External Sampling

36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part IWe initiate the study of sublinear-time algorithms in the external memory model [1]. In this model, the data is stored in blocks of a certain size B, and the algorithm is charged a unit cost for each block access. This model is well-studied, since it reflects the computational issues occurring when the (massive) input is stored on a disk. Since each block access operates on B data elements in parallel, many problems have external memory algorithms whose number of block accesses is only a small fraction (e.g. 1/B) of their main memory complexity. However, to the best of our knowledge, no such reduction in complexity is known for any sublinear-time algorithm. One plausible explanation is that the vast majority of sublinear-time algorithms use random sampling and thus exhibit no locality of reference. This state of affairs is quite unfortunate, since both sublinear-time algorithms and the external memory model are important approaches to dealing with massive data sets, and ideally they should be combined to achieve best performance. In this paper we show that such combination is indeed possible. In particular, we consider three well-studied problems: testing of distinctness, uniformity and identity of an empirical distribution induced by data. For these problems we show random-sampling-based algorithms whose number of block accesses is up to a factor of 1/√B smaller than the main memory complexity of those problems. We also show that this improvement is optimal for those problems. Since these problems are natural primitives for a number of sampling-based algorithms for other problems, our tools improve the external memory complexity of other problems as well.David & Lucile Packard Foundation (Fellowship)Center for Massive Data Algorithmics (MADALGO)Marie Curie (International Reintegration Grant 231077)National Science Foundation (U.S.) (Grant 0514771)National Science Foundation (U.S.) (Grant 0728645)National Science Foundation (U.S.) (Grant 0732334)Symantec Research Labs (Research Fellowship

Implicit Decomposition for Write-Efficient Connectivity Algorithms

The future of main memory appears to lie in the direction of new technologies that provide strong capacity-to-performance ratios, but have write operations that are much more expensive than reads in terms of latency, bandwidth, and energy. Motivated by this trend, we propose sequential and parallel algorithms to solve graph connectivity problems using significantly fewer writes than conventional algorithms. Our primary algorithmic tool is the construction of an $o(n)$-sized "implicit decomposition" of a bounded-degree graph $G$ on $n$ nodes, which combined with read-only access to $G$ enables fast answers to connectivity and biconnectivity queries on $G$. The construction breaks the linear-write "barrier", resulting in costs that are asymptotically lower than conventional algorithms while adding only a modest cost to querying time. For general non-sparse graphs on $m$ edges, we also provide the first $o(m)$ writes and $O(m)$ operations parallel algorithms for connectivity and biconnectivity. These algorithms provide insight into how applications can efficiently process computations on large graphs in systems with read-write asymmetry

New Algorithms and Lower Bounds for Sequential-Access Data Compression

This thesis concerns sequential-access data compression, i.e., by algorithms that read the input one or more times from beginning to end. In one chapter we consider adaptive prefix coding, for which we must read the input character by character, outputting each character's self-delimiting codeword before reading the next one. We show how to encode and decode each character in constant worst-case time while producing an encoding whose length is worst-case optimal. In another chapter we consider one-pass compression with memory bounded in terms of the alphabet size and context length, and prove a nearly tight tradeoff between the amount of memory we can use and the quality of the compression we can achieve. In a third chapter we consider compression in the read/write streams model, which allows us passes and memory both polylogarithmic in the size of the input. We first show how to achieve universal compression using only one pass over one stream. We then show that one stream is not sufficient for achieving good grammar-based compression. Finally, we show that two streams are necessary and sufficient for achieving entropy-only bounds.Comment: draft of PhD thesi

Echo State Networks for Proactive Caching in Cloud-Based Radio Access Networks with Mobile Users

In this paper, the problem of proactive caching is studied for cloud radio access networks (CRANs). In the studied model, the baseband units (BBUs) can predict the content request distribution and mobility pattern of each user, determine which content to cache at remote radio heads and BBUs. This problem is formulated as an optimization problem which jointly incorporates backhaul and fronthaul loads and content caching. To solve this problem, an algorithm that combines the machine learning framework of echo state networks with sublinear algorithms is proposed. Using echo state networks (ESNs), the BBUs can predict each user's content request distribution and mobility pattern while having only limited information on the network's and user's state. In order to predict each user's periodic mobility pattern with minimal complexity, the memory capacity of the corresponding ESN is derived for a periodic input. This memory capacity is shown to be able to record the maximum amount of user information for the proposed ESN model. Then, a sublinear algorithm is proposed to determine which content to cache while using limited content request distribution samples. Simulation results using real data from Youku and the Beijing University of Posts and Telecommunications show that the proposed approach yields significant gains, in terms of sum effective capacity, that reach up to 27.8% and 30.7%, respectively, compared to random caching with clustering and random caching without clustering algorithm.Comment: Accepted in the IEEE Transactions on Wireless Communication

Space and Time Efficient Parallel Graph Decomposition, Clustering, and Diameter Approximation

We develop a novel parallel decomposition strategy for unweighted, undirected graphs, based on growing disjoint connected clusters from batches of centers progressively selected from yet uncovered nodes. With respect to similar previous decompositions, our strategy exercises a tighter control on both the number of clusters and their maximum radius. We present two important applications of our parallel graph decomposition: (1) $k$-center clustering approximation; and (2) diameter approximation. In both cases, we obtain algorithms which feature a polylogarithmic approximation factor and are amenable to a distributed implementation that is geared for massive (long-diameter) graphs. The total space needed for the computation is linear in the problem size, and the parallel depth is substantially sublinear in the diameter for graphs with low doubling dimension. To the best of our knowledge, ours are the first parallel approximations for these problems which achieve sub-diameter parallel time, for a relevant class of graphs, using only linear space. Besides the theoretical guarantees, our algorithms allow for a very simple implementation on clustered architectures: we report on extensive experiments which demonstrate their effectiveness and efficiency on large graphs as compared to alternative known approaches.Comment: 14 page

Equivalence Classes and Conditional Hardness in Massively Parallel Computations

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model