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DOI: 10.4230/LIPIcs.ESA.2025.22

Constructing Long Paths in Graph Streams

Authors: Christian Konrad and Chhaya Trehan

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

Abstract

In the graph stream model of computation, an algorithm processes the edges of an n-vertex input graph in one or more sequential passes while using a memory that is sublinear in the input size. The streaming model poses significant challenges for algorithmically constructing long paths. Many known algorithms that are tasked with extending an existing path as a subroutine require an entire pass over the input to add a single additional edge. This raises a fundamental question: Are multiple passes inherently necessary to construct paths of non-trivial lengths, or can a single pass suffice? To address this question, we systematically study the Longest Path problem in the one-pass streaming model. In this problem, given a desired approximation factor α, the objective is to compute a path of length at least lp(G)/α, where lp(G) is the length of a longest path in the input graph G. We study the problem in the insertion-only and the insertion-deletion streaming models, and we give algorithms as well as space lower bounds for both undirected and directed graphs. Our results are: 1) We show that for undirected graphs, in both the insertion-only and the insertion-deletion models, there are semi-streaming algorithms, i.e., algorithms that use space O(n poly log n), that compute a path of length at least d/3 with high probability, where d is the average degree of the input graph. These algorithms can also yield an α-approximation to Longest Path using space Õ(n²/α). 2) Next, we show that such a result cannot be achieved for directed graphs, even in the insertion-only model. We show that computing a (n^{1-o(1)})-approximation to Longest Path in directed graphs in the insertion-only model requires space Ω(n²). This result is in line with recent results that demonstrate that processing directed graphs is often significantly harder than undirected graphs in the streaming model. 3) We further complement our results with two additional lower bounds. First, we show that semi-streaming space is insufficient for small constant factor approximations to Longest Path for undirected graphs in the insertion-only model. Last, in undirected graphs in the insertion-deletion model, we show that computing an α-approximation requires space Ω(n²/α³).

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Christian Konrad and Chhaya Trehan. Constructing Long Paths in Graph Streams. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{konrad_et_al:LIPIcs.ESA.2025.22,
  author =	{Konrad, Christian and Trehan, Chhaya},
  title =	{{Constructing Long Paths in Graph Streams}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{22:1--22:19},
  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.2025.22},
  URN =		{urn:nbn:de:0030-drops-244902},
  doi =		{10.4230/LIPIcs.ESA.2025.22},
  annote =	{Keywords: Longest Path Problem, Streaming Algorithms, One-way Two-party Communication Complexity}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2022.25

When You Come at the King You Best Not Miss

Authors: Oded Lachish, Felix Reidl, and Chhaya Trehan

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract

A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [Jian Shen et al., 2003] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least Ω(n^{4/3}) arcs and provided a strategy that queries at most O(n^{3/2}). While this upper bound has not yet been improved for the original problem, [Biswas et al., 2017] proved that with O(n^{4/3}) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n^{4/3} polylog n) queries, we can identify a (1/2+2/17)-king. To achieve this goal we use a novel structural result for tournaments.

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Oded Lachish, Felix Reidl, and Chhaya Trehan. When You Come at the King You Best Not Miss. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lachish_et_al:LIPIcs.FSTTCS.2022.25,
  author =	{Lachish, Oded and Reidl, Felix and Trehan, Chhaya},
  title =	{{When You Come at the King You Best Not Miss}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{25:1--25:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.25},
  URN =		{urn:nbn:de:0030-drops-174177},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.25},
  annote =	{Keywords: Digraphs, tournaments, kings, query complexity}
}

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APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.51

(1+ε)-Approximate Shortest Paths in Dynamic Streams

Authors: Michael Elkin and Chhaya Trehan

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ-1 versus n^{1+1/κ}, for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1+ε in the stretch for weighted graphs, and an additional factor of log^{O(1)}n in the space.) The only existing solution of the second type uses n^{1/2 - O(1/κ)} passes over the stream (for space O(n^{1+1/κ})), and applies only to unweighted graphs. In this paper we show that (1+ε)-approximate single-source shortest paths can be computed with Õ(n^{1+1/κ}) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n^{1/κ}). We achieve these results by devising efficient dynamic streaming constructions of (1 + ε, β)-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [S. Ganguly, 2007; Graham Cormode and D. Firmani, 2013], we believe that it is of independent interest.

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Michael Elkin and Chhaya Trehan. (1+ε)-Approximate Shortest Paths in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 51:1-51:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{elkin_et_al:LIPIcs.APPROX/RANDOM.2022.51,
  author =	{Elkin, Michael and Trehan, Chhaya},
  title =	{{(1+\epsilon)-Approximate Shortest Paths in Dynamic Streams}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{51:1--51:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.51},
  URN =		{urn:nbn:de:0030-drops-171733},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.51},
  annote =	{Keywords: Shortest Paths, Dynamic Streams, Approximate Distances, Spanners, Hopsets}
}

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Documents authored by Trehan, Chhaya

Constructing Long Paths in Graph Streams

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When You Come at the King You Best Not Miss

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(1+ε)-Approximate Shortest Paths in Dynamic Streams

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