Cutoff Phenomenon for Random Walks on Kneser Graphs

The cutoff phenomenon for an ergodic Markov chain describes a sharp transition in the convergence to its stationary distribution, over a negligible period of time, known as cutoff window. We study the cutoff phenomenon for simple random walks on Kneser graphs, which is a family of ergodic Markov chains. Given two integers $n$ and $k$, the Kneser graph $K(2n+k,n)$ is defined as the graph with vertex set being all subsets of $\{1,\ldots,2n+k\}$ of size $n$ and two vertices $A$ and $B$ being connected by an edge if $A\cap B =\emptyset$. We show that for any $k=O(n)$, the random walk on $K(2n+k,n)$ exhibits a cutoff at $\frac{1}{2}\log_{1+k/n}{(2n+k)}$ with a window of size $O(\frac{n}{k})$

Tight Bounds for Randomized Load Balancing on Arbitrary Network Topologies

We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. The goal is to achieve a distribution where all nodes have nearly the same number of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains, whose convergence is fairly well-understood in terms of their spectral gap. However, in many applications, load items cannot be divided arbitrarily, and we need to deal with the discrete case where the load is composed of indivisible tokens. This discretization entails a non-linear behavior due to its rounding errors, which makes this analysis much harder than in the continuous case. We investigate several randomized protocols for different communication models in the discrete case. As our main result, we prove that for any regular network in the matching model, all nodes have the same load up to an additive constant in (asymptotically) the same number of rounds as required in the continuous case. This generalizes and tightens the previous best result, which only holds for expander graphs, and demonstrates that there is almost no difference between the discrete and continuous cases. Our results also provide a positive answer to the question of how well discrete load balancing can be approximated by (continuous) Markov chains, which has been posed by many researchers.Comment: 74 pages, 4 figure

Cover Time and Broadcast Time

We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows: begin{itemize} item For any graph $G=(V,E)$ of size $n$ and minimum degree $delta$, we have $mathcal{R}(G)= mathcal{O}(frac{|E|}{delta} cdot log n)$, where $mathcal{R}(G)$ denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. item For any $delta$-regular (or almost $delta$-regular) graph $G$ it holds that $mathcal{R}(G) = Omega(frac{delta^2}{n} cdot frac{1}{log n})$. Together with our upper bound on $mathcal{R}(G)$, this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree $Theta(n)$, since then the cover time equals the broadcast time multiplied by $n$ (neglecting logarithmic factors). item Conversely, for any $delta$ we construct almost $delta$-regular graphs that satisfy $mathcal{R}(G) = mathcal{O}(max { sqrt{n},delta } cdot log^2 n)$. Since any regular expander satisfies $mathcal{R}(G) = Theta(n)$, the strong relationship given above does not hold if $delta$ is polynomially smaller than $n$. end{itemize} Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)

Quasirandom Rumor Spreading: An Experimental Analysis

We empirically analyze two versions of the well-known "randomized rumor spreading" protocol to disseminate a piece of information in networks. In the classical model, in each round each informed node informs a random neighbor. In the recently proposed quasirandom variant, each node has a (cyclic) list of its neighbors. Once informed, it starts at a random position of the list, but from then on informs its neighbors in the order of the list. While for sparse random graphs a better performance of the quasirandom model could be proven, all other results show that, independent of the structure of the lists, the same asymptotic performance guarantees hold as for the classical model. In this work, we compare the two models experimentally. This not only shows that the quasirandom model generally is faster, but also that the runtime is more concentrated around the mean. This is surprising given that much fewer random bits are used in the quasirandom process. These advantages are also observed in a lossy communication model, where each transmission does not reach its target with a certain probability, and in an asynchronous model, where nodes send at random times drawn from an exponential distribution. We also show that typically the particular structure of the lists has little influence on the efficiency.Comment: 14 pages, appeared in ALENEX'0

An Improved Drift Theorem for Balanced Allocations

In the balanced allocations framework, there are $m$ jobs (balls) to be allocated to $n$ servers (bins). The goal is to minimize the gap, the difference between the maximum and the average load. Peres, Talwar and Wieder (RSA 2015) used the hyperbolic cosine potential function to analyze a large family of allocation processes including the $(1+\beta)$-process and graphical balanced allocations. The key ingredient was to prove that the potential drops in every step, i.e., a drift inequality. In this work we improve the drift inequality so that (i) it is asymptotically tighter, (ii) it assumes weaker preconditions, (iii) it applies not only to processes allocating to more than one bin in a single step and (iv) to processes allocating a varying number of balls depending on the sampled bin. Our applications include the processes of (RSA 2015), but also several new processes, and we believe that our techniques may lead to further results in future work.Comment: This paper refines and extends the content on the drift theorem and applications in arXiv:2203.13902. It consists of 38 pages, 7 figures, 1 tabl

Balanced Allocations in Batches: The Tower of Two Choices

In balanced allocations, the goal is to place $m$ balls into $n$ bins, so as to minimize the gap (difference of max to average load). The One-Choice process places each ball to a bin sampled independently and uniformly at random. The Two-Choice process places balls in the least loaded of two sampled bins. Finally, the $(1+\beta)$-process mixes these processes, meaning each ball is allocated using Two-Choice with probability $\beta\in(0,1)$, and using One-Choice otherwise. Despite Two-Choice being optimal in the sequential setting, it has been observed in practice that it does not perform well in a parallel environment, where load information may be outdated. Following [BCEFN12], we study such a parallel setting where balls are allocated in batches of size $b$, and balls within the same batch are allocated with the same strategy and based on the same load information. For small batch sizes $b\in[n,n\log n]$, it was shown in [LS22a] that Two-Choice achieves an asymptotically optimal gap among all processes with a constant number of samples. In this work, we focus on larger batch sizes $b\in[n\log n,n^3]$. It was proved in [LS22c] that Two-Choice leads to a gap of $\Theta(b/n)$. As our main result, we prove that the gap reduces to $O(\sqrt{(b/n)\cdot\log n})$, if one runs the $(1+\beta)$-process with an appropriately chosen $\beta$ (in fact this result holds for a larger class of processes). This not only proves the phenomenon that Two-Choice is not the best (leading to the formation of "towers" over previously light bins), but also that mixing two processes (One-Choice and Two-Choice) leads to a process which achieves a gap that is asymptotically smaller than both. We also derive a matching lower bound of $\Omega(\sqrt{(b/n)\cdot\log n})$ for any allocation process, which demonstrates that the above $(1+\beta)$-process is asymptotically optimal.Comment: 36 pages;6 figures; 2 tables. arXiv admin note: text overlap with arXiv:2203.1390

Quasirandom Load Balancing

We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm surprisingly closely approximates the idealized process (where the tokens are divisible) on important network topologies. On d-dimensional torus graphs with n nodes it deviates from the idealized process only by an additive constant. In contrast to that, the randomized rounding approach of Friedrich and Sauerwald (2009) can deviate up to Omega(polylog(n)) and the deterministic algorithm of Rabani, Sinclair and Wanka (1998) has a deviation of Omega(n^{1/d}). This makes our quasirandom algorithm the first known algorithm for this setting which is optimal both in time and achieved smoothness. We further show that also on the hypercube our algorithm has a smaller deviation from the idealized process than the previous algorithms.Comment: 25 page

The Support of Open Versus Closed Random Walks

A closed random walk of length ? on an undirected and connected graph G = (V,E) is a random walk that returns to the start vertex at step ?, and its properties have been recently related to problems in different mathematical fields, e.g., geometry and combinatorics (Jiang et al., Annals of Mathematics \u2721) and spectral graph theory (McKenzie et al., STOC \u2721). For instance, in the context of analyzing the eigenvalue multiplicity of graph matrices, McKenzie et al. show that, with high probability, the support of a closed random walk of length ? ? 1 is ?(?^{1/5}) on any bounded-degree graph, and leaves as an open problem whether a stronger bound of ?(?^{1/2}) holds for any regular graph. First, we show that the support of a closed random walk of length ? is at least ?(?^{1/2} / ?{log n}) for any regular or bounded-degree graph on n vertices. Secondly, we prove for every ? ? 1 the existence of a family of bounded-degree graphs, together with a start vertex such that the support is bounded by O(?^{1/2}/?{log n}). Besides addressing the open problem of McKenzie et al., these two results also establish a subtle separation between closed random walks and open random walks, for which the support on any regular (or bounded-degree) graph is well-known to be ?(?^{1/2}) for all ? ? 1. For irregular graphs, we prove that even if the start vertex is chosen uniformly, the support of a closed random walk may still be O(log ?). This rules out a general polynomial lower bound in ? for all graphs. Finally, we apply our results on random walks to obtain new bounds on the multiplicity of the second largest eigenvalue of the adjacency matrices of graphs