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Báo cáo toán học: " Periodic Sorting Using Minimum Delay, Recursively Constructed"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Periodic Sorting Using Minimum Delay, Recursively Constructed . | Periodic Sorting Using Minimum Delay Recursively Constructed Merging Networks Edward A. Bender Center for Communications Research 4320 Westerra Court San Diego CA 92121 USA ed@ccrwest.org S. Gill Williamson Department of Computer Science and Engineering University of California San Diego La Jolla CA 92093-0114 USA gwilliamson@ucsd.edu Submitted December 6 1996 Submitted in revised form August 25 1997 Accepted December 9 1997 Abstract Let a and 3 be a partition of 1 . ng into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that whenever the input sequence x1 x2 . xn is such that the subsequence in the positions a and the subsequence in the positions 3 are each sorted the output sequence will be sorted. We study the class of recursively constructed merging networks and characterize those with delay log2 ng the best possible delay for all merging networks . When n is a power of 2 we show that at least 3n 2-1 of these nets are log-periodic sorters that is they sort any input sequence after log2 n passes through the net. Two of these have appeared previously in the literature. 1991 AMS Classification Number. Primary 68P10 THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1998 R5 2 1. Introduction This paper is divided into two main parts in two ways. First Sections 1-5 contain the concepts and results and Sections 6-10 contain the proofs. Second one part of the paper deals with merging and the other with sorting. The concepts mentioned in the present section will be made precise in the next section. In software terms a merging network is a program with no branching looping or arithmetic other than the replacement of a pair of values x y with c x y min x y max x y called a comparator. In hardware terms a merging network is a branch-free and feedback-free circuit whose only logic units are comparators. Given two interleaved sorted sequences a merging net sorts the entire sequence. A sorting net is like a