Balanced Allocations: A Simple Proof for the Heavily Loaded Case

We provide a relatively simple proof that the expected gap between the maximum load and the average load in the two choice process is bounded by $(1+o(1))\log \log n$, irrespective of the number of balls thrown. The theorem was first proven by Berenbrink et al. Their proof uses heavy machinery from Markov-Chain theory and some of the calculations are done using computers. In this manuscript we provide a significantly simpler proof that is not aided by computers and is self contained. The simplification comes at a cost of weaker bounds on the low order terms and a weaker tail bound for the probability of deviating from the expectation

Parallel Load Balancing on Constrained Client-Server Topologies

We study parallel \emph{Load Balancing} protocols for a client-server distributed model defined as follows. There is a set \sC of $n$ clients and a set \sS of $n$ servers where each client has (at most) a constant number $d \geq 1$ of requests that must be assigned to some server. The client set and the server one are connected to each other via a fixed bipartite graph: the requests of client $v$ can only be sent to the servers in its neighborhood $N(v)$. The goal is to assign every client request so as to minimize the maximum load of the servers. In this setting, efficient parallel protocols are available only for dense topolgies. In particular, a simple symmetric, non-adaptive protocol achieving constant maximum load has been recently introduced by Becchetti et al \cite{BCNPT18} for regular dense bipartite graphs. The parallel completion time is \bigO(\log n) and the overall work is \bigO(n), w.h.p. Motivated by proximity constraints arising in some client-server systems, we devise a simple variant of Becchetti et al's protocol \cite{BCNPT18} and we analyse it over almost-regular bipartite graphs where nodes may have neighborhoods of small size. In detail, we prove that, w.h.p., this new version has a cost equivalent to that of Becchetti et al's protocol (in terms of maximum load, completion time, and work complexity, respectively) on every almost-regular bipartite graph with degree $\Omega(\log^2n)$. Our analysis significantly departs from that in \cite{BCNPT18} for the original protocol and requires to cope with non-trivial stochastic-dependence issues on the random choices of the algorithmic process which are due to the worst-case, sparse topology of the underlying graph

Random walks which prefer unvisited edges : exploring high girth even degree expanders in linear time.

Let G = (V,E) be a connected graph with |V | = n vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge-process (or E -process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E -process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap 1 -λmax of the transition matrix of a simple random walk on G. A vertex v is ℓ -good, if any even degree subgraph containing all edges incident with v contains at least ℓ vertices. A graph G is ℓ -good, if every vertex has the ℓ -good property. Let G be an even degree ℓ -good expander of bounded maximum degree. Any E -process on G has vertex cover time equation image This is to be compared with the Ω(nlog n) lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on-line by an adversary. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2013 As no walk based process can cover an n vertex graph in less than n - 1 steps, the cover time of the E -process is of optimal order when ℓ =Θ (log n). With high probability random r -regular graphs, r ≥ 4 even, have ℓ =Ω (log n). Thus the vertex cover time of the E -process on such graphs is Θ(n)

Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

How to Spread a Rumor: Call Your Neighbors or Take a Walk?

We study the problem of randomized information dissemination in networks. We compare the now standard PUSH-PULL protocol, with agent-based alternatives where information is disseminated by a collection of agents performing independent random walks. In the VISIT-EXCHANGE protocol, both nodes and agents store information, and each time an agent visits a node, the two exchange all the information they have. In the MEET-EXCHANGE protocol, only the agents store information, and exchange their information with each agent they meet. We consider the broadcast time of a single piece of information in an $n$-node graph for the above three protocols, assuming a linear number of agents that start from the stationary distribution. We observe that there are graphs on which the agent-based protocols are significantly faster than PUSH-PULL, and graphs where the converse is true. We attribute the good performance of agent-based algorithms to their inherently fair bandwidth utilization, and conclude that, in certain settings, agent-based information dissemination, separately or in combination with PUSH-PULL, can significantly improve the broadcast time. The graphs considered above are highly non-regular. Our main technical result is that on any regular graph of at least logarithmic degree, PUSH-PULL and VISIT-EXCHANGE have the same asymptotic broadcast time. The proof uses a novel coupling argument which relates the random choices of vertices in PUSH-PULL with the random walks in VISIT-EXCHANGE. Further, we show that the broadcast time of MEET-EXCHANGE is asymptotically at least as large as the other two's on all regular graphs, and strictly larger on some regular graphs. As far as we know, this is the first systematic and thorough comparison of the running times of these very natural information dissemination protocols.The authors would like to thank Thomas Sauerwald and Nicol\'{a}s Rivera for helpful discussions. This research was undertaken, in part, thanks to funding from the ANR Project PAMELA (ANR-16-CE23-0016-01), the NSF Award Numbers CCF-1461559, CCF-0939370 and CCF-18107, the Gates Cambridge Scholarship programme, and the ERC grant DYNAMIC MARCH

Almost Logarithmic-Time Space Optimal Leader Election in Population Protocols

The model of population protocols refers to a large collection of simple indistinguishable entities, frequently called {\em agents}. The agents communicate and perform computation through pairwise interactions. We study fast and space efficient leader election in population of cardinality $n$ governed by a random scheduler, where during each time step the scheduler uniformly at random selects for interaction exactly one pair of agents. We propose the first $o(\log^2 n)$-time leader election protocol. Our solution operates in expected parallel time $O(\log n\log\log n)$ which is equivalent to $O(n \log n\log\log n)$ pairwise interactions. This is the fastest currently known leader election algorithm in which each agent utilises asymptotically optimal number of $O(\log\log n)$ states. The new protocol incorporates and amalgamates successfully the power of assorted {\em synthetic coins} with variable rate {\em phase clocks}

A simple approach for adapting continuous load balancing processes to discrete settings

We consider the neighbourhood load balancing problem. Given a network of processors and an arbitrary distribution of tasks over the network, the goal is to balance load by exchanging tasks between neighbours. In the continuous model, tasks can be arbitrarily divided and perfectly balanced state can always be reached. This is not possible in the discrete model where tasks are non-divisible. In this paper we consider the problem in a very general setting, where the tasks can have arbitrary weights and the nodes can have different speeds. Given a continuous load balancing algorithm that balances the load perfectly in (Formula presented.) rounds, we convert the algorithm into a discrete version. This new algorithm is deterministic and balances the load in  (Formula presented.) rounds so that the difference between the average and the maximum load is at most (Formula presented.) , where d is the maximum degree of the network and  (Formula presented.) is the maximum weight of any task. For general graphs, these bounds are asymptotically lower compared to the previous results. The proposed conversion scheme can be applied to a wide class of continuous processes, including first and second order diffusion, dimension exchange, and random matching processes. For the case of identical tasks, we present a randomized version of our algorithm that balances the load up to a discrepancy of  (Formula presented.) provided that the initial load on every node is large enough.Hoda Akbari, Petra Berenbrink and Thomas Sauerwald work was supported by an NSERC Discovery Grant “Analysis of Randomized Algorithms”

Communication Complexity of Quasirandom Rumor Spreading

We consider rumor spreading on random graphs and hypercubes in the quasirandom phone call model. In this model, every node has a list of neighbors whose order is specified by an adversary. In step i every node opens a channel to its ith neighbor (modulo degree) on that list, beginning from a randomly chosen starting position. Then, the channels can be used for bi-directional communication in that step. The goal is to spread a message efficiently to all nodes of the graph.For random graphs (with sufficiently many edges) we present an address-oblivious algorithm with runtime O(logn) that uses at most O(nloglogn) message transmissions. For hypercubes of dimension logn we present an address-oblivious algorithm with runtime O(logn) that uses at most O(n(loglogn)2) message transmissions.Together with a result of Elsässer (Proc. of SPAA’06, pp. 148–157, 2006), our results imply that for random graphs the communication complexity of the quasirandom phone call model is significantly smaller than that of the standard phone call model