Tight bounds on the size of fault-tolerant merging and sorting networks with destructive faults

We study networks that can sort n items even when a large number of the comparators in the network are faulty. We restrict attention to networks that consist of registers, comparators, and replicators. (Replicators are used to copy an item from one register to another, and they are assumed to be fault-free.) We consider the scenario of bot h random and worst-case comparator faults, and we follow the general model of destructive comparator failure proposed by Assaf and Upfal [2] in which the two outputs of a faulty comparator can fail independently of each other. In the case of random faults, Assaf and Upfal [2] showed how to construct a network with O(n log2 n) comparators that (with high probability) can sort n items even if a constant fraction of the comparators are faulty, Whether or not the bound on the number of comparators can be improved (to, say, O(n log n)) for sorting (or merging) has remained an interesting open question. We resolve this question in the paper by prov“This research is supported by AF05R Contract F49620-92-J-0125, DARPA Contract NOO014-91-J-1698, and DARPA Contract NOO01492-J-1799. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and ita date appear, and notice ia given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. ACM-SPAA’93-6193 /Velen,Germany. 01993 ACM 0-89791 -59921931000610030 . ..$1 .50 ing that any n-item sorting or merging network which can tolerate a constant fraction of random failures must have Q(n log2 n) comparators. In the case of worst-case faults, we show that $l(kn log n) comparators are necessary to construct a sorting or merging network that can tolerate up to k worst-case faults. We also show that this bound is tight for k = O(log n). The lower bound is particularly significant since it formally proves that the cost of being tolerant to worstcase failures is very high. Both the lower bound for random faults and the lower bound for worst-case faults are the first nontrivial lower bounds on the size of a fault-tolerant sorting or merging network.