We study the problem of mapping the N nodes of a data structure on M memory modules so that they can be accessed in parallel by templates i.e. distinct sets of nodes. In literature several algorithms are available for arrays (accessed by rows, columns, diagonals and subarrays) and trees (accessed by subtrees, root-to-leaf paths, levels, etc.). Although some mapping algorithms for arrays allow conflict-free access to several templates at once (for example rows and columns), no mapping algorithm is known for efficiently accessing subtree, path and level templates in complete binary trees. In our paper, we, first, prove that any mapping algorithm that is conflict-free for tree/level template has \Omega\Gamma M= log M ) conflicts when access is done according to path template and vice versa. Therefore, no mapping algorithm can be found that is conflict-free on both path and tree (or path and level) templates. Our main result is an algorithm for mapping complete binary trees wi...

We study the problem of mapping the N nodes of a complete t-ary 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, root-to-leaf 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 S-node 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...

Abstract. We study the problem of mapping tree-structured 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 tree-vertices stored on the same module. Our attack on this problem abstracts it to a search for the smallest c-perfect 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

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, root-to-leaf 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...

We study the problem of mapping the N nodes of a data structure on M memory modules so that they can be accessed in parallel by templates i.e. distinct sets of nodes. In literature several algorithms are available for arrays (accessed by rows, columns, diagonals and subarrays) and trees (accessed by subtrees, root-to-leaf paths, etc.). Although some mapping algorithms for arrays allow conflict-free access to several templates at once (for example rows and columns), no mapping algorithm is known for efficiently accessing both subtree and rootto -leaf path templates in complete binary trees. We prove that any mapping algorithm that is conflict-free for one of these two templates has\Omega\Gamma M= log M ) conflicts on the other. Therefore, no mapping algorithm can be found that is conflict-free on both templates. We give an algorithm for mapping complete binary trees with N = 2 M \Gamma 1 nodes on M memory modules in such a way that: ffl the number of conflicts for accessing a subtre...