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Toward a Universal Mapping Algorithm for Accessing Trees in Parallel Memory Systems
, 1998
"... We study the problem of mapping the N nodes of a complete tary tree on M memory modules so that they can be accessed in parallel by templates, i.e. distinct sets of nodes. Typical templates for accessing trees are subtrees, roottoleaf paths, or levels which will be referred to as elementary templ ..."
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Cited by 4 (3 self)
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We study the problem of mapping the N nodes of a complete tary tree on M memory modules so that they can be accessed in parallel by templates, i.e. distinct sets of nodes. Typical templates for accessing trees are subtrees, roottoleaf paths, or levels which will be referred to as elementary templates. In this paper, we first propose a new mapping algorithm for accessing both paths and subtrees of size M with an optimal number of conflicts (i.e., only one conflict) when the number of memory modules is limited to M . We also propose another mapping algorithm for a composite template, say V (as versatile), such that its size is not fixed and an instance of V is composed of any combination of c instances of elementary templates. The number of conflicts for accessing an Snode instance of template V is O ` S p M log M + c ' and the memory load is 1 + o(1) where load is defined as the ratio between the maximum and minimum number of data items mapped onto each memory module. 1. In...
Mappings for ConflictFree Access of Paths in Bidimensional Arrays, Circular Lists, and Complete Trees
 JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING
, 1999
"... Since the divergence between the processor speed and the memory access rate is progressively increasing, an efficient partition of the main memory into multibanks is useful to improve the overall system performance. The effectiveness of the multibank partition can be degraded by memory conflicts, ..."
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Cited by 4 (2 self)
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Since the divergence between the processor speed and the memory access rate is progressively increasing, an efficient partition of the main memory into multibanks is useful to improve the overall system performance. The effectiveness of the multibank partition can be degraded by memory conflicts, that occur when there are many references to the same memory bank while accessing the same memory pattern. Therefore, mapping schemes are needed to distribute data in such a way that data can be retrieved via regular patterns without conflicts. In this paper, the problem of conflictfree access of arbitrary paths in bidimensional arrays, circular lists and complete trees is considered for the first time and reduced to variants of graphcoloring problems. Balanced and fast mappings are proposed which require an optimal number of colors (i.e., memory banks). The solution for bidimensional arrays is based on a combinatorial object similar to a Latin Square. The functions that map an array node or a circular list node to a memory bank can be calculated in constant time. As for complete trees, the mapping of a tree node to a memory bank takes time that grows logarithmically with the number of nodes of the tree.
Principles of Highlevel Petri Nets
 Lectures on Petri nets I: Basic models, Advances in Petri nets, LNCS 1491
, 1998
"... Abstract. We study the problem of mapping treestructured data to an ensemble of parallel memory modules. We are given a “conflict tolerance” c, and we seek the smallest ensemble that will allow us to store any nvertex rooted binary tree with no more than c treevertices stored on the same module. O ..."
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Abstract. We study the problem of mapping treestructured data to an ensemble of parallel memory modules. We are given a “conflict tolerance” c, and we seek the smallest ensemble that will allow us to store any nvertex rooted binary tree with no more than c treevertices stored on the same module. Our attack on this problem abstracts it to a search for the smallest cperfect universal graph for complete binary trees. We construct such a graph which witnesses that only O � c (1−1/c) · 2 (n+1)/(c+1) � memory modules are needed to obtain the required bound on conflicts, and we prove that Ω � 2 (n+1)/(c+1) � memory modules are necessary. These bounds are tight to within constant factors when c is fixed—as it is with the motivating application. 1
Versatile Access to Parallel Memory Systems
 Proc. of 1998 Wkshp. on Distributed Data and Structures
"... In this paper, we present a survey of results about the problem of mapping the N items of a data structure on M memory modules so that items can be accessed in parallel by templates i.e. distinct sets of nodes. In particular, we present some results that allow to access several different templates a ..."
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Cited by 2 (2 self)
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In this paper, we present a survey of results about the problem of mapping the N items of a data structure on M memory modules so that items can be accessed in parallel by templates i.e. distinct sets of nodes. In particular, we present some results that allow to access several different templates at once, i.e. we focus on versatile mapping algorithms (for a comprehensive survey of other related results see [14] in this volume). In particular, we present some of the algorithms in literature for accessing arrays (by rows, columns, diagonals and subarrays) and trees (accessed by subtrees, roottoleaf paths, levels and compositions thereof). 1 Introduction In this paper we present a survey of results related to the problem of mapping a data structure to M memory modules when it is known how the access is required i.e. templates of memory access are known. The problem is to minimize the conflicts on the memory modules, i.e. simultaneous access to the same module to retrieve different dat...
BoundedCollision MemoryMapping Schemes for Data Structures with Applications to Parallel Memories
"... Abstract—Techniques are developed for mapping structured data to an ensemble of parallel memory modules in a way that limits the number of conflicts, i.e., simultaneous accesses by distinct processors to the same memory module. The techniques determine, for any given conflict tolerance c, the smalle ..."
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Abstract—Techniques are developed for mapping structured data to an ensemble of parallel memory modules in a way that limits the number of conflicts, i.e., simultaneous accesses by distinct processors to the same memory module. The techniques determine, for any given conflict tolerance c, the smallest ensemble that allows one to store any nnode data structure “of type X ” in such a way that no more than c nodes of a structure are stored on the same module. This goal is achieved by determining the smallest cperfect universal graphs for data structures “of type X. ” Such a graph is the smallest graph that contains a homomorphic image of each nnode structure “of type X, ” with each node of the image holding c nodes of the structure. In the current paper, “type X ” refers to rooted binary trees and three arraylike structures: chaotic arrays, ragged arrays, and rectangular arrays. For each of these families of data structures, the number of memory modules needed to achieve conflict tolerance c is determined to within constant factors. Index Terms—Parallel memory systems, data structures for parallel systems, boundedconflict parallel memory access, data mapping, parallel architectures, parallel systems, data structures, graph labeling. Ç
The CPU speed has been traditionally increasing at a much faster rat...
"... We study conflictfree data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if ..."
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We study conflictfree data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflictfree access to star templates, i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadthfirst search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the startemplate access problem on two specific host graphs – tori and hypercubes – that are also popular interconnection network topologies. The proposed conflictfree mappings on these graphs are fast, use an optimal or provably good