Depth Optimal Sorting Networks Resistant to k Passive Faults

• Authors: Marek Piotrów
• Venue: SIAM Journal on Computing
• Year: 2004

We study the problem of constructing a sorting network that is tolerant to faults and whose running time (i.e., depth) is as small as possible. We consider the scenario of worst-case comparator faults and follow the model of passive comparator failure proposed by Yao and Yao SIAM J. Comput., 14 (1985), pp. 120--128], in which a faulty comparator outputs its inputs directly without comparison. Our main result is the first construction of an N-input k-fault-tolerant sorting network with an asymptotically optimal depth $\theta$(log N + k). That improves over the result of Leighton and Ma [Proceedings of the 5th Annual ACM Symposium on Parallel Algorithms and Architectures, Velen, Germany, 1993, ACM, New York, pp. 30--41], whose network is of depth O(log N + klog\frac{log N}{log k})$.Actually, we present a fault-tolerant correction network that can be added after any N-input sorting network to correct its output in the presence of at most k faulty comparators. Since the depth of the network is O(log N + k) and the constants hidden behind the "O" notation are small, the construction can be of practical use.Developing the techniques necessary to show the main result, we construct a fault-tolerant network for the insertion problem. As a by-product, we get an N-input O(log N)-depth INSERT-network that is tolerant to random faults, thereby answering a question posed by Ma in his Ph. D. thesis [Fault-Tolerant Sorting Network, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1994].The results are based on a new notion of constant delay comparator networks, that is, networks in which each register is used (compared) only in a period of time of a constant length. Copies of such networks can be pipelined with only a constant increase in the total depth per copy.