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

Tight Load Balancing via Randomized Local Search

We consider the following balls-into-bins process with $n$ bins and $m$ balls: each ball is equipped with a mutually independent exponential clock of rate 1. Whenever a ball's clock rings, the ball samples a random bin and moves there if the number of balls in the sampled bin is smaller than in its current bin. This simple process models a typical load balancing problem where users (balls) seek a selfish improvement of their assignment to resources (bins). From a game theoretic perspective, this is a randomized approach to the well-known Koutsoupias-Papadimitriou model, while it is known as randomized local search (RLS) in load balancing literature. Up to now, the best bound on the expected time to reach perfect balance was $O\left({(\ln n)}^2+\ln(n)\cdot n^2/m\right)$ due to Ganesh, Lilienthal, Manjunath, Proutiere, and Simatos (Load balancing via random local search in closed and open systems, Queueing Systems, 2012). We improve this to an asymptotically tight $O\left(\ln(n)+n^2/m\right)$. Our analysis is based on the crucial observation that performing "destructive moves" (reversals of RLS moves) cannot decrease the balancing time. This allows us to simplify problem instances and to ignore "inconvenient moves" in the analysis.Comment: 24 pages, 3 figures, preliminary version appeared in proceedings of 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS'17

Utilitarian resource assignment

This paper studies a resource allocation problem introduced by Koutsoupias and Papadimitriou. The scenario is modelled as a multiple-player game in which each player selects one of a finite number of known resources. The cost to the player is the total weight of all players who choose that resource, multiplied by the ``delay'' of that resource. Recent papers have studied the Nash equilibria and social optima of this game in terms of the $L_\infty$ cost metric, in which the social cost is taken to be the maximum cost to any player. We study the $L_1$ variant of this game, in which the social cost is taken to be the sum of the costs to the individual players, rather than the maximum of these costs. We give bounds on the size of the coordination ratio, which is the ratio between the social cost incurred by selfish behavior and the optimal social cost; we also study the algorithmic problem of finding optimal (lowest-cost) assignments and Nash Equilibria. Additionally, we obtain bounds on the ratio between alternative Nash equilibria for some special cases of the problem.Comment: 19 page

Palindrome Recognition In The Streaming Model

In the Palindrome Problem one tries to find all palindromes (palindromic substrings) in a given string. A palindrome is defined as a string which reads forwards the same as backwards, e.g., the string "racecar". A related problem is the Longest Palindromic Substring Problem in which finding an arbitrary one of the longest palindromes in the given string suffices. We regard the streaming version of both problems. In the streaming model the input arrives over time and at every point in time we are only allowed to use sublinear space. The main algorithms in this paper are the following: The first one is a one-pass randomized algorithm that solves the Palindrome Problem. It has an additive error and uses $O(\sqrt n$) space. The second algorithm is a two-pass algorithm which determines the exact locations of all longest palindromes. It uses the first algorithm as the first pass. The third algorithm is again a one-pass randomized algorithm, which solves the Longest Palindromic Substring Problem. It has a multiplicative error using only $O(\log(n))$ space. We also give two variants of the first algorithm which solve other related practical problems

A new analytical method for parallel, diffusion-type load balancing

We propose a new proof technique which can be used to analyze many parallel load balancing algorithms. The technique is designed to handle concurrent load balancing actions, which are often the main obstacle in the analysis. We demonstrate the usefulness of the approach by analyzing various natural diffusion-type protocols. Our results are similar to, or better than, previously existing ones, while our proofs are much easier. The key idea is to first sequentialize the original, concurrent load transfers, analyze this new, sequential system, and then to bound the gap between both.

Bounds on the Voter Model in Dynamic Networks

In the voter model, each node of a graph has an opinion, and in every round each node chooses independently a random neighbour and adopts its opinion. We are interested in the consensus time, which is the first point in time where all nodes have the same opinion. We consider dynamic graphs in which the edges are rewired in every round (by an adversary) giving rise to the graph sequence $G_1, G_2, \dots$, where we assume that $G_i$ has conductance at least $\phi_i$. We assume that the degrees of nodes don't change over time as one can show that the consensus time can become super-exponential otherwise. In the case of a sequence of $d$-regular graphs, we obtain asymptotically tight results. Even for some static graphs, such as the cycle, our results improve the state of the art. Here we show that the expected number of rounds until all nodes have the same opinion is bounded by $O(m/(d_{min} \cdot \phi))$, for any graph with $m$ edges, conductance $\phi$, and degrees at least $d_{min}$. In addition, we consider a biased dynamic voter model, where each opinion $i$ is associated with a probability $P_i$, and when a node chooses a neighbour with that opinion, it adopts opinion $i$ with probability $P_i$ (otherwise the node keeps its current opinion). We show for any regular dynamic graph, that if there is an $\epsilon>0$ difference between the highest and second highest opinion probabilities, and at least $\Omega(\log n)$ nodes have initially the opinion with the highest probability, then all nodes adopt w.h.p. that opinion. We obtain a bound on the convergences time, which becomes $O(\log n/\phi)$ for static graphs

Parallel Balanced Allocations: The Heavily Loaded Case

We study parallel algorithms for the classical balls-into-bins problem, in which $m$ balls acting in parallel as separate agents are placed into $n$ bins. Algorithms operate in synchronous rounds, in each of which balls and bins exchange messages once. The goal is to minimize the maximal load over all bins using a small number of rounds and few messages. While the case of $m=n$ balls has been extensively studied, little is known about the heavily loaded case. In this work, we consider parallel algorithms for this somewhat neglected regime of $m\gg n$. The naive solution of allocating each ball to a bin chosen uniformly and independently at random results in maximal load $m/n+\Theta(\sqrt{m/n\cdot \log n})$ (for $m\geq n \log n$) w.h.p. In contrast, for the sequential setting Berenbrink et al (SIAM J. Comput 2006) showed that letting each ball join the least loaded bin of two randomly selected bins reduces the maximal load to $m/n+O(\log\log m)$ w.h.p. To date, no parallel variant of such a result is known. We present a simple parallel threshold algorithm that obtains a maximal load of $m/n+O(1)$ w.h.p. within $O(\log\log (m/n)+\log^* n)$ rounds. The algorithm is symmetric (balls and bins all "look the same"), and balls send $O(1)$ messages in expectation per round. The additive term of $O(\log^* n)$ in the complexity is known to be tight for such algorithms (Lenzen and Wattenhofer Distributed Computing 2016). We also prove that our analysis is tight, i.e., algorithms of the type we provide must run for $\Omega(\min\{\log\log (m/n),n\})$ rounds w.h.p. Finally, we give a simple asymmetric algorithm (i.e., balls are aware of a common labeling of the bins) that achieves a maximal load of $m/n + O(1)$ in a constant number of rounds w.h.p. Again, balls send only a single message per round, and bins receive $(1+o(1))m/n+O(\log n)$ messages w.h.p

Improved Analysis of Deterministic Load-Balancing Schemes

We consider the problem of deterministic load balancing of tokens in the discrete model. A set of $n$ processors is connected into a $d$-regular undirected network. In every time step, each processor exchanges some of its tokens with each of its neighbors in the network. The goal is to minimize the discrepancy between the number of tokens on the most-loaded and the least-loaded processor as quickly as possible. Rabani et al. (1998) present a general technique for the analysis of a wide class of discrete load balancing algorithms. Their approach is to characterize the deviation between the actual loads of a discrete balancing algorithm with the distribution generated by a related Markov chain. The Markov chain can also be regarded as the underlying model of a continuous diffusion algorithm. Rabani et al. showed that after time $T = O(\log (Kn)/\mu)$, any algorithm of their class achieves a discrepancy of $O(d\log n/\mu)$, where $\mu$ is the spectral gap of the transition matrix of the graph, and $K$ is the initial load discrepancy in the system. In this work we identify some natural additional conditions on deterministic balancing algorithms, resulting in a class of algorithms reaching a smaller discrepancy. This class contains well-known algorithms, eg., the Rotor-Router. Specifically, we introduce the notion of cumulatively fair load-balancing algorithms where in any interval of consecutive time steps, the total number of tokens sent out over an edge by a node is the same (up to constants) for all adjacent edges. We prove that algorithms which are cumulatively fair and where every node retains a sufficient part of its load in each step, achieve a discrepancy of $O(\min\{d\sqrt{\log n/\mu},d\sqrt{n}\})$ in time $O(T)$. We also show that in general neither of these assumptions may be omitted without increasing discrepancy. We then show by a combinatorial potential reduction argument that any cumulatively fair scheme satisfying some additional assumptions achieves a discrepancy of $O(d)$ almost as quickly as the continuous diffusion process. This positive result applies to some of the simplest and most natural discrete load balancing schemes.Comment: minor corrections; updated literature overvie

Efficient randomised broadcasting in random regular networks with applications in peer-to-peer systems

We consider broadcasting in random d-regular graphs by using a simple modification of the random phone call model introduced by Karp et al. (Proceedings of the FOCS ’00, 2000). In the phone call model, in every time step, each node calls a randomly chosen neighbour to establish a communication channel to this node. The communication channels can then be used bi-directionally to transmit messages. We show that, if we allow every node to choose four distinct neighbours instead of one, then the average number of message transmissions per node required to broadcast a message efficiently decreases exponentially. Formally, we present an algorithm that has time complexity O(logn) and uses O(nloglogn) transmissions per message. In contrast, we show for the standard model that every distributed algorithm in a restricted address-oblivious model that broadcasts a message in time O(logn) requires Ω(nlogn/logd) message transmissions. Our algorithm efficiently handles limited communication failures, only requires rough estimates of the number of nodes, and is robust against limited changes in the size of the network. Our results have applications in peer-to-peer networks and replicated databases. Preliminary version published in the Proceedings of the 27th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2008)