7,603 research outputs found
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
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
Balanced Allocation on Graphs: A Random Walk Approach
In this paper we propose algorithms for allocating sequential balls into
bins that are interconnected as a -regular -vertex graph , where
can be any integer.Let be a given positive integer. In each round
, , ball picks a node of uniformly at random and
performs a non-backtracking random walk of length 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 has a sufficiently large girth
and . Then we establish an upper bound for the maximum number
of balls at any bin after allocating balls by the algorithm, called {\it
maximum load}, in terms of with high probability. We also show that the
upper bound is at most an factor above the lower bound that is
proved for the algorithm. In particular, we show that if we set , for every constant , and
has girth at least , then the maximum load attained by the
algorithm is bounded by with high probability.Finally, we
slightly modify the algorithm to have similar results for balanced allocation
on -regular graph with and sufficiently large girth
More Analysis of Double Hashing for Balanced Allocations
With double hashing, for a key , one generates two hash values and
, and then uses combinations for
to generate multiple hash values in the range from the initial two.
For balanced allocations, keys are hashed into a hash table where each bucket
can hold multiple keys, and each key is placed in the least loaded of
choices. It has been shown previously that asymptotically the performance of
double hashing and fully random hashing is the same in the balanced allocation
paradigm using fluid limit methods. Here we extend a coupling argument used by
Lueker and Molodowitch to show that double hashing and ideal uniform hashing
are asymptotically equivalent in the setting of open address hash tables to the
balanced allocation setting, providing further insight into this phenomenon. We
also discuss the potential for and bottlenecks limiting the use this approach
for other multiple choice hashing schemes.Comment: 13 pages ; current draft ; will be submitted to conference shortl
Counting Connected Graphs Asymptotically
We find the asymptotic number of connected graphs with vertices and
edges when approach infinity, reproving a result of Bender,
Canfield and McKay. We use the {\em probabilistic method}, analyzing
breadth-first search on the random graph for an appropriate edge
probability . Central is analysis of a random walk with fixed beginning and
end which is tilted to the left.Comment: 23 page
Fast and Powerful Hashing using Tabulation
Randomized algorithms are often enjoyed for their simplicity, but the hash
functions employed to yield the desired probabilistic guarantees are often too
complicated to be practical. Here we survey recent results on how simple
hashing schemes based on tabulation provide unexpectedly strong guarantees.
Simple tabulation hashing dates back to Zobrist [1970]. Keys are viewed as
consisting of characters and we have precomputed character tables
mapping characters to random hash values. A key
is hashed to . This schemes is
very fast with character tables in cache. While simple tabulation is not even
4-independent, it does provide many of the guarantees that are normally
obtained via higher independence, e.g., linear probing and Cuckoo hashing.
Next we consider twisted tabulation where one input character is "twisted" in
a simple way. The resulting hash function has powerful distributional
properties: Chernoff-Hoeffding type tail bounds and a very small bias for
min-wise hashing. This also yields an extremely fast pseudo-random number
generator that is provably good for many classic randomized algorithms and
data-structures.
Finally, we consider double tabulation where we compose two simple tabulation
functions, applying one to the output of the other, and show that this yields
very high independence in the classic framework of Carter and Wegman [1977]. In
fact, w.h.p., for a given set of size proportional to that of the space
consumed, double tabulation gives fully-random hashing. We also mention some
more elaborate tabulation schemes getting near-optimal independence for given
time and space.
While these tabulation schemes are all easy to implement and use, their
analysis is not
Target Assignment in Robotic Networks: Distance Optimality Guarantees and Hierarchical Strategies
We study the problem of multi-robot target assignment to minimize the total
distance traveled by the robots until they all reach an equal number of static
targets. In the first half of the paper, we present a necessary and sufficient
condition under which true distance optimality can be achieved for robots with
limited communication and target-sensing ranges. Moreover, we provide an
explicit, non-asymptotic formula for computing the number of robots needed to
achieve distance optimality in terms of the robots' communication and
target-sensing ranges with arbitrary guaranteed probabilities. The same bounds
are also shown to be asymptotically tight.
In the second half of the paper, we present suboptimal strategies for use
when the number of robots cannot be chosen freely. Assuming first that all
targets are known to all robots, we employ a hierarchical communication model
in which robots communicate only with other robots in the same partitioned
region. This hierarchical communication model leads to constant approximations
of true distance-optimal solutions under mild assumptions. We then revisit the
limited communication and sensing models. By combining simple rendezvous-based
strategies with a hierarchical communication model, we obtain decentralized
hierarchical strategies that achieve constant approximation ratios with respect
to true distance optimality. Results of simulation show that the approximation
ratio is as low as 1.4
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