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Documents authored by Saneian, Mohammad

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.121
Streaming Edge Coloring with Asymptotically Optimal Colors

Authors: Mohammad Saneian and Soheil Behnezhad

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

Abstract
Given a graph G, an edge-coloring is an assignment of colors to edges of G such that any two edges sharing an endpoint receive different colors. By Vizing’s celebrated theorem, any graph of maximum degree Δ needs at least Δ and at most (Δ + 1) colors to be properly edge colored. In this paper, we study edge colorings in the streaming setting. The edges arrive one by one in an arbitrary order. The algorithm takes a single pass over the input and must output a solution using a much smaller space than the input size. Since the output of edge coloring is as large as its input, the assigned colors should also be reported in a streaming fashion. The streaming edge coloring problem has been studied in a series of works over the past few years. The main challenge is that the algorithm cannot "remember" all the color assignments that it returns. To ensure the validity of the solution, existing algorithms use many more colors than Vizing’s bound. Namely, in n-vertex graphs, the state-of-the-art algorithm with Õ(n s) space requires O(Δ²/s + Δ) colors. Note, in particular, that for an asymptotically optimal O(Δ) coloring, this algorithm requires Ω(nΔ) space which is as large as the input. Whether such a coloring can be achieved with sublinear space has been left open. In this paper, we answer this question in the affirmative. We present a randomized algorithm that returns an asymptotically optimal O(Δ) edge coloring using Õ(n √{Δ}) space. More generally, our algorithm returns a proper O(Δ^{1.5}/s + Δ) edge coloring with Õ(n s) space, improving prior algorithms for the whole range of s.
Cite as

Mohammad Saneian and Soheil Behnezhad. Streaming Edge Coloring with Asymptotically Optimal Colors. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 121:1-121:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{saneian_et_al:LIPIcs.ICALP.2024.121,
  author =	{Saneian, Mohammad and Behnezhad, Soheil},
  title =	{{Streaming Edge Coloring with Asymptotically Optimal Colors}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{121:1--121:20},
  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.2024.121},
  URN =		{urn:nbn:de:0030-drops-202640},
  doi =		{10.4230/LIPIcs.ICALP.2024.121},
  annote =	{Keywords: Streaming, Edge coloring, Adversarial order}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.8
Simple Streaming Algorithms for Edge Coloring

Authors: Mohammad Ansari, Mohammad Saneian, and Hamid Zarrabi-Zadeh

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We revisit the classical edge coloring problem for general graphs in the streaming model. In this model, the input graph is presented as a stream of edges, and the algorithm must report colors assigned to the edges in a streaming fashion, using a memory of size O(n polylog n) on graphs of up to O(n²) edges. In ESA 2019 and SOSA 2021, two elegant randomized algorithms were presented for this problem in the adversarial edge arrival model, where the latest one colors any input graph using O(Δ²/s) colors with high probability in Õ(ns) space. In this short note, we propose two extremely simple streaming algorithms that achieve the same color and space bounds deterministically. Besides being surprisingly simple, our algorithms do not use randomness at all, and are very simple to analyze.
Cite as

Mohammad Ansari, Mohammad Saneian, and Hamid Zarrabi-Zadeh. Simple Streaming Algorithms for Edge Coloring. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 8:1-8:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ansari_et_al:LIPIcs.ESA.2022.8,
  author =	{Ansari, Mohammad and Saneian, Mohammad and Zarrabi-Zadeh, Hamid},
  title =	{{Simple Streaming Algorithms for Edge Coloring}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{8:1--8:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.8},
  URN =		{urn:nbn:de:0030-drops-169468},
  doi =		{10.4230/LIPIcs.ESA.2022.8},
  annote =	{Keywords: Edge coloring, streaming model, adversarial order}
}
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