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16
Direct BulkSynchronous Parallel Algorithms
 JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING
, 1992
"... We describe a methodology for constructing parallel algorithms that are transportable among parallel computers having different numbers of processors, different bandwidths of interprocessor communication and different periodicity of global synchronisation. We do this for the bulksynchronous paralle ..."
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Cited by 171 (27 self)
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We describe a methodology for constructing parallel algorithms that are transportable among parallel computers having different numbers of processors, different bandwidths of interprocessor communication and different periodicity of global synchronisation. We do this for the bulksynchronous parallel (BSP) model, which abstracts the characteristics of a parallel machine into three numerical parameters p, g, and L, corresponding to processors, bandwidth, and periodicity respectively. The model differentiates memory that is local to a processor from that which is not, but, for the sake of universality, does not differentiate network proximity. The advantages of this model in supporting shared memory or PRAM style programming have been treated elsewhere. Here we emphasise the viability of an alternative direct style of programming where, for the sake of efficiency the programmer retains control of memory allocation. We show that optimality to within a multiplicative factor close to one ca...
Simulating uniform hashing in constant time and optimal space
, 2003
"... Many algorithms and data structures employing hashing have been analyzed under the uniform hashing assumption, i.e., the assumption that hash functions behave like truly random functions. Starting with the discovery of universal hash functions, many researchers have studied to what extent this theo ..."
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Cited by 11 (3 self)
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Many algorithms and data structures employing hashing have been analyzed under the uniform hashing assumption, i.e., the assumption that hash functions behave like truly random functions. Starting with the discovery of universal hash functions, many researchers have studied to what extent this theoretical ideal can be realized by hash functions that do not take up too much space and can be evaluated quickly. In this paper we present an almost ideal solution to this problem: A hash function h: U → V that, on any set of n inputs, behaves like a truly random function with high probability, can be evaluated in constant time on a RAM, and can be stored in (1 + ɛ)n lg V  + O(n + lg lg U) bits. Here ɛ can be chosen to be any positive constant, so this essentially matches the entropy lower bound. For many hashing schemes this is the first hash function that makes their uniform hashing analysis come true, with high probability, without incurring overhead in time or space.
Hash and displace: Efficient evaluation of minimal perfect hash functions
 In Workshop on Algorithms and Data Structures
, 1999
"... A new way of constructing (minimal) perfect hash functions is described. The technique considerably reduces the overhead associated with resolving buckets in twolevel hashing schemes. Evaluating a hash function requires just one multiplication and a few additions apart from primitive bit operations ..."
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Cited by 9 (1 self)
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A new way of constructing (minimal) perfect hash functions is described. The technique considerably reduces the overhead associated with resolving buckets in twolevel hashing schemes. Evaluating a hash function requires just one multiplication and a few additions apart from primitive bit operations. The number of accesses to memory is two, one of which is to a fixed location. This improves the probe performance of previous minimal perfect hashing schemes, and is shown to be optimal. The hash function description (“program”) for a set of size n occupies O(n) words, and can be constructed in expected O(n) time. 1
Topics in Parallel and Distributed Computation
"... With advances in communication technology, the introduction of multipleinstruction multipledata parallel computers and the increasing interest in neural networks, the fields of parallel and distributed computation have received increasing attention in recent years. We study in this work the bulksy ..."
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Cited by 5 (2 self)
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With advances in communication technology, the introduction of multipleinstruction multipledata parallel computers and the increasing interest in neural networks, the fields of parallel and distributed computation have received increasing attention in recent years. We study in this work the bulksynchronous parallel model, that attempts to bridge the software and hardware worlds with respect to parallel computing. It offers a high level abstraction of the hardware with the purpose of allowing parallel programs to run efficiently on diverse hardware platforms. We examine direct algorithms on this model and also give simulations of other models of parallel computation on this one as well as on models that bypass it. While the term parallel computation refers to the execution of a single or a set of closely coupled tasks by a set of processors, the term distributed computation refers to more loosely coupled or uncoupled tasks being executed at different locations. In a distributed computing environment it is sometimes necessary that one computer send to the remaining ones various pieces of information. The term broadcasting is used to describe the dissemination of information from one computer to the others in such an environment. We examine various classes of random graphs with respect to broadcasting and establish results related to the minimum time required to perform broadcasting from any vertex of such graphs.
6.897: Advanced data structures (Spring 2005), Lecture 3, February 8
, 2005
"... Recall from last lecture that we are looking at the documentretrieval problem. The problem can be stated as follows: Given a set of texts T1, T2,..., Tk and a pattern P, determine the distinct texts in which the patterns occurs. In particular, we are allowed to preprocess the texts in order to be a ..."
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Cited by 3 (0 self)
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Recall from last lecture that we are looking at the documentretrieval problem. The problem can be stated as follows: Given a set of texts T1, T2,..., Tk and a pattern P, determine the distinct texts in which the patterns occurs. In particular, we are allowed to preprocess the texts in order to be able to answer the query faster. Our preprocessing choice was the use of a single suffix tree, in which all the suffixes of all the texts appear, each suffix ending with a distinct symbol that determines the text in which the suffix appears. In order to answer the query we reduced the problem to rangemin queries, which in turn was reduced to the least common ancestor (LCA) problem on the cartesian tree of an array of numbers. The cartesian tree is constructed recursively by setting its root to be the minimum element of the array and recursively constructing its two subtrees using the left and right partitions of the array. The rangemin query of an interval [i, j] is then equivalent to finding the LCA of the two nodes of the cartesian tree that correspond to i and j. In this lecture we continue to see how we can solve the LCA problem on any static tree. This will involve a reduction of the LCA problem back to the rangemin query problem (!) and then a
Lower Bound Techniques for Data Structures
, 2008
"... We describe new techniques for proving lower bounds on datastructure problems, with the following broad consequences:
â¢ the first Î©(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989;
â¢ for static data structures, the first separation between linea ..."
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Cited by 1 (0 self)
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We describe new techniques for proving lower bounds on datastructure problems, with the following broad consequences:
â¢ the first Î©(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989;
â¢ for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show Î©(lg n/ lg lg n) bounds when the space is O(n Â· polylog n).
Using these techniques, we analyze a variety of central datastructure problems, and obtain improved lower bounds for the following:
â¢ the partialsums problem (a fundamental application of augmented binary search trees);
â¢ the predecessor problem (which is equivalent to IP lookup in Internet routers);
â¢ dynamic trees and dynamic connectivity;
â¢ orthogonal range stabbing;
â¢ orthogonal range counting, and orthogonal range reporting;
â¢ the partial match problem (searching with wildcards);
â¢ (1 + Îµ)approximate near neighbor on the hypercube;
â¢ approximate nearest neighbor in the lâ metric.
Our new techniques lead to surprisingly nontechnical proofs. For several problems, we obtain simpler proofs for bounds that were already known.
A New Tradeoff for Deterministic Dictionaries
, 2000
"... . We consider dictionaries over the universe U = f0; 1g w on a unitcost RAM with word size w and a standard instruction set. We present a linear space deterministic dictionary with membership queries in time (log log n) O(1) and updates in time (log n) O(1) , where n is the size of the se ..."
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Cited by 1 (0 self)
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. We consider dictionaries over the universe U = f0; 1g w on a unitcost RAM with word size w and a standard instruction set. We present a linear space deterministic dictionary with membership queries in time (log log n) O(1) and updates in time (log n) O(1) , where n is the size of the set stored. This is the rst such data structure to simultaneously achieve query time (log n) o(1) and update time O(2 (log n) c ) for a constant c < 1. 1 Introduction Among the most fundamental data structures is the dictionary. A dictionary stores a subset S of a universe U , oering membership queries of the form \x 2 S?". The result of a membership query is either 'no' or a piece of satellite data associated with x. Updates of the set are supported via insertion and deletion of single elements. Several performance measures are of interest for dictionaries: The amount of space used, the time needed to answer queries, and the time needed to perform updates. The most ecient dictionar...
OneProbe Search
, 2002
"... We consider dictionaries that perform lookups by probing a single word of memory, knowing only the size of the data structure. We describe a randomized dictionary where a lookup returns the correct answer with probability 1 − ɛ, and otherwise returns “don’t know”. The lookup procedure uses an expa ..."
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Cited by 1 (0 self)
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We consider dictionaries that perform lookups by probing a single word of memory, knowing only the size of the data structure. We describe a randomized dictionary where a lookup returns the correct answer with probability 1 − ɛ, and otherwise returns “don’t know”. The lookup procedure uses an expander graph to select the memory location to probe. Recent explicit expander constructions are shown to yield space usage far smaller than what would be required using a deterministic lookup procedure. Our data structure supports efficient deterministic updates, exhibiting new probabilistic guarantees on dictionary running time.
3.5Way Cuckoo Hashing for the Price of 2andaBit
"... Abstract. The study of hashing is closely related to the analysis of balls and bins; items are hashed to memory locations much as balls are thrown into bins. In particular, Azar et. al. [2] considered putting each ball in the lessfull of two random bins. This lowers the probability that a bin excee ..."
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Abstract. The study of hashing is closely related to the analysis of balls and bins; items are hashed to memory locations much as balls are thrown into bins. In particular, Azar et. al. [2] considered putting each ball in the lessfull of two random bins. This lowers the probability that a bin exceeds a certain load from exponentially small to doubly exponential, giving maximum load log log n + O(1) with high probability. Cuckoo hashing [20] draws on this idea. Each item is hashed to two buckets of capacity k. If both are full, then the insertion procedure moves previouslyinserted items to their alternate buckets to make space for the new item. In a natural implementation, the buckets are represented by partitioning a fixed array of memory into nonoverlapping blocks of size k. An item is hashed to two such blocks and may be stored at any location within either one. We analyze a simple twist in which each item is hashed to two arbitrary sizek memory blocks. (So consecutive blocks are no longer disjoint, but rather overlap by k − 1 locations.) This twist increases the space utilization from 1 − (2/e + o(1)) k to 1 − (1/e + o(1)) 1.59k in general. For k = 2, the new method improves utilization from 89.7 % to 96.5%, yet lookups access only two items at each of two random locations. This result is surprising because the opposite happens in the noncuckoo setting; if items are not moved during later insertions, then shifting from nonoverlapping to overlapping blocks makes the distribution less uniform. 1
On Dynamic Range Reporting in One Dimension Christian Worm Mortensen ∗ IT U. Copenhagen
, 2005
"... We consider the problem of maintaining a dynamic set of integers and answering queries of the form: report a point (equivalently, all points) in a given interval. Range searching is a natural and fundamental variant of integer search, and can be solved using predecessor search. However, for a RAM wi ..."
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We consider the problem of maintaining a dynamic set of integers and answering queries of the form: report a point (equivalently, all points) in a given interval. Range searching is a natural and fundamental variant of integer search, and can be solved using predecessor search. However, for a RAM with wbit words, we show how to perform updates in O(lg w) time and answer queries in O(lg lg w) time. The update time is identical to the van Emde Boas structure, but the query time is exponentially faster. Existing lower bounds show that achieving our query time for predecessor search requires doublyexponentially slower updates. We present some arguments supporting the conjecture that our solution is optimal. Our solution is based on a new and interesting recursion idea which is “more extreme” that the van Emde Boas recursion. Whereas van Emde Boas uses a simple recursion (repeated halving) on each path in a trie, we use a nontrivial, van Emde Boaslike recursion on every such path. Despite this, our algorithm is quite clean when seen from the right angle. To achieve linear space for our data structure, we solve a problem which is of independent interest. We develop the first scheme for dynamic perfect hashing requiring sublinear space. This gives a dynamic Bloomier filter (an approximate storage scheme for sparse vectors) which uses low space. We strengthen previous lower bounds to show that these results are optimal. 1