[Paper Review] On the push&pull protocol for rumour spreading

Hüseyin Acan, Andrea Collevecchio+2 more|arXiv (Cornell University)|Nov 4, 2014

Optimization and Search Problems29 references7 citations

TL;DR

This paper analyzes the asynchronous push&pull protocol for rumor spreading in graphs, proving that both average and guaranteed spread times are O(n log n) and Ω(log n), respectively, with tight bounds up to constant factors. It establishes analytical relationships between synchronous and asynchronous versions, showing the asynchronous spread time is within an O(log n) factor of the synchronous one, and provides examples where asynchronous times are logarithmic while synchronous times are polynomially large.

ABSTRACT

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$ knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the rumour (a push operation), and if $x$ does not know the rumour and $y$ knows it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$ is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of $G$ is the smallest time $t$ such that with probability at least $1-1/n$, after time $t$ all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times $1,2,\dots$, has been studied extensively. We prove the following results for any $n$-vertex graph: In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is $O(n\log n)$. In the asynchronous version, both the average and guaranteed spread times are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove theoretical relationships between the guaranteed spread times in the two versions. Firstly, in all graphs the guaranteed spread time in the asynchronous version is within an $O(\log n)$ factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the synchronous spread time to the asynchronous spread time is $O(n^{2/3})$.

Motivation & Objective

• To analyze the average and guaranteed spread times of the asynchronous push&pull protocol in general undirected graphs.
• To establish tight upper and lower bounds on spread times, showing they are Θ(log n) and O(n log n) respectively.
• To compare the spread times between the synchronous and asynchronous versions of the protocol.
• To identify graph families where asynchronous performance is significantly better than synchronous performance.
• To provide the first analytical relationships between synchronous and asynchronous rumor spreading times.

Proposed method

• Uses an exponential clock model where each vertex has an independent Poisson clock of rate 1, triggering push or pull operations when it rings.
• Models the spread process as a sequence of blocks of communications, with block sizes S1, S2, ..., and defines Nb as the minimal number of blocks to cover 2N vertices.
• Introduces a 'lazy scenario' coupling to stochastically dominate the synchronous protocol, enabling comparison via block-based analysis.
• Applies Chernoff bounds to show that the sum of block sizes exceeds 8gsta(G)n with high probability, leading to a bound on Nb.
• Uses conditional probability and repetition analysis to bound the probability that a block exceeds a certain size, ensuring sufficient progress in rumor propagation.
• Employs a secondary block-splitting rule at special vertices to prevent misleading dependencies in the analysis.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the average and guaranteed spread times in the asynchronous push&pull protocol for any n-vertex graph?
• RQ2 How do the spread times in the asynchronous and synchronous versions of the protocol compare in terms of order of magnitude?
• RQ3 Are there graphs where the asynchronous protocol achieves logarithmic spread time while the synchronous protocol has polynomial spread time?
• RQ4 Can the ratio between synchronous and asynchronous spread times be bounded, and if so, what is the tightest such bound?
• RQ5 What is the relationship between the conductance/vertex expansion of a graph and the resulting spread time in the asynchronous model?

Key findings

• The average spread time in the asynchronous push&pull protocol is at most linear in n, even when only pull operations are used.
• The guaranteed spread time in both versions is O(n log n), and this bound is tight up to constant factors.
• The asynchronous spread time is Ω(log n), and this lower bound is tight for certain graph families.
• The asynchronous spread time is within an O(log n) factor of the synchronous spread time, and this factor is tight.
• There exist graphs where the asynchronous spread time is Θ(log n), but the synchronous spread time is Θ(n^c) for some c > 0, demonstrating a significant performance gap.
• For any graph, the ratio of the synchronous spread time to the asynchronous spread time is O(n^{2/3}), establishing a universal upper bound on their discrepancy.