Stereoscopic families of permutations, and their applications

A stereoscopic family of permutations maps an mdimensional mesh into several 1-dimensional lines, in a way that jointly preserves distance information. Specifically, consider any two points and denote their distance on the m-dimensional mesh by d. Then the distance between their images, on the line on which these images are closest together, is O(dm). We initiate a systematic study of stereoscopic families of permutations. We show a construction of these families that involves the use of m + 1 images. We also show that under some additional restrictions (namely, adjacent points on the image lines originate at points which are not too far away on the mesh), three images are necessary in order to construct such a family for the 2-dimensional mesh. We present two applications for stereoscopic families of permutations. One application is an algorithm for routing on the mesh that guarantees delivery of each packet within a number of steps that depends upon the distance between this packet’s source and destination, but is independent of the size of the mesh. Our algorithm is exceptionally simple, involves no queues, and can be used in dynamic settings in which packets are continuously generated. Another application is an extension of the construction of nonexpansive hash functions of Linial and Sasson (STOC 96) from the case of one dimensional metrics to arbitrary dimensions.