Bounds for algebraic gossip on graphs1. Issue 2 (27th December 2012)

• Record Type: Journal Article
• Authors: Borokhovich, Michael; Avin, Chen; Lotker, Zvi
• Is Part Of: Random structures & algorithms. Volume 45:Issue 2(2014)
• Page Start: 185
• Page End: 217
• Publication Date: 2012-12-27
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20480
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading: Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size <italic>n</italic> is <italic>O</italic>(Δ<italic>n</italic>) where Δ is the maximum degree of the graph. This leads to a tight bound of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4nzqtd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20480:rsa20480-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Θ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> for bounded degree graphs and an upper bound of <italic>O</italic>(<italic>n</italic><sup>2</sup>) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4nzqj8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20480:rsa20480-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 185–217, 2014</p> </abstract>