Paging with Dynamic Memory Capacity

We study a generalization of the classic paging problem that allows the amount of available memory to vary over time - capturing a fundamental property of many modern computing realities, from cloud computing to multi-core and energy-optimized processors. It turns out that good performance in the "classic" case provides no performance guarantees when memory capacity fluctuates: roughly speaking, moving from static to dynamic capacity can mean the difference between optimality within a factor 2 in space and time, and suboptimality by an arbitrarily large factor. More precisely, adopting the competitive analysis framework, we show that some online paging algorithms, despite having an optimal (h,k)-competitive ratio when capacity remains constant, are not (3,k)-competitive for any arbitrarily large k in the presence of minimal capacity fluctuations. In this light it is surprising that several classic paging algorithms perform remarkably well even if memory capacity changes adversarially - in fact, even without taking those changes into explicit account! In particular, we prove that LFD still achieves the minimum number of faults, and that several classic online algorithms such as LRU have a "dynamic" (h,k)-competitive ratio that is the best one can achieve without knowledge of future page requests, even if one had perfect knowledge of future capacity fluctuations. Thus, with careful management, knowing/predicting future memory resources appears far less crucial to performance than knowing/predicting future data accesses. We characterize the optimal "dynamic" (h,k)-competitive ratio exactly, and show it has a somewhat complex expression that is almost but not quite equal to the "classic" ratio k/(k-h+1), thus proving a strict if minuscule separation between online paging performance achievable in the presence or absence of capacity fluctuations

Sublinear algorithms for local graph centrality estimation

We study the complexity of local graph centrality estimation, with the goal of approximating the centrality score of a given target node while exploring only a sublinear number of nodes/arcs of the graph and performing a sublinear number of elementary operations. We develop a technique, that we apply to the PageRank and Heat Kernel centralities, for building a low-variance score estimator through a local exploration of the graph. We obtain an algorithm that, given any node in any graph of $m$ arcs, with probability $(1-\delta)$ computes a multiplicative $(1\pm\epsilon)$-approximation of its score by examining only $\tilde{O}(\min(m^{2/3} \Delta^{1/3} d^{-2/3},\, m^{4/5} d^{-3/5}))$ nodes/arcs, where $\Delta$ and $d$ are respectively the maximum and average outdegree of the graph (omitting for readability $\operatorname{poly}(\epsilon^{-1})$ and $\operatorname{polylog}(\delta^{-1})$ factors). A similar bound holds for computational complexity. We also prove a lower bound of $\Omega(\min(m^{1/2} \Delta^{1/2} d^{-1/2}, \, m^{2/3} d^{-1/3}))$ for both query complexity and computational complexity. Moreover, our technique yields a $\tilde{O}(n^{2/3})$ query complexity algorithm for the graph access model of [Brautbar et al., 2010], widely used in social network mining; we show this algorithm is optimal up to a sublogarithmic factor. These are the first algorithms yielding worst-case sublinear bounds for general directed graphs and any choice of the target node.Comment: 29 pages, 1 figur

Matching on the Line Admits No o(?log n)-Competitive Algorithm

We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line exceeds ?{log?(n +1)}/15 for all n = 2?-1: i ? ?, settling a 25-year-old open question

A Dynamically Partitionable Compressed Cache

The effective size of an L2 cache can be increased by using a dictionary-based compression scheme. Naive application of this idea performs poorly since the data values in a cache greatly vary in their “compressibility.” The novelty of this paper is a scheme that dynamically partitions the cache into sections of different compressibilities. While compression is often researched in the context of a large stream, in this work it is applied repeatedly on smaller cache-line sized blocks so as to preserve the random access requirement of a cache. When a cache-line is brought into the L2 cache or the cache-line is to be modified, the line is compressed using a dynamic, LZW dictionary. Depending on the compression, it is placed into the relevant partition. The partitioning is dynamic in that the ratio of space allocated to compressed and uncompressed varies depending on the actual performance, Certain SPEC-2000 benchmarks using a compressed L2 cache show an 80reduction in L2 miss-rate when compared to using an uncompressed L2 cache of the same area, taking into account all area overhead associated with the compression circuitry. For other SPEC-2000 benchmarks, the compressed cache performs as well as a traditional cache that is 4.3 times as large as the compressed cache in terms of hit rate, The adaptivity ensures that, in terms of miss rates, the compressed cache never performs worse than a traditional cache.Singapore-MIT Alliance (SMA

On Approximating the Stationary Distribution of Time-reversible Markov Chains

Approximating the stationary probability of a state in a Markov chain through Markov chain Monte Carlo techniques is, in general, inefficient. Standard random walk approaches require tilde{O}(tau/pi(v)) operations to approximate the probability pi(v) of a state v in a chain with mixing time tau, and even the best available techniques still have complexity tilde{O}(tau^1.5 / pi(v)^0.5); and since these complexities depend inversely on pi(v), they can grow beyond any bound in the size of the chain or in its mixing time. In this paper we show that, for time-reversible Markov chains, there exists a simple randomized approximation algorithm that breaks this "small-pi(v) barrier"