17,693 research outputs found
Multiple choice allocations with small maximum loads
The idea of using multiple choices to improve allocation schemes is now well understood and is often illustrated by the following example. Suppose balls are allocated to bins with each ball choosing a bin independently and uniformly at random. The \emph{maximum load}, or the number of balls in the most loaded bin, will then be approximately with high probability. Suppose now the balls are allocated sequentially by placing a ball in the least loaded bin among the bins chosen independently and uniformly at random. Azar, Broder,
Karlin, and Upfal showed that in this scenario, the maximum load drops to , with high probability, which is an exponential improvement over the previous case.
In this thesis we investigate multiple choice allocations from a slightly different perspective. Instead of minimizing the maximum load, we fix the bin capacities and focus on maximizing the number of balls that can be allocated without overloading any bin. In the process that we consider we have balls and bins. Each ball chooses bins independently and uniformly at random. \emph{Is it possible to assign each ball to one of its choices such that the no bin receives more than balls?} For all and we give a critical value, , such that when this is not the case.
In case such an allocation exists, \emph{how quickly can we find it?} Previous work on total allocation time for case and has analyzed a \emph{breadth first strategy} which is shown to be linear only in expectation. We give a simple and efficient algorithm which we also call \emph{local search allocation}(LSA) to find an allocation for all and . Provided the number of balls are below (but arbitrarily close to) the theoretical achievable load threshold, we give a \emph{linear} bound for the total allocation time that holds with high probability.
We demonstrate, through simulations, an order of magnitude improvement for total and maximum allocation times when compared to the state of the art method.
Our results find applications in many areas including hashing, load balancing, data management, orientability of random hypergraphs and maximum matchings in a special class of bipartite graphs.Die Idee, mehrere Wahlmöglichkeiten zu benutzen, um Zuordnungsschemas zu verbessern, ist mittlerweile gut verstanden und wird oft mit Hilfe des folgenden Beispiels illustriert: Man nehme an, dass n Kugeln auf n BehĂ€lter verteilt werden und jede Kugel unabhĂ€ngig und gleichverteilt per Zufall ihren BehĂ€lter wĂ€hlt. Die maximale Auslastung, bzw. die Anzahl an Kugeln im meist befĂŒllten BehĂ€lter, wird dann mit hoher Wahrscheinlichkeit schĂ€tzungsweise sein. Alternativ können die Kugeln sequenziell zugeordnet werden, indem jede Kugel k â„ 2 BehĂ€lter unabhĂ€ngig und gleichverteilt zufĂ€llig auswĂ€hlt und in dem am wenigsten befĂŒllten dieser k BehĂ€lter platziert wird. Azar, Broder, Karlin, and Upfal haben gezeigt, dass in diesem Szenario die maximale Auslastung mit hoher Wahrscheinlichkeit auf sinkt, was eine exponentielle Verbesserung des vorhergehenden Falls darstellt.
In dieser Doktorarbeit untersuchen wir solche Zuteilungschemas von einem etwas anderen Standpunkt. Statt die maximale Last zu minimieren, ïŹxieren wir die KapazitĂ€ten der BehĂ€lter und konzentrieren uns auf die Maximierung der Anzahl der Kugeln, die ohne Ăberlastung eines BehĂ€lters zugeteilt werden können. In dem von uns betrachteten Prozess haben wir m = bcnc Kugeln und n BehĂ€lter. Jede Kugel wĂ€hlt unabhĂ€ngig und gleichverteilt zufĂ€llig k BehĂ€lter. Ist es möglich, jeder Kugel einen BehĂ€lter ihrer Wahl zuzuordnen, so dass kein BehĂ€lter mehr als Kugeln erhĂ€lt? FĂŒr alle k â„ 3 und â„ 2 geben wir einen kritischen Wert , an sodass fĂŒr c c {k,\ell}^*\ell = 1\ell = 1$ ïŹndet. Sofern die Anzahl der Kugeln unter (aber beliebig nahe an) der theoretisch erreichbaren Lastschwelle ist, zeigen wir eine lineare Schranke fĂŒr die Gesamtzuordnungszeit, die mit hoher Wahrscheinlichkeit gilt. Anhand von Simulationen demonstrieren wir eine Verbesserung der Gesamt- und Maximalzuordnungszeiten um eine GröĂenordnung im Vergleich zu anderen aktuellen Methoden.
Unsere Ergebnisse ïŹnden Anwendung in vielen Bereichen einschlieĂlich Hashing, Lastbalancierung, Datenmanagement, Orientierbarkeit von zufĂ€lligen Hypergraphen und maximale Paarungen in einer speziellen Klasse von bipartiten Graphen
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
Traffic-Driven Spectrum Allocation in Heterogeneous Networks
Next generation cellular networks will be heterogeneous with dense deployment
of small cells in order to deliver high data rate per unit area. Traffic
variations are more pronounced in a small cell, which in turn lead to more
dynamic interference to other cells. It is crucial to adapt radio resource
management to traffic conditions in such a heterogeneous network (HetNet). This
paper studies the optimization of spectrum allocation in HetNets on a
relatively slow timescale based on average traffic and channel conditions
(typically over seconds or minutes). Specifically, in a cluster with base
transceiver stations (BTSs), the optimal partition of the spectrum into
segments is determined, corresponding to all possible spectrum reuse patterns
in the downlink. Each BTS's traffic is modeled using a queue with Poisson
arrivals, the service rate of which is a linear function of the combined
bandwidth of all assigned spectrum segments. With the system average packet
sojourn time as the objective, a convex optimization problem is first
formulated, where it is shown that the optimal allocation divides the spectrum
into at most segments. A second, refined model is then proposed to address
queue interactions due to interference, where the corresponding optimal
allocation problem admits an efficient suboptimal solution. Both allocation
schemes attain the entire throughput region of a given network. Simulation
results show the two schemes perform similarly in the heavy-traffic regime, in
which case they significantly outperform both the orthogonal allocation and the
full-frequency-reuse allocation. The refined allocation shows the best
performance under all traffic conditions.Comment: 13 pages, 11 figures, accepted for publication by JSAC-HC
Probabilistic game approaches for network cost allocation
In a restructured power market, the network cost is to be allocated between multiple players utilizing the system in varying capacities. Cooperative game approaches based on Shapley value and Nucleolus provide stable models for embedded cost allocation of power networks. Varying network usage necessitates the introduction of probabilistic approaches to cooperative games. This paper proposes a variety of probabilistic cooperative game approaches. These have variably been modeled based upon the probability of existence of players, the probability of existence of coalitions, and the probability of players joining a particular coalition along with their joining in a particular sequence. Application of these approaches to power networks reflects the system usage in a more justified way. Consistent and stable results qualify the application of probabilistic cooperative game approaches for cost allocation of power networks.Cooperative games, embedded cost allocation, probabilistic games, transmission pricing
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
The densest subgraph problem in sparse random graphs
We determine the asymptotic behavior of the maximum subgraph density of large
random graphs with a prescribed degree sequence. The result applies in
particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of
Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in
extending the notion of balanced loads from finite graphs to their local weak
limits, using unimodularity. This is a new illustration of the objective method
described by Aldous and Steele [In Probability on Discrete Structures (2004)
1-72 Springer].Comment: Published at http://dx.doi.org/10.1214/14-AAP1091 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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