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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an n-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors.
Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in Õ(n) space a proper O(Δ²)-coloring w.h.p. (here Δ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] showed that it is possible to deterministically compute an O(Δ²/s)-coloring in O(ns) space. However, none of the improvements succeeded in reducing the number of colors to O(Δ^{2-ε}) while keeping the same space bound of Õ(n). In particular, no progress was made on the question of whether computing an O(Δ)-coloring is possible with roughly O(n) space, which was stated in [Behnezhad et al., 2019] to be an interesting open problem.
In this paper we bypass the quadratic bound by presenting a new randomized Õ(n)-space algorithm that uses Õ(Δ^{1.5}) colors.

Shiri Chechik, Doron Mukhtar, and Tianyi Zhang. Streaming Edge Coloring with Subquadratic Palette Size. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 40:1-40:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chechik_et_al:LIPIcs.ICALP.2024.40, author = {Chechik, Shiri and Mukhtar, Doron and Zhang, Tianyi}, title = {{Streaming Edge Coloring with Subquadratic Palette Size}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {40:1--40:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.40}, URN = {urn:nbn:de:0030-drops-201831}, doi = {10.4230/LIPIcs.ICALP.2024.40}, annote = {Keywords: graph algorithms, streaming algorithms, edge coloring} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter D of the underlying network is constant.
We show that for the case where D = 1 there is, quite surprisingly, a very simple algorithm that solves the reachability problem in 1(!) round. In contrast, for networks with D = 2, we show that any distributed algorithm (possibly randomized) for this problem requires Omega(sqrt{n/ log{n}}) rounds. Our results therefore completely resolve (up to a small polylog factor) the complexity of the single-source reachability problem for a wide range of diameters.
Furthermore, we show that when D = 1, it is even possible to get an almost 3 - approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just 2 rounds. We also prove a stronger lower bound of Omega(sqrt{n}) for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is 2. As far as we know this is the first lower bound that achieves Omega(sqrt{n}) for this problem.

Shiri Chechik and Doron Mukhtar. Reachability and Shortest Paths in the Broadcast CONGEST Model. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chechik_et_al:LIPIcs.DISC.2019.11, author = {Chechik, Shiri and Mukhtar, Doron}, title = {{Reachability and Shortest Paths in the Broadcast CONGEST Model}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.11}, URN = {urn:nbn:de:0030-drops-113183}, doi = {10.4230/LIPIcs.DISC.2019.11}, annote = {Keywords: Distributed algorithms, Broadcast CONGEST, distance estimation, small diameter} }

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