Space-Efficient Parallel Algorithms for Combinatorial Search Problems

We present space-efficient parallel strategies for two fundamental combinatorial search problems, namely, backtrack search and branch-and-bound, both involving the visit of an $n$-node tree of height $h$ under the assumption that a node can be accessed only through its father or its children. For both problems we propose efficient algorithms that run on a $p$-processor distributed-memory machine. For backtrack search, we give a deterministic algorithm running in $O(n/p+h\log p)$ time, and a Las Vegas algorithm requiring optimal $O(n/p+h)$ time, with high probability. Building on the backtrack search algorithm, we also derive a Las Vegas algorithm for branch-and-bound which runs in $O((n/p+h\log p \log n)h\log^2 n)$ time, with high probability. A remarkable feature of our algorithms is the use of only constant space per processor, which constitutes a significant improvement upon previous algorithms whose space requirements per processor depend on the (possibly huge) tree to be explored.Comment: Extended version of the paper in the Proc. of 38th International Symposium on Mathematical Foundations of Computer Science (MFCS

The input/output complexity of sorting and related problems

The input/output complexity of sorting and related problem

Average-case analysis of algorithms and data structures

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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

A simple optimal randomized algorithm for sorting on the PDM

The Parallel Disks Model (PDM) has been proposed to alleviate the I/O bottleneck that arises in the processing of massive data sets. Sorting has been extensively studied on the PDM model due to the fundamental nature of the problem. Several randomized algorithms are known for sorting. Most of the prior algorithms suffer from undue complications in memory layouts, implementation, or lack of tight analysis. In this paper we present a simple randomized algorithm that sorts in optimal time with high probablity and has all the desirable features for practical implementation