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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 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

A (ϕ,ε)-expander decomposition of a graph G (with n vertices and m edges) is a partition of V into clusters V₁,…,V_k with conductance Φ(G[V_i]) ≥ ϕ, such that there are at most ε m inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized Õ(m/ϕ) time algorithm for computing a (ϕ, ϕlog²n)-expander decomposition. This improves upon the (ϕ, ϕlog³n)-expander decomposition also obtained in Õ(m/ϕ) time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal.
One crucial component of SW’s algorithm is a non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW.

Daniel Agassy, Dani Dorfman, and Haim Kaplan. Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agassy_et_al:LIPIcs.ICALP.2023.9, author = {Agassy, Daniel and Dorfman, Dani and Kaplan, Haim}, title = {{Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {9:1--9:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.9}, URN = {urn:nbn:de:0030-drops-180619}, doi = {10.4230/LIPIcs.ICALP.2023.9}, annote = {Keywords: Exapander Decomposition, Cut-Matching Game} }

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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:** 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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