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Register a new userBalanced allocation: Memory performance tradeoffs
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Added by
Itai Benjamini
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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Suppose we sequentially put $n$ balls into $n$ bins. If we put each ball into a random bin then the heaviest bin will contain ${sim}log n/loglog n$ balls with high probability. However, Azar, Broder, Karlin and Upfal [SIAM J. Comput. 29 (1999) 180--200] showed that if each time we choose two bins at random and put the ball in the least loaded bin among the two, then the heaviest bin will contain only ${sim}loglog n$ balls with high probability. How much memory do we need to implement this scheme? We need roughly $logloglog n$ bits per bin, and $nlogloglog n$ bits in total. Let us assume now that we have limited amount of memory. For each ball, we are given two random bins and we have to put the ball into one of them. Our goal is to minimize the load of the heaviest bin. We prove that if we have $n^{1-delta}$ bits then the heaviest bin will contain at least $Omega(deltalog n/loglog n)$ balls with high probability. The bound is tight in the communication complexity model.
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Grimmett and McDiarmid suggested a simple heuristic for finding stable sets in random graphs. They showed that the heuristic finds a stable set of size $simlog_2 n$ (with high probability) on a $G(n, 1/2)$ random graph. We determine the asymptotic distribution of the size of the stable set found by the algorithm.
Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix $A$, one may look at the spectrum of $psi(A)$ for a properly chosen $psi$. The issue is that the spectrum of $A$ might be contaminated by non-informational top eigenvalues, e.g., due to scale` variations in the data, and the application of $psi$ aims to remove these. Designing a good functional $psi$ (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, $$psi(A) = 1((I+A)^r ge1),$$ where $A$ denotes the adjacency matrix and $r$ is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance $r$, and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse ErdH{o}s-Renyi ensemble, which has no spectral gap, it is shown that graph powering produces a `maximal spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to cite{massoulie-STOC}, settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts cite{massoulie-STOC,Mossel_SBM2,bordenave,colin3}. This is illustrated on a geometric block model that is dense in cliques.
The Load-Balanced Router architecture has received a lot of attention because it does not require centralized scheduling at the internal switch fabrics. In this paper we reexamine the architecture, motivated by its potential to turn off multiple components and thereby conserve energy in the presence of low traffic. We perform a detailed analysis of the queue and delay performance of a Load-Balanced Router under a simple random routing algorithm. We calculate probabilistic bounds for queue size and delay, and show that the probabilities drop exponentially with increasing queue size or delay. We also demonstrate a tradeoff in energy consumption against the queue and delay performance.
In recent years much effort has been concentrated towards achieving polynomial time lower bounds on algorithms for solving various well-known problems. A useful technique for showing such lower bounds is to prove them conditionally based on well-studied hardness assumptions such as 3SUM, APSP, SETH, etc. This line of research helps to obtain a better understanding of the complexity inside P. A related question asks to prove conditional space lower bounds on data structures that are constructed to solve certain algorithmic tasks after an initial preprocessing stage. This question received little attention in previous research even though it has potential strong impact. In this paper we address this question and show that surprisingly many of the well-studied hard problems that are known to have conditional polynomial time lower bounds are also hard when concerning space. This hardness is shown as a tradeoff between the space consumed by the data structure and the time needed to answer queries. The tradeoff may be either smooth or admit one or more singularity points. We reveal interesting connections between different space hardness conjectures and present matching upper bounds. We also apply these hardness conjectures to both static and dynamic problems and prove their conditional space hardness. We believe that this novel framework of polynomial space conjectures can play an important role in expressing polynomial space lower bounds of many important algorithmic problems. Moreover, it seems that it can also help in achieving a better understanding of the hardness of their corresponding problems in terms of time.
This paper initiates the study of the classic balanced graph partitioning problem from an online perspective: Given an arbitrary sequence of pairwise communication requests between $n$ nodes, with patterns that may change over time, the objective is to service these requests efficiently by partitioning the nodes into $ell$ clusters, each of size $k$, such that frequently communicating nodes are located in the same cluster. The partitioning can be updated dynamically by migrating nodes between clusters. The goal is to devise online algorithms which jointly minimize the amount of inter-cluster communication and migration cost. The problem features interesting connections to other well-known online problems. For example, scenarios with $ell=2$ generalize online paging, and scenarios with $k=2$ constitute a novel online variant of maximum matching. We present several lower bounds and algorithms for settings both with and without cluster-size augmentation. In particular, we prove that any deterministic online algorithm has a competitive ratio of at least $k$, even with significant augmentation. Our main algorithmic contributions are an $O(k log{k})$-competitive deterministic algorithm for the general setting with constant augmentation, and a constant competitive algorithm for the maximum matching variant.
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