Balls into bins via local search: Cover time and maximum load

© 2015 Wiley Periodicals, Inc. Abstract-We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m=n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is vertex transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m≥n.ETH Zurich Postdoctoral Fellowship Program Marie Curie Career Integration. Grant Number: PCIG13‐GA‐2013‐618588 DSRELI

Balls into bins via local search: Cover time and maximum load

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m = n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m > n.Comment: arXiv admin note: text overlap with arXiv:1207.212

Locally Optimal Load Balancing

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.Comment: 19 pages, 11 figure

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

Stationary Distribution Analysis of a Queueing Model with Local Choice

The paper deals with load balancing between one-server queues on a circle by a local choice policy. Each one-server queue has a Poissonian arrival of customers. When a customer arrives at a queue, he joins the least loaded queue between this queue and the next one, ties solved at random. Service times have exponential distribution. The system is stable if the arrival-to-service rate ratio called load is less than one. When the load tends to zero, we derive the first terms of the expansion in this parameter for the stationary probabilities that a queue has 0 to 3 customers. We investigate the error, comparing these expansion results to numerical values obtained by simulations. Then we provide the asymptotics, as the load tends to zero, for the stationary probabilities of the queue length, for a fixed number of queues. It quantifies the difference between policies with this local choice, no choice and the choice between two queues chosen at random

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Balanced Allocation on Hypergraphs

We consider a variation of balls-into-bins which randomly allocates $m$ balls into $n$ bins. Following Godfrey's model (SODA, 2008), we assume that each ball $t$, $1\le t\le m$, comes with a hypergraph $\mathcal{H}^{(t)}=\{B_1,B_2,\ldots,B_{s_t}\}$, and each edge $B\in\mathcal{H}^{(t)}$ contains at least a logarithmic number of bins. Given $d\ge 2$, our $d$-choice algorithm chooses an edge $B\in \mathcal{H}^{(t)}$, uniformly at random, and then chooses a set $D$ of $d$ random bins from the selected edge $B$. The ball is allocated to a least-loaded bin from $D$, with ties are broken randomly. We prove that if the hypergraphs $\mathcal{H}^{(1)},\ldots, \mathcal{H}^{(m)}$ satisfy a \emph{balancedness} condition and have low \emph{pair visibility}, then after allocating $m=\Theta(n)$ balls, the maximum number of balls at any bin, called the \emph{maximum load}, is at most $\log_d\log n+O(1)$, with high probability. The balancedness condition enforces that bins appear almost uniformly within the hyperedges of $\mathcal{H}^{(t)}$, $1\le t\le m$, while the pair visibility condition measures how frequently a pair of bins is chosen during the allocation of balls. Moreover, we establish a lower bound for the maximum load attained by the balanced allocation for a sequence of hypergraphs in terms of pair visibility, showing the relevance of the visibility parameter to the maximum load. In Godfrey's model, each ball is forced to probe all bins in a randomly selected hyperedge and the ball is then allocated in a least-loaded bin. Godfrey showed that if each $\mathcal{H}^{(t)}$, $1\le t\le m$, is balanced and $m=O(n)$, then the maximum load is at most one, with high probability. However, we apply the power of $d$ choices paradigm, and only query the load information of $d$ random bins per ball, while achieving very slow growth in the maximum load

Sampling from discrete distributions and computing Fréchet distances

In the first part of this thesis, we study the fundamental problem of sampling from a discrete probability distribution. Specifically, given non-negative numbers p_1,...,p_n the task is to draw i with probability proportional to p_i. We extend the classic solution to this problem, Walker's alias method, in various directions: We improve its space requirements, we solve the special case of sorted input, we study sampling natural distributions on a bounded precision machine, and as an application we speed up sampling a model from physics. The second part of this thesis belongs to the area of computational geometry and deals with algorithms for the Fréchet distance, which is a popular measure of similarity of two curves and can be computed in quadratic time (ignoring logarithmic factors). We provide the first conditional lower bound for this problem: No polynomial factor improvement over the quadratic running time is possible unless the Strong Exponential Time Hypothesis fails. We also present an improved approximation algorithm for realistic input curves.Im ersten Teil dieser Dissertation untersuchen wir das fundamentale Problem des Ziehens einer Zufallsvariablen von einer gegebenen diskreten Wahrscheinlichkeitsverteilung. Die Aufgabe ist, gegeben nichtnegative Zahlen p_1,...,p_n, eine Zahl i mit Wahrscheinlichkeit proportional zu p_i zu ziehen. Wir erweitern die klassische Lösung dieses Problems, Walkers Aliasmethode, in verschiedene Richtungen: Wir verbessern ihren Speicherbedarf, wir lösen den Spezialfall von sortierter Eingabe, wir untersuchen das Ziehen von natürlichen Verteilungen auf Maschinen mit beschränkter Präzision, und als Anwendung beschleunigen wir die Simulation eines physikalischen Modells. Der zweite Teil dieser Dissertation gehört zum Gebiet der Computergeometrie und beschäftigt sich mit Algorithmen für die Fréchetdistanz, die ein beliebtes Ähnlichkeitsmaß zweier Kurven ist und in quadratischer Zeit berechnet werden kann (bis auf logarithmische Faktoren). Wir zeigen die erste bedingte untere Schranke für dieses Problem: Keine Verbesserung um einen polynomiellen Faktor ist möglich unter der starken Exponentialzeithypothese. Zudem präsentieren wir einen verbesserten Approximationsalgorithmus für realistische Eingabekurven