Design of Efficient Regular Arrays for Matrix Multiplication by Two-Step Regularization

215 given k and n, one can choose any value of nz in the range from 2 to n to construct a multiway merging network. However, use of certain values of m results in less delay than others for the same value of n. One choice of the value of m is m = n. In this case, can achieve this time bound for the given value of n, are m = 2' with c < [log, n1. We have also assumed that there are n keys in each ascending sequence and m divides n for simplicity and clarity of exposition. In general, a multiway merging network can be constructed to merge k sorted lists of different lengths with the ith list having n, keys not necessarily equal to n. One way of doing this is to construct a k-way merger of size n first using keys with the value of positive infinity to fill the difference, and then remove these positive infinity keys together with the comparison-exchange elements associated with them, resulting in a simpler k-way merging network. Thus, there are no restrictions on the property of n,, k, and m. A sorter can be constructed from k-way mergers: the keys are combined k at a time to form ordered lists of length k; these lists are merged k at a time to form ordered lists of length k 2 , etc., until all keys are merged into one ordered list. To sort k p keys using the mergers requires kp-' mergers of size 1 followed by ICp-' mergers of size k followed by kp-3 mergers of size k2 followed by kp-4 mergers of size k 3 , etc., etc. The longest path will go through steps. V. CONCLUSION The multiway merge described in this paper merges k ascending sequences into an ascending sequence for any integer k. This differs from existing merging networks that merge only two ascending sequences into one. Furthermore, the k-way merge represents a complete generalization of the odd-even merge, when k = 2. In this case, it uses m small two-way mergers to merge two ascending sequences into one, where m is not restricted to 2. The odd-even merge is a special case of the k-way merge, when k = 2 and m = 2. The multiway merges does not reduce the number of comparators nor does it reduce the delay over the odd-even …