3 Search Results for "Sundaresan, Janani"

Document
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.ITCS.2023.65
Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments

Authors: Varun Gupta, Ravishankar Krishnaswamy, Sai Sandeep, and Janani Sundaresan

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
In this paper we study two fully-dynamic multi-dimensional vector load balancing problems with recourse. The adversary presents a stream of n job insertions and deletions, where each job j is a vector in ℝ^d_{≥ 0}. In the vector scheduling problem, the algorithm must maintain an assignment of the active jobs to m identical machines to minimize the makespan (maximum load on any dimension on any machine). In the vector bin packing problem, the algorithm must maintain an assignment of active jobs into a number of bins of unit capacity in all dimensions, to minimize the number of bins currently used. In both problems, the goal is to maintain solutions that are competitive against the optimal solution for the active set of jobs, at every time instant. The algorithm is allowed to change the assignment from time to time, with the secondary objective of minimizing the amortized recourse, which is the average cardinality of the change of the assignment per update to the instance. For the vector scheduling problem, we present two simple algorithms. The first is a randomized algorithm with an O(1) amortized recourse and an O(log d/log log d) competitive ratio against oblivious adversaries. The second algorithm is a deterministic algorithm that is competitive against adaptive adversaries but with a slightly higher competitive ratio of O(log d) and a per-job recourse guarantee bounded by Õ(log n + log d log OPT). We also prove a sharper instance-dependent recourse guarantee for the deterministic algorithm. For the vector bin packing problem, we make the so-called small jobs assumption that the size of all jobs in all the coordinates is O(1/log d) and present a simple O(1)-competitive algorithm with O(log n) recourse against oblivious adversaries. For both problems, the main challenge is to determine when and how to migrate jobs to maintain competitive solutions. Our central idea is that for each job, we make these decisions based only on the active set of jobs that are "earlier" than this job in some ordering ≺ of the jobs.
Cite as

Varun Gupta, Ravishankar Krishnaswamy, Sai Sandeep, and Janani Sundaresan. Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gupta_et_al:LIPIcs.ITCS.2023.65,
  author =	{Gupta, Varun and Krishnaswamy, Ravishankar and Sandeep, Sai and Sundaresan, Janani},
  title =	{{Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{65:1--65:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.65},
  URN =		{urn:nbn:de:0030-drops-175685},
  doi =		{10.4230/LIPIcs.ITCS.2023.65},
  annote =	{Keywords: Vector Scheduling, Vector Load Balancing}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.48
Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting

Authors: Sepehr Assadi and Hoai-An Nguyen

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

Abstract
The h-index is a metric used to measure the impact of a user in a publication setting, such as a member of a social network with many highly liked posts or a researcher in an academic domain with many highly cited publications. Specifically, the h-index of a user is the largest integer h such that at least h publications of the user have at least h units of positive feedback. We design an algorithm that, given query access to the n publications of a user and each publication’s corresponding positive feedback number, outputs a (1± ε)-approximation of the h-index of this user with probability at least 1-δ in time O(n⋅ln(1/δ) / (ε²⋅h)), where h is the actual h-index which is unknown to the algorithm a-priori. We then design a novel lower bound technique that allows us to prove that this bound is in fact asymptotically optimal for this problem in all parameters n,h,ε, and δ. Our work is one of the first in sublinear time algorithms that addresses obtaining asymptotically optimal bounds, especially in terms of the error and confidence parameters. As such, we focus on designing novel techniques for this task. In particular, our lower bound technique seems quite general - to showcase this, we also use our approach to prove an asymptotically optimal lower bound for the problem of estimating the number of triangles in a graph in sublinear time, which now is also optimal in the error and confidence parameters. This latter result improves upon prior lower bounds of Eden, Levi, Ron, and Seshadhri (FOCS'15) for this problem, as well as multiple follow-up works that extended this lower bound to other subgraph counting problems.
Cite as

Sepehr Assadi and Hoai-An Nguyen. Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.APPROX/RANDOM.2022.48,
  author =	{Assadi, Sepehr and Nguyen, Hoai-An},
  title =	{{Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{48:1--48:20},
  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.2022.48},
  URN =		{urn:nbn:de:0030-drops-171708},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.48},
  annote =	{Keywords: Sublinear time algorithms, h-index, asymptotically optimal bounds, lower bounds, subgraph counting}
}
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