An Efficient Algorithm for Multi-indexed Recurrence Relations with Applications to Compression of Multidimensional Signals

Parallel solution of 1-indexed recurrence relations has received much attention, but no e ort has been made to design parallel algorithms for multiindexed recurrence relations. Multi-indexed relations arise in JPEG-standard DPCM compression of images, and thus merits attention. In this paper we design and analyze a parallel algorithm for solving q-indexed recurrence relations of arbitrary order and arbitrary number of indices q. The approach is dimension shifting, which involves reducing the number of indices to 1 while clustering the terms of the relation into vectors. This is accomplished by means of special vector operators that we de ne and study. Our parallel algorithm for doing the dimension shifting and for solving the resulting 1-indexed recurrence relation takes O(logN) time on meshes of partitionable buses or hypercubes, where N is the data input-output size.