Pii: S0898-1221(02)00264-x

-When solving problems in such areas as image processing, computer vision, and computational geometry, within a parallel architecture, the mesh structure often stands out as a natural choice. In this paper, we analyze a scalable measurement of the average distance between two arbitrary but fixed processors in a network structure, which is useful in providing a more global characterization of its data transmission behavior. In particular, we provide a closed-form expression for the average distance between two processors in the case of a traditional mesh. For a more complex case, where a traditional mesh is augmented with some additional diagonal links, after providing a telling expression for the average distance between its two processors, we prove that this latter expression cannot be represented in a closed-form format, with respect to a fairly general class of "standard operations", namely, the class of the hypergeometric terms. Besides suggesting another global measurement of the communication behavior for general computer networks, and deriving concrete results for some "popular" mesh structures, this paper provides both positive and negative results regarding the derivation of closed-form expressions for combinatorial quantities, thus, is also theoretically interesting. We also believe that some of the general techniques used in this paper should be applicable elsewhere when closed-form expression needs to be derived, or its existence is in question. © 2002 Elsevier Science Ltd. All rights reserved. 1. I N T R O D U C T I O N W h e n so lv ing p r o b l e m s fall ing in to such areas as image process ing , c o m p u t e r v is ion a n d g raph ics , a n d c o m p u t a t i o n a l geomet ry , w i t h a paral le l a rch i t ec tu re , t h e m e s h s t r u c t u r e 1, i.e., a 2-D a r r ay 1In this paper, terms such as network structure, network, structure, and graph will be used interchangeably. The terms of processors and nodes will also be used interchangeably. The author wishes to express his gratitude to the Marion and Jasper Whiting Foundation and Plymouth State College for their generous financial support to allow the author to attend the ACM 2000 Symposium on Applied Computing, held in Lake Come, Italy, in March, 2000, where he reported some of the ideas contained in this paper. The author also wishes to express his gratitude to the DIMACS center at Rutgers University and Plymouth State College for their generous financial support to allow the author to attend the DCI2000 Workshop on Distance and Centrality Concepts in Graph, in July, 2000, where he discussed some of the ideas contained in this paper. The author particularly wants to express his gratitude to one of the participants for his helpful comments regarding effective paths in the case of augmented mesh structures. The author also wishes to express his gratitude to an anonymous reviewer for his/her comments on an earlier version of this paper. The author did not agree with many of the comments the reviewer made, but these comments definitely helped the author in revising this paper, which, in the author's opinion, has led to a better expression of some of the ideas contained within this paper. Some of the material contained in this paper appeared in [2]. 0898-1221/02/$ see front matter (~) 2002 Elsevier Science Ltd, All rights reserved. Typeset by .Aj~-TEX PII: S0898-1221(02)00264-X