Nearly optimal algorithms for broadcast on d-dimensional all-port and wormhole-routed torus

We present nearly optimal algorithms for broadcast on a d-dimensional n×n x...x n torus that supports all-port communication and wormhole routing. Let T(n) denote the number of communication steps performed to broadcast a message. We present algorithms that achieve the following performance: (i) T(n)=d[log_2d+1n]+1 when n is odd, and (ii) T(n)=d[log_2d+1(n-1)]+[d/2]+1 when n is even. The lower bound is [dlog_2d+1n]. Our algorithm is the first one that works for arbitrary n, and is optimal up to the constant term. Previous algorithms with comparable performance only work for the case that either d is 2 or 3, or n is a power of 2d+1. For special cases such as d=2 or 3, we can improve the algorithm so that T(n)=d[log_2d+1n] for any n