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CONTENTION RESOLUTION IN HASHING BASED SHARED MEMORY SIMULATIONS
, 2000
"... In this paper we study the problem of simulating shared memory on the distributed memory machine (DMM). Our approach uses multiple copies of shared memory cells, distributed among the memory modules of the DMM via universal hashing. The main aim is to design strategies that resolve contention at th ..."
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Cited by 9 (3 self)
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In this paper we study the problem of simulating shared memory on the distributed memory machine (DMM). Our approach uses multiple copies of shared memory cells, distributed among the memory modules of the DMM via universal hashing. The main aim is to design strategies that resolve contention at the memory modules. Extending results and methods from random graphs and very fast randomized algorithms, we present new simulation techniques that enable us to improve the previously best results exponentially. In particular, we show that an nprocessor CRCW PRAM can be simulated by an nprocessor DMM with delay O(log log log n log ∗ n), with high probability. Next we describe a general technique that can be used to turn these simulations into timeprocessor optimal ones, in the case of EREW PRAMs to be simulated. We obtain a timeprocessor optimal simulation of an (n log log log n log ∗ n)processor EREW PRAM on an nprocessor DMM with delay O(log log log n log ∗ n), with high probability. When an (n log log log n log ∗ n)processor CRCW PRAM is simulated, the delay is only by a log ∗ n factor larger. We further demonstrate that the simulations presented can not be significantly improved using our techniques. We show an Ω(log log log n / log log log log n) lower bound on the expected delay for a class of PRAM simulations, called topological simulations, that covers all previously known simulations as well as the simulations presented in the paper.
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
Lineartime algorithms to color topological graphs
, 2005
"... We describe a lineartime algorithm for 4coloring planar graphs. We indeed give an O(V + E + χ  + 1)time algorithm to Ccolor Vvertex Eedge graphs embeddable on a 2manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) ..."
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We describe a lineartime algorithm for 4coloring planar graphs. We indeed give an O(V + E + χ  + 1)time algorithm to Ccolor Vvertex Eedge graphs embeddable on a 2manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) time, to find the exact chromatic number of a maximal planar graph (one with E = 3V − 6) and a coloring achieving it. Finally, there is a lineartime algorithm to 5color a graph embedded on any fixed surface M except that an Mdependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultraparallelizable deterministic computational complexity class “NC. ” We also have more practical planar 4coloring algorithms that can run on pointer machines in O(V log V) randomized time and O(V) space, and a very simple deterministic O(V)time coloring algorithm for planar graphs which conjecturally uses 4 colors.
COMBINATORICA Bolyai Society – SpringerVerlag COMBINATORICA 18 (1) (1998) 121–132 NONEXPANSIVE HASHING
, 1996
"... In a nonexpansive hashing scheme, similar inputs are stored in memory locations which are close. We develop a nonexpansive hashing scheme wherein any set of size O(R 1−ε) from a large universe may be stored in a memory of size R (any ε>0, and R>R 0(ɛ)), and where retrieval takes O(1) operati ..."
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In a nonexpansive hashing scheme, similar inputs are stored in memory locations which are close. We develop a nonexpansive hashing scheme wherein any set of size O(R 1−ε) from a large universe may be stored in a memory of size R (any ε>0, and R>R 0(ɛ)), and where retrieval takes O(1) operations. We explain how to use nonexpansive hashing schemes for efficient storage and retrieval of noisy data. A dynamic version of this hashing scheme is presented as well. 1.
A reliable randomized algorithm for the . . .
, 1997
"... The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest p ..."
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The following two computational problems are studied: Duplicate grouping: Assume that n items are given, each of which is labeled by an integer key from the set 0,..., U � 1 4. Store the items in an array of size n such that items with the same key occupy a contiguous segment of the array. Closest pair: Assume that a multiset of n points in the ddimensional Euclidean space is given, where d � 1 is a fixed integer. Each point is represented as a dtuple of integers in the range 0,..., U � 14 Ž or of arbitrary real numbers.. Find a closest pair, i.e., a pair of points whose distance is minimal over all such pairs.