansen, Thomas Dueholm and Zwick, Uri}, title = {{Random-Edge Is Slower Than Random-Facet on Abstract Cubes}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.51}, URN = {urn:nbn:de:0030-drops-63316}, doi = {10.4230/LIPIcs.ICALP.2016.51}, annote = {Keywords: Linear programming, the Simplex Algorithm, Pivoting rules, Acyclic Unique Sink Orientations} }

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

Gabow and Tarjan showed that the Bottleneck Path (BP) problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree (BST) problem, i.e., finding a directed spanning tree rooted at a given vertex whose largest edge weight is minimized, can both be solved deterministically in O(m * log^*(n)) time, where m is the number of edges and n is the number of vertices in the graph. We present a slightly improved randomized algorithm for these problems with an expected running time of O(m * beta(m,n)), where beta(m,n) = min{k >= 1 | log^{(k)}n <= m/n } <= log^*(n) - log^*(m/n)+1. This is the first improvement for these problems in over 25 years. In particular, if m >= n * log^{(k)} * n, for some constant k, the expected running time of the new algorithm is O(m). Our algorithm, as that of Gabow and Tarjan, work in the comparison model. We also observe that in the word-RAM model, both problems can be solved deterministically in O(m) time. Finally, we solve an open problem of Andersson et al., giving a deterministic O(m)-time comparison-based algorithm for solving deterministic 2-player turn-based zero-sum terminal payoff games, also known as Deterministic Graphical Games (DGG).

Shiri Chechik, Haim Kaplan, Mikkel Thorup, Or Zamir, and Uri Zwick. Bottleneck Paths and Trees and Deterministic Graphical Games. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chechik_et_al:LIPIcs.STACS.2016.27, author = {Chechik, Shiri and Kaplan, Haim and Thorup, Mikkel and Zamir, Or and Zwick, Uri}, title = {{Bottleneck Paths and Trees and Deterministic Graphical Games}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {27:1--27:13}, 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.2016.27}, URN = {urn:nbn:de:0030-drops-57283}, doi = {10.4230/LIPIcs.STACS.2016.27}, annote = {Keywords: bottleneck paths, comparison model, deterministic graphical games} }

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