78,978 research outputs found
Balanced Allocations and Double Hashing
Double hashing has recently found more common usage in schemes that use
multiple hash functions. In double hashing, for an item , one generates two
hash values and , and then uses combinations for to generate multiple hash values from the initial two. We
first perform an empirical study showing that, surprisingly, the performance
difference between double hashing and fully random hashing appears negligible
in the standard balanced allocation paradigm, where each item is placed in the
least loaded of choices, as well as several related variants. We then
provide theoretical results that explain the behavior of double hashing in this
context.Comment: Further updated, small improvements/typos fixe
Parallel Balanced Allocations: The Heavily Loaded Case
We study parallel algorithms for the classical balls-into-bins problem, in
which balls acting in parallel as separate agents are placed into 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 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 . The naive solution of
allocating each ball to a bin chosen uniformly and independently at random
results in maximal load (for ) 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 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 w.h.p. within rounds. The algorithm
is symmetric (balls and bins all "look the same"), and balls send
messages in expectation per round. The additive term of 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 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 in a
constant number of rounds w.h.p. Again, balls send only a single message per
round, and bins receive messages w.h.p
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
On monotonic core allocations for coalitional games whith veto players
We characterize a monotonic core concept defined on the class of veto balanced games. We also discuss what restricted versions of monotonicity are possible when selecting core allocations. We introduce a family of monotonic core concepts for veto balanced games and we show that, in general, the nucleolus per capita is not monotonic.monotonicity, core, nucleolus per capita, TU games
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
, 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
- VALUATION EQULIBRIUM REVISITED
This paper extends the notion of valuation equilibrium which applies to market economies involving the choice of a public environment. Unlike some other recent work, it is assumed here that consumers and firms evaluate alternative environments taking market prices as given (hence this notion is closer to that of competitive equilibria). It is shown that valuation equilibria with balanced tax schemas yield eficient allocations and that eficient allocations can be decentralized as valuation equilibria, with tax schemas that may be unbalanced.valuation equilibrium, non-convexities, public goods
On the Existence of Anonymous and Balanced Mechanisms Implementing the Lindahl Allocations
In this note, we discuss the existence of anonymous and balanced mechanisms to implement the Lindahl allocations. We obtain an impossibility result for the class of mechanisms defining an homeomorphism between the message space and the allocation space.Lindahl equilibrium ; economic mechanism
On extensions of the core and the anticore of transferable utility games
We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the L1-center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The L1-center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single-valued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.Transferable utility game; core; anticore; core extension; min-prenucleolus
The Power of Filling in Balanced Allocations
It is well known that if balls (jobs) are placed sequentially into
bins (servers) according to the One-Choice protocol choose a single bin in
each round and allocate one ball to it then, for , the gap between
the maximum and average load diverges. Many refinements of the One-Choice
protocol have been studied that achieve a gap that remains bounded by a
function of , for any . However most of these variations, such as
Two-Choice, are less sample-efficient than One-Choice, in the sense that for
each allocated ball more than one sample is needed (in expectation).
We introduce a new class of processes which are primarily characterized by
"filling" underloaded bins. A prototypical example is the Packing process: At
each round we only take one bin sample, if the load is below the average load,
then we place as many balls until the average load is reached; otherwise, we
place only one ball. We prove that for any process in this class the gap
between the maximum and average load is for any number of
balls . For the Packing process, we also prove a matching lower bound. We
also prove that the Packing process is more sample-efficient than One-Choice,
that is, it allocates on average more than one ball per sample. Finally, we
also demonstrate that the upper bound of on the gap can
be extended to the Caching process (a.k.a. memory protocol) studied by
Mitzenmacher, Prabhakar and Shah (2002).Comment: This paper refines and extends the content on filling processes in
arXiv:2110.10759. It consists of 31 pages, 6 figures, 2 table
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