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Volume

LIPIcs, Volume 265

21st International Symposium on Experimental Algorithms (SEA 2023)

SEA 2023, July 24-26, 2023, Barcelona, Spain

Editors: Loukas Georgiadis

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DOI: 10.4230/LIPIcs.ITCS.2026.39

Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions

Authors: Keerti Choudhary, Amit Kumar, and Lakshay Saggi

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs (s₁,t₁) and (s₂,t₂), decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an O(mn log n)-time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known O(m⁵n)-time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in O(mn² log n)-time. In addition, we extend our techniques to a more general setting: given two terminal pairs (s₁, t₁) and (s₂, t₂) in a directed graph, find the minimum possible number of vertex intersections between any shortest path from s₁ to t₁ and s₂ to t₂. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an O(m² n³)-time algorithm for directed graphs with positive edge weights, and an O(m+n)-time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same O(m+n)-time. This is somewhat surprising, as there is no known o(mn) time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).

Cite as

Keerti Choudhary, Amit Kumar, and Lakshay Saggi. Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{choudhary_et_al:LIPIcs.ITCS.2026.39,
  author =	{Choudhary, Keerti and Kumar, Amit and Saggi, Lakshay},
  title =	{{Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{39:1--39:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.39},
  URN =		{urn:nbn:de:0030-drops-253267},
  doi =		{10.4230/LIPIcs.ITCS.2026.39},
  annote =	{Keywords: Disjoint paths, Disjoint shortest paths, Algebraic graph algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.5

Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs

Authors: Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

This paper addresses the problem of designing fault-tolerant data structures for the (s,t)-max-flow and (s,t)-min-cut problems in unweighted directed graphs. Given a directed graph G = (V, E) with a designated source s, sink t, and an (s,t)-max-flow of value λ, we present constructions for max-flow and min-cut sensitivity oracles, and introduce the concept of a fault-tolerant flow family, which may be of independent interest. Our main contributions are as follows. 1) Fault-Tolerant Flow Family: We construct a family ℬ of 2λ+1 (s,t)-flows such that for every edge e, ℬ contains an (s,t)-max-flow of G-e. This covering property is tight up to constants for single failures and provably cannot extend to comparably small families for k ≥ 2, where we show an Ω(n) lower bound on the family size, independent of λ. 2) Max-Flow Sensitivity Oracle: Using the fault-tolerant flow family, we construct a single as well as dual-edge sensitivity oracle for (s,t)-max-flow that requires only O(λ n) space. Given any set F of up to two failing edges, the oracle reports the updated max-flow value in G-F in O(n) time. Additionally, for the single-failure case, the oracle can determine in constant time whether the flow through an edge x changes when another edge e fails. 3) Min-Cut Sensitivity Oracle for Dual Failures: Recently, Baswana et al. (ICALP’22) designed an O(n²)-sized oracle for answering (s,t)-min-cut size queries under dual edge failures in constant time, along with a matching lower bound. We extend this by focusing on graphs with small min-cut values λ, and present a more compact oracle of size O(λ n) that answers such min-cut size queries in constant time and reports the corresponding (s,t)-min-cut partition in O(n) time. We also show that the space complexity of our oracle is asymptotically optimal in this setting. 4) Min-Cut Sensitivity Oracle for Multiple Failures: We extend our results to the general case of k edge failures. For any graph with (s,t)-min-cut of size λ, we construct a k-fault-tolerant min-cut oracle with space complexity O_{λ,k}(n log n) that answers min-cut size queries in O_{λ,k}(log n) time. This also leads to improved fault-tolerant (s,t)-reachability oracles, achieving O(n log n) space and O(log n) query time for up to k = O(1) edge failures.

Cite as

Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi. Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ahi_et_al:LIPIcs.ITCS.2026.5,
  author =	{Ahi, Mridul and Choudhary, Keerti and Pande, Shlok and Pushpraj and Saggi, Lakshay},
  title =	{{Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{5:1--5:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.5},
  URN =		{urn:nbn:de:0030-drops-252920},
  doi =		{10.4230/LIPIcs.ITCS.2026.5},
  annote =	{Keywords: Fault tolerance, Data structures, Minimum cuts, Maximum flows}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.8

Parameterized Algorithms for the Drone Delivery Problem

Authors: Simon Bartlmae, Andreas Hene, Joshua Könen, and Heiko Röglin

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Timely delivery and optimal routing remain fundamental challenges in the modern logistics industry. Building on prior work that considers single-package delivery across networks using multiple types of collaborative agents with restricted movement areas (e.g., drones or trucks), we examine the complexity of the problem under structural and operational constraints. Our focus is on minimizing total delivery time by coordinating agents that differ in speed and movement range across a graph. This problem formulation aligns with the recently proposed Drone Delivery Problem with respect to delivery time (DDT), introduced by Erlebach et al. [ISAAC 2022]. We first resolve an open question posed by Erlebach et al. [ISAAC 2022] by showing that even when the delivery network is a path graph, DDT admits no polynomial-time approximation within any polynomially encodable factor a(n), unless P=NP. Additionally, we identify the intersection graph of the agents, where nodes represent agents and edges indicate an overlap of the movement areas of two agents, as an important structural concept. For path graphs, we show that DDT becomes tractable when parameterized by the treewidth w of the intersection graph, and we present an exact FPT algorithm with running time f(w)⋅poly(n,k), for some computable function f. For general graphs, we give an FPT algorithm with running time f(Δ,w)⋅poly(n,k), where Δ is the maximum degree of the intersection graph. In the special case where the intersection graph is a tree, we provide a simple polynomial-time algorithm.

Cite as

Simon Bartlmae, Andreas Hene, Joshua Könen, and Heiko Röglin. Parameterized Algorithms for the Drone Delivery Problem. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bartlmae_et_al:LIPIcs.ISAAC.2025.8,
  author =	{Bartlmae, Simon and Hene, Andreas and K\"{o}nen, Joshua and R\"{o}glin, Heiko},
  title =	{{Parameterized Algorithms for the Drone Delivery Problem}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.8},
  URN =		{urn:nbn:de:0030-drops-249162},
  doi =		{10.4230/LIPIcs.ISAAC.2025.8},
  annote =	{Keywords: Complexity, Delivery, FPT algorithms, Graph Theory}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.10

Connected Partitions via Connected Dominating Sets

Authors: Aikaterini Niklanovits, Kirill Simonov, Shaily Verma, and Ziena Zeif

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The classical theorem due to Győri and Lovász states that any k-connected graph G admits a partition into k connected subgraphs, where each subgraph has a prescribed size and contains a prescribed vertex, as long as the total size of target subgraphs is equal to the size of G. However, this result is notoriously evasive in terms of efficient constructions, and it is still unknown whether such a partition can be computed in polynomial time, even for k = 5. We make progress towards an efficient constructive version of the Győri-Lovász theorem by considering a natural strengthening of the k-connectivity requirement. Specifically, we show that the desired connected partition can be found in polynomial time, if G contains k disjoint connected dominating sets. As a consequence of this result, we give several efficient approximate and exact constructive versions of the original Győri-Lovász theorem: - On general graphs, a Győri-Lovász partition with k parts can be computed in polynomial time when the input graph has connectivity Ω(k ⋅ log² n); - On convex bipartite graphs, connectivity of 4k is sufficient; - On biconvex graphs and interval graphs, connectivity of k is sufficient, meaning that our algorithm gives a "true" constructive version of the theorem on these graph classes.

Cite as

Aikaterini Niklanovits, Kirill Simonov, Shaily Verma, and Ziena Zeif. Connected Partitions via Connected Dominating Sets. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{niklanovits_et_al:LIPIcs.ESA.2025.10,
  author =	{Niklanovits, Aikaterini and Simonov, Kirill and Verma, Shaily and Zeif, Ziena},
  title =	{{Connected Partitions via Connected Dominating Sets}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.10},
  URN =		{urn:nbn:de:0030-drops-244785},
  doi =		{10.4230/LIPIcs.ESA.2025.10},
  annote =	{Keywords: Gy\H{o}ri-Lov\'{a}sz theorem, connected dominating sets, graph classes}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.55

Safe Sequences via Dominators in DAGs for Path-Covering Problems

Authors: Francisco Sena, Romeo Rizzi, and Alexandru I. Tomescu

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

A path-covering problem on a directed acyclic graph (DAG) requires finding a set of source-to-sink paths that cover all the nodes, all the arcs, or subsets thereof, and additionally they are optimal with respect to some function. In this paper we study safe sequences of nodes or arcs, namely sequences that appear in some path of every path cover of a DAG. We show that safe sequences admit a simple characterization via cutnodes. Moreover, we establish a connection between maximal safe sequences and leaf-to-root paths in the source- and sink-dominator trees of the DAG, which may be of independent interest in the extensive literature on dominators. With dominator trees, safe sequences admit an O(n)-size representation and a linear-time output-sensitive enumeration algorithm running in time O(m + o), where n and m are the number of nodes and arcs, respectively, and o is the total length of the maximal safe sequences. We then apply maximal safe sequences to simplify Integer Linear Programs (ILPs) for two path-covering problems, LeastSquares and MinPathError, which are at the core of RNA transcript assembly problems from bioinformatics. On various datasets, maximal safe sequences can be computed in under 0.1 seconds per graph, on average, and ILP solvers whose search space is reduced in this manner exhibit significant speed-ups. For example on graphs with a large width, average speed-ups are in the range 50-250× for MinPathError and in the range 80-350× for LeastSquares. Optimizing ILPs using safe sequences can thus become a fast building block of practical RNA transcript assembly tools, and more generally, of path-covering problems.

Cite as

Francisco Sena, Romeo Rizzi, and Alexandru I. Tomescu. Safe Sequences via Dominators in DAGs for Path-Covering Problems. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sena_et_al:LIPIcs.ESA.2025.55,
  author =	{Sena, Francisco and Rizzi, Romeo and Tomescu, Alexandru I.},
  title =	{{Safe Sequences via Dominators in DAGs for Path-Covering Problems}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{55:1--55:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.55},
  URN =		{urn:nbn:de:0030-drops-245230},
  doi =		{10.4230/LIPIcs.ESA.2025.55},
  annote =	{Keywords: directed acyclic graph, path cover, dominator tree, integer linear programming, least squares, minimum path error}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.26

Faster Dynamic 2-Edge Connectivity in Directed Graphs

Authors: Loukas Georgiadis, Konstantinos Giannis, and Giuseppe F. Italiano

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Let G be a directed graph with n vertices and m edges. We present a deterministic algorithm that maintains the 2-edge-connected components of G under a sequence of m edge insertions, with a total running time of O(n² log n). This significantly improves upon the previous best bound of O(mn) for graphs that are not very sparse. After each insertion, our algorithm supports the following queries with asymptotically optimal efficiency: - Test in constant time whether two query vertices v and w are 2-edge-connected in G. - Report in O(n) time all the 2-edge-connected components of G. Our approach builds on the recent framework of Georgiadis, Italiano, and Kosinas [FOCS 2024] for computing the 3-edge-connected components of a directed graph in linear time, which leverages the minset-poset technique of Gabow [TALG 2016]. Additionally, we provide a deterministic decremental algorithm for maintaining 2-edge-connectivity in strongly connected directed graphs. Given a sequence of m edge deletions, our algorithm maintains the 2-edge-connected components in total time n^(2+o(1)), while supporting the same queries as the incremental algorithm. This result assumes that the edges of a fixed spanning tree of G and of its reverse graph G^R are not deleted. Previously, the best known bound for the decremental problem was O(mn log n), obtained by a randomized algorithm without restrictions on the deletions. In contrast to prior dynamic algorithms for 2-edge-connectivity in directed graphs, our method avoids the incremental computation of dominator trees, thereby circumventing the known conditional lower bound of Ω(mn).

Cite as

Loukas Georgiadis, Konstantinos Giannis, and Giuseppe F. Italiano. Faster Dynamic 2-Edge Connectivity in Directed Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2025.26,
  author =	{Georgiadis, Loukas and Giannis, Konstantinos and Italiano, Giuseppe F.},
  title =	{{Faster Dynamic 2-Edge Connectivity in Directed Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{26:1--26:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.26},
  URN =		{urn:nbn:de:0030-drops-244945},
  doi =		{10.4230/LIPIcs.ESA.2025.26},
  annote =	{Keywords: Connectivity, dynamic algorithms, directed graphs}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.36

Efficient Contractions of Dynamic Graphs - With Applications

Authors: Monika Henzinger, Evangelos Kosinas, Robin Münk, and Harald Räcke

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

A non-trivial minimum cut (NMC) sparsifier is a multigraph Ĝ that preserves all non-trivial minimum cuts of a given undirected graph G. We introduce a flexible data structure for fully dynamic graphs that can efficiently provide an NMC sparsifier upon request at any point during the sequence of updates. We employ simple dynamic forest data structures to achieve a fast from-scratch construction of the sparsifier at query time. Based on the strength of the adversary and desired type of time bounds, the data structure comes with different guarantees. Specifically, let G be a fully dynamic simple graph with n vertices and minimum degree δ. Then our data structure supports an insertion/deletion of an edge to/from G in n^o(1) worst-case time. Furthermore, upon request, it can return w.h.p. an NMC sparsifier of G that has O(n/δ) vertices and O(n) edges, in Ô(n) time. The probabilistic guarantees hold against an adaptive adversary. Alternatively, the update and query times can be improved to Õ(1) and Õ(n) respectively, if amortized-time guarantees are sufficient, or if the adversary is oblivious. Throughout the paper, we use Õ to hide polylogarithmic factors and Ô to hide subpolynomial (i.e., n^o(1)) factors. We discuss two applications of our new data structure. First, it can be used to efficiently report a cactus representation of all minimum cuts of a fully dynamic simple graph. Building this cactus for the NMC sparsifier instead of the original graph allows for a construction time that is sublinear in the number of edges. Against an adaptive adversary, we can with high probability output the cactus representation in worst-case Ô(n) time. Second, our data structure allows us to efficiently compute the maximal k-edge-connected subgraphs of undirected simple graphs, by repeatedly applying a minimum cut algorithm on the NMC sparsifier. Specifically, we can compute with high probability the maximal k-edge-connected subgraphs of a simple graph with n vertices and m edges in Õ(m+n²/k) time. This improves the best known time bounds for k = Ω(n^{1/8}) and naturally extends to the case of fully dynamic graphs.

Cite as

Monika Henzinger, Evangelos Kosinas, Robin Münk, and Harald Räcke. Efficient Contractions of Dynamic Graphs - With Applications. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2025.36,
  author =	{Henzinger, Monika and Kosinas, Evangelos and M\"{u}nk, Robin and R\"{a}cke, Harald},
  title =	{{Efficient Contractions of Dynamic Graphs - With Applications}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{36:1--36:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.36},
  URN =		{urn:nbn:de:0030-drops-245047},
  doi =		{10.4230/LIPIcs.ESA.2025.36},
  annote =	{Keywords: Graph Algorithms, Cut Sparsifiers, Dynamic Algorithms}
}

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Invited Talk

DOI: 10.4230/LIPIcs.MFCS.2025.2

Higher Connectivity in Directed Graphs (Invited Talk)

Authors: Giuseppe F. Italiano

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

The computation of edge-connected components in directed and undirected graphs is a well studied problem that is motivated by several applications (see, e.g., [Hiroshi Nagamochi and Toshihide Ibaraki, 2008]). Let G = (V,E) be a strongly connected directed graph with m edges and n vertices. An edge e ∈ E is a strong bridge if G ⧵ e is not strongly connected. More generally, a set of edges C ⊆ E is a cut if G ⧵ C is not strongly connected. If |C| = k then we refer to C as a k-sized cut of G. Hence, a strong bridge is a 1-sized cut of G. A digraph G is k-edge-connected if it has no (k-1)-cuts. We say that two vertices v and w are k-edge-connected, and we denote this relation by v ↔_{k} w, if there are k edge-disjoint directed paths from v to w and k edge-disjoint directed paths from w to v. (Note that a path from v to w and a path from w to v need not be edge-disjoint). By Menger’s theorem [Karl Menger, 1927], v ↔_{k} w if and only if the removal of any set of at most k-1 edges leaves v and w in the same strongly connected component. We define a k-edge-connected component of a digraph G = (V,E) as a maximal subset U ⊆ V such that u ↔_{k} v for all u, v ∈ U. The k-edge-connected components of G form a partition of V, since v ↔_{k} w is an equivalence relation [Loukas Georgiadis et al., 2016]. Connectivity-related problems are known to be much more difficult in directed graphs than in undirected graphs (see, e.g., [Harold N. Gabow, 2016; Monika Henzinger et al., 2020; Ken-Ichi Kawarabayashi and Mikkel Thorup, 2018]). Indeed, there is a fundamental difference in the structure of the cuts in the two scenarios. Specifically, it has been established more than 60 years ago [Gomory and Hu, 1961] that edge cuts in undirected graphs have a nice structure, as defined by the Gomory-Hu tree (or cut tree), which plays a special role in identifying, for any k, the k-edge-connected components of undirected graphs. Furthermore, many efficient algorithms for computing Gomory-Hu trees are available (see e.g., [Amir Abboud et al., 2021; Amir Abboud et al., 2022; Amir Abboud et al., 2023; Chen et al., 2022; Hariharan et al., 2007; Li et al., 2022]). On the contrary, in directed graphs edge cuts have a more complicated structure, and it was proved by Benczúr [Benczúr, 1995] that in this case cut trees do not even exist. It is thus not surprising that, while it is known how to compute the k-edge-connected components of undirected graphs in linear time for k ≤ 5 [Harold N. Gabow, 2000; Zvi Galil and Giuseppe F. Italiano, 1991; Loukas Georgiadis et al., 2021; John E. Hopcroft and Robert E. Tarjan, 1973; Kosinas, 2024; Wojciech Nadara et al., 2021; Hiroshi Nagamochi and Toshihide Ibaraki, 1992; Robert E. Tarjan, 1972; Yung H. Tsin, 2009], the situation is more challenging for directed graphs, where linear-time algorithms are only known for k ≤ 2 [Robert E. Tarjan, 1972; Loukas Georgiadis et al., 2020]. Also, as argued in [Loukas Georgiadis et al., 2023], there is a substantial increase in the inherent difficulty of the problem of computing k-edge-connected components in digraphs for k = 3 compared to k = 2. Indeed, for k = 2 any pair of vertices s,t that are not 2-edge-connected can be separated by only O(n) s-t min-cuts of size 1, for which we can define a total order [Giuseppe F. Italiano et al., 2012]. For k = 3, any pair of vertices s,t that are 2-edge-connected but not 3-edge-connected, can be separated by as many as O(n²) s-t min-cuts of size 2, which are also not totally ordered. This makes it difficult to explore the effect of removing each such cut of size 2 on the strong connectivity of the graph, similar to what was done for the case of k = 2 [Loukas Georgiadis et al., 2020]. Until recently, the best-known bound for computing the k-edge-connected components of a digraph, for constant k ≥ 3, was O(mn) by Nagamochi and Watanabe [Hiroshi Nagamochi and Toshimasa Watanabe, 1993]. Georgiadis et al. [Loukas Georgiadis et al., 2023] presented a randomized (Monte-Carlo) algorithm that computes the 3-edge-connected components of a digraph with m edges in Õ(m^{3/2}) time. Their algorithm involves a nontrivial extension of the framework of [Forster et al., 2020; Nanongkai et al., 2019] for deciding whether a digraph is (k+1)-edge-connected. It applies a local search procedure [Shiri Chechik et al., 2017; Forster et al., 2020] for identifying 2-in or 2-out sets, i.e., vertex sets S ⊆ V such that there are at most 2 edges from V ⧵ S to S or from S to V⧵ S. After finding such a set S, [Loukas Georgiadis et al., 2023] applies an efficient graph operation for replacing S with a gadget of small size that preserves the pairwise connectivity among the vertices of V ⧵ S. As in [Forster et al., 2020; Nanongkai et al., 2019], local search is initiated from sampled edges, but the overall scheme is more complicated to guarantee that enough 2-in sets or 2-out sets are identified that separate vertices that are not 3-edge-connected. Recently, Georgiadis, Italiano and Kosinas [Georgiadis et al., 2024] improved significantly the bound of [Loukas Georgiadis et al., 2023] by showing how to compute the 3-edge-connected components of a digraph in linear time with a deterministic algorithm. Their algorithm differs substantially from [Loukas Georgiadis et al., 2023], as it is based on a new characterization of 2-sized cuts in digraphs, which requires new techniques and a suitable combination of the notions of 2-connectivity-light graphs [Loukas Georgiadis et al., 2023] and of maximally edge-disjoint strongly divergent spanning trees [Loukas Georgiadis and Robert E. Tarjan, 2015; Robert E. Tarjan, 1976]. In particular, Georgiadis, Italiano and Kosinas [Georgiadis et al., 2024] showed how to modify the minset-poset technique of Gabow [Harold N. Gabow, 2016], in order to find the 3-edge-connected components of a digraph with m edges in O(m) time. In the invited talk, I will survey some of this recent work on higher connectivity on directed graphs.

Cite as

Giuseppe F. Italiano. Higher Connectivity in Directed Graphs (Invited Talk). In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 2:1-2:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{italiano:LIPIcs.MFCS.2025.2,
  author =	{Italiano, Giuseppe F.},
  title =	{{Higher Connectivity in Directed Graphs}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{2:1--2:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.2},
  URN =		{urn:nbn:de:0030-drops-241096},
  doi =		{10.4230/LIPIcs.MFCS.2025.2},
  annote =	{Keywords: Connectivity, Directed graphs, Graph algorithms}
}

Document

DOI: 10.4230/LIPIcs.SEA.2025.9

Incremental Reachability Index

Authors: Laurent Bulteau, Pierre-Yves David, Florian Horn, and Euxane Tran-Girard

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract

We study the reachability problem in append-only DAGs: given two nodes u and v, is there a path from u to v? While the problem is linear in general, it can be answered faster by using a precomputed index, which gives a compressed representation of the transitive closure of the graph. Index algorithms are evaluated on three dimensions: the query time that the algorithm needs to answer whether there is a path from one node to another, the memory that the index uses per node, and the indexing time that is required to update the index when a node is added to the graph. In this paper, we combine Jagadish’s static index [Jagadish, 1990] with Felsner’s online chain-decomposition algorithm [Stefan Felsner, 1997] to create an incremental index: data associated with a node is immutable, guaranteeing that queries are answered properly even if new nodes are inserted while the query is processed. Its query time is constant, but its index size is heavily dependent on the graph width, and as such is not competitive with recent indexing algorithms (2-hop, tree-chain, ...). We also propose a version of that incremental algorithm with a much lighter index. In the most compressed version, the query time becomes O(log n). However, constant-time queries can be retained depending on the desired time/memory trade-off.

Cite as

Laurent Bulteau, Pierre-Yves David, Florian Horn, and Euxane Tran-Girard. Incremental Reachability Index. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bulteau_et_al:LIPIcs.SEA.2025.9,
  author =	{Bulteau, Laurent and David, Pierre-Yves and Horn, Florian and Tran-Girard, Euxane},
  title =	{{Incremental Reachability Index}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.9},
  URN =		{urn:nbn:de:0030-drops-232477},
  doi =		{10.4230/LIPIcs.SEA.2025.9},
  annote =	{Keywords: Directed acyclic graphs, reachability, append-only, index}
}

Document

DOI: 10.4230/LIPIcs.SEA.2025.10

Algorithm Engineering of SSSP with Negative Edge Weights

Authors: Alejandro Cassis, Andreas Karrenbauer, André Nusser, and Paolo Luigi Rinaldi

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract

Computing shortest paths is one of the most fundamental algorithmic graph problems. It is known since decades that this problem can be solved in near-linear time if all weights are nonnegative. A recent break-through by [Aaron Bernstein et al., 2022] presented a randomized near-linear time algorithm for this problem. A subsequent improvement in [Karl Bringmann et al., 2023] significantly reduced the number of logarithmic factors and thereby also simplified the algorithm. It is surprising and exciting that both of these algorithms are combinatorial and do not contain any fundamental obstacles for being practical. We launch the, to the best of our knowledge, first extensive investigation towards a practical implementation of [Karl Bringmann et al., 2023]. To this end, we give an accessible overview of the algorithm and discuss what adaptions are necessary to obtain a fast algorithm in practice. We manifest these adaptions in an efficient implementation. We test our implementation on a benchmark data set that is adapted to be more difficult for our implementation in order to allow for a fair comparison. As in [Karl Bringmann et al., 2023] as well as in our implementation there are multiple parameters to tune, we empirically evaluate their effect and thereby determine the best choices. Our implementation is then extensively compared to one of the state-of-the-art algorithms for this problem [Andrew V. Goldberg and Tomasz Radzik, 1993]. On the hardest instance type, we are faster by up to almost two orders of magnitude.

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Alejandro Cassis, Andreas Karrenbauer, André Nusser, and Paolo Luigi Rinaldi. Algorithm Engineering of SSSP with Negative Edge Weights. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cassis_et_al:LIPIcs.SEA.2025.10,
  author =	{Cassis, Alejandro and Karrenbauer, Andreas and Nusser, Andr\'{e} and Rinaldi, Paolo Luigi},
  title =	{{Algorithm Engineering of SSSP with Negative Edge Weights}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.10},
  URN =		{urn:nbn:de:0030-drops-232486},
  doi =		{10.4230/LIPIcs.SEA.2025.10},
  annote =	{Keywords: Single Source Shortest Paths, Negative Weights, Near-Linear Time}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.110

An Optimal 3-Fault-Tolerant Connectivity Oracle

Authors: Evangelos Kosinas

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

We present an optimal oracle for answering connectivity queries in undirected graphs in the presence of at most three vertex failures. Specifically, we show that we can process a graph G in O(n+m) time, in order to build a data structure that occupies O(n) space, which can be used in order to answer queries of the form "given a set F of at most three vertices, and two vertices x and y not in F, are x and y connected in G⧵ F?" in constant time, where n and m denote the number of vertices and edges, respectively, of G. The idea is to rely on the DFS-based framework introduced by Kosinas [ESA'23], for handling connectivity queries in the presence of multiple vertex failures. Our technical contribution is to show how to appropriately extend the toolkit of the DFS-based parameters, in order to optimally handle up to three vertex failures. Our approach has the interesting property that it does not rely on a compact representation of vertex cuts, and has the potential to provide optimal solutions for more vertex failures. Furthermore, we show that the DFS-based framework can be easily extended in order to answer vertex-cut queries, and the number of connected components in the presence of multiple vertex failures. In the case of three vertex failures, we can answer such queries in O(log n) time.

Cite as

Evangelos Kosinas. An Optimal 3-Fault-Tolerant Connectivity Oracle. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 110:1-110:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kosinas:LIPIcs.ICALP.2025.110,
  author =	{Kosinas, Evangelos},
  title =	{{An Optimal 3-Fault-Tolerant Connectivity Oracle}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{110:1--110:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.110},
  URN =		{urn:nbn:de:0030-drops-234879},
  doi =		{10.4230/LIPIcs.ICALP.2025.110},
  annote =	{Keywords: Graphs, Connectivity, Fault-Tolerant, Oracles}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.92

Fully Dynamic Algorithms for Transitive Reduction

Authors: Gramoz Goranci, Adam Karczmarz, Ali Momeni, and Nikos Parotsidis

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

Given a directed graph G, a transitive reduction G^t of G (first studied by Aho, Garey, Ullman [SICOMP `72]) is a minimal subgraph of G that preserves the reachability relation between every two vertices in G. In this paper, we study the computational complexity of transitive reduction in the dynamic setting. We obtain the first fully dynamic algorithms for maintaining a transitive reduction of a general directed graph undergoing updates such as edge insertions or deletions. Our first algorithm achieves O(m+n log n) amortized update time, which is near-optimal for sparse directed graphs, and can even support extended update operations such as inserting a set of edges all incident to the same vertex, or deleting an arbitrary set of edges. Our second algorithm relies on fast matrix multiplication and achieves O(m+ n^{1.585}) worst-case update time.

Cite as

Gramoz Goranci, Adam Karczmarz, Ali Momeni, and Nikos Parotsidis. Fully Dynamic Algorithms for Transitive Reduction. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 92:1-92:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{goranci_et_al:LIPIcs.ICALP.2025.92,
  author =	{Goranci, Gramoz and Karczmarz, Adam and Momeni, Ali and Parotsidis, Nikos},
  title =	{{Fully Dynamic Algorithms for Transitive Reduction}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{92:1--92:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.92},
  URN =		{urn:nbn:de:0030-drops-234697},
  doi =		{10.4230/LIPIcs.ICALP.2025.92},
  annote =	{Keywords: Spectral sparsification, Dynamic algorithms, (Directed) hypergraphs, Data structures}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2025.69

Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology

Authors: Dmitriy Morozov and Luis Scoccola

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes changes. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of 0-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of 0-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

Cite as

Dmitriy Morozov and Luis Scoccola. Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 69:1-69:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{morozov_et_al:LIPIcs.SoCG.2025.69,
  author =	{Morozov, Dmitriy and Scoccola, Luis},
  title =	{{Computing Betti Tables and Minimal Presentations of Zero-Dimensional Persistent Homology}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{69:1--69:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.69},
  URN =		{urn:nbn:de:0030-drops-232219},
  doi =		{10.4230/LIPIcs.SoCG.2025.69},
  annote =	{Keywords: Multiparameter persistence, Zero-dimensional homology, Minimal presentation, Betti table}
}

Document

DOI: 10.4230/LIPIcs.CPM.2025.19

The Trie Measure, Revisited

Authors: Jarno N. Alanko, Ruben Becker, Davide Cenzato, Travis Gagie, Sung-Hwan Kim, Bojana Kodric, and Nicola Prezza

Published in: LIPIcs, Volume 331, 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)

Abstract

In this paper, we study the following problem: given n subsets S₁, … , S_n of an integer universe U = {0,… , u-1}, having total cardinality N = ∑_{i = 1}ⁿ |S_i|, find a prefix-free encoding enc : U → {0,1}^+ minimizing the so-called trie measure, i.e., the total number of edges in the n binary tries T₁, … , T_n, where T_i is the trie packing the encoded integers {enc(x):x ∈ S_i}. We first observe that this problem is equivalent to that of merging u sets with the cheapest sequence of binary unions, a problem which in [Ghosh et al., ICDCS 2015] is shown to be NP-hard. Motivated by the hardness of the general problem, we focus on particular families of prefix-free encodings. We start by studying the fixed-length shifted encoding of [Gupta et al., Theoretical Computer Science 2007]. Given a parameter 0 ≤ a < u, this encoding sends each x ∈ U to (x + a) mod u, interpreted as a bit-string of log u bits. We develop the first efficient algorithms that find the value of a minimizing the trie measure when this encoding is used. Our two algorithms run in O(u + Nlog u) and O(Nlog² u) time, respectively. We proceed by studying ordered encodings (a.k.a. monotone or alphabetic), and describe an algorithm finding the optimal such encoding in O(N+u³) time. Within the same running time, we show how to compute the best shifted ordered encoding, provably no worse than both the optimal shifted and optimal ordered encodings. We provide implementations of our algorithms and discuss how these encodings perform in practice.

Cite as

Jarno N. Alanko, Ruben Becker, Davide Cenzato, Travis Gagie, Sung-Hwan Kim, Bojana Kodric, and Nicola Prezza. The Trie Measure, Revisited. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alanko_et_al:LIPIcs.CPM.2025.19,
  author =	{Alanko, Jarno N. and Becker, Ruben and Cenzato, Davide and Gagie, Travis and Kim, Sung-Hwan and Kodric, Bojana and Prezza, Nicola},
  title =	{{The Trie Measure, Revisited}},
  booktitle =	{36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-369-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{331},
  editor =	{Bonizzoni, Paola and M\"{a}kinen, Veli},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2025.19},
  URN =		{urn:nbn:de:0030-drops-231135},
  doi =		{10.4230/LIPIcs.CPM.2025.19},
  annote =	{Keywords: Succinct data structures, degenerate strings, integer encoding}
}

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