20,320 research outputs found
Tight bounds for parallel randomized load balancing
Given a distributed system of n balls and n bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest non-adaptive and symmetric algorithm achieving a constant maximum bin load requires Θ(loglogn) rounds, and any such algorithm running for r∈O(1) rounds incurs a bin load of Ω((logn/loglogn)1/r). In this work, we explore the fundamental limits of the general problem. We present a simple adaptive symmetric algorithm that achieves a bin load of 2 in log∗n+O(1) communication rounds using O(n) messages in total. Our main result, however, is a matching lower bound of (1−o(1))log∗n on the time complexity of symmetric algorithms that guarantee small bin loads. The essential preconditions of the proof are (i) a limit of O(n) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls need not be globally consistent. In order to show that our technique yields indeed tight bounds, we provide for each assumption an algorithm violating it, in turn achieving a constant maximum bin load in constant time.German Research Foundation (DFG, reference number Le 3107/1-1)Society of Swiss Friends of the Weizmann Institute of ScienceSwiss National Fun
Tight Load Balancing via Randomized Local Search
We consider the following balls-into-bins process with bins and
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 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
. 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
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
A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase-Type Service Times
In this paper, we provide a matrix-analytic solution for randomized load
balancing models (also known as \emph{supermarket models}) with phase-type (PH)
service times. Generalizing the service times to the phase-type distribution
makes the analysis of the supermarket models more difficult and challenging
than that of the exponential service time case which has been extensively
discussed in the literature. We first describe the supermarket model as a
system of differential vector equations, and provide a doubly exponential
solution to the fixed point of the system of differential vector equations.
Then we analyze the exponential convergence of the current location of the
supermarket model to its fixed point. Finally, we present numerical examples to
illustrate our approach and show its effectiveness in analyzing the randomized
load balancing schemes with non-exponential service requirements.Comment: 24 page
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