D-dimensional Range Search on Multicomputers Ecole Normale Supérieure De Lyon D-dimensional Range Search on Multicomputers D-dimensional Range Search on Multicomputers

Given a set L of n points in the d-dimensional Cartesian space E d , and a query specifying a domain q in E d , the Range Search problem consists in identifying the subset R(q) of the points of L contained in q. The Range Tree data-structure represents a particularly good balance between storage space and search time. The structure requires O(n log d?1 n) space and construction time, but supports an O(log d n) time search algorithm. In this paper, we describe a set of eecient scalable algorithms for the construction and manipulation of a distributed analog to the sequential range tree data structure. We then show how to perform O(n) independent range searches on a distributed range tree, in parallel. These parallel construction and search algorithms are both optimal in the Coarse Grained Multicomputer model (also referred to as the weak-CREW BSP model), in the sense that their running times are the sequential time divided by the number of processors , plus a constant number of parallel communication rounds (i.e., h-relations in BSP context). R esum e Prenons un ensemble de n points dans un espace cart esien E d , ainsi qu'une requ^ ete q d eenissant un domaine de E d. On appelle Range Search le probl eme consistant a identiier le sous-ensemble r(q) de points de L contenu dans q. Le Range Tree est une structure de donn ees oorant un excellent compromis entre la taille m emoire n ecessaire a son stockage et le temps de r eponse au probl eme. Il n ecessite O(n log d?1 n) emplacements m emoire et permet de traiter une requ^ ete en O(log d n) unit es de temps. Dans cet article, nous pr esentons des algorithmes eecaces et ex-tensibles pour construire et manipuler une version distribu ee du Range Tree comparable a la version s equentielle. Nous montrons comment r epondre a O(n) requ^ etes ind ependantes en parall ele. Les algorithmes de construction du Range Tree distribu e et de r eponses aux requ^ etes sont optimaux dans le mod ele de machine a gros grains (aussi appel e mod ele weak CREW-BSP), optimal signiiant que le temps d'ex ecution en parall ele est egal au temps d'ex ecution en s equentiel divis e par le nombre de processeurs, auquel on doit ajouter le temps n ecessaire a un nombre constant d' etapes de …