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Determinism vs. Nondeterminism in Multiparty Communication Complexity

Danny Dolev, Tomás Feder

1992
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SIAM journal on computing (Print)
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A given Boolean function has its input distributed among many parties . The aim is to determine which parties to talk to and what information to exchange in order to evaluate the function while minimizing the total communication . This paper shows that it is possible to evaluate the Boolean function deterministically with only a polynomial increase in communication and number of parties accessed with respect to the information lower bound given by the nondeterministic communication complexity
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... the function . Key words . communication complexity, multiparty communication 1 . Introduction . Our model of multiparty communication complexity is motivated by two basic earlier models . The two-party communication model assumes that each of two processors has a part of the input, and the aim is to compute a function on the input minimizing the amount of communication . In the decision tree model, the input is distributed among many memory locations, and the aim is to compute a function on the input while minimizing the number of memory locations examined . Our multiparty communication model extends these two basic models by assuming that the input is distributed among many processors ; here the goal is to minimize both communication and number of processors accessed . Two-party communication has been extensively studied . The main issues studied were the relative power of determinism, nondeterminism, and randomization . Yao [19] introduced the tool of minimum fooling set (or crossing sequence) as a measure for the amount of information that needs to be exchanged for a given input partitioned among the two parties . The same technique was widely used in [2], [7], [9], [11], [13] . The decision tree model has been studied in several contexts [3], [10], [12], [15], [16], [17] . An area that inspired research in this direction is the study of graph properties (see [14] , for example) . The main focus in these studies is how to minimize the fraction of the input that must be examined in order to verify a given property . Here again we are interested in the relative power of determinism, nondeterminism, and randomization . The basic issue is how to decide what input locations to examine . Similar reduction ideas appear in the proof of Theorem 1 in [1] . In the multiparty communication model, when a large amount of information is distributed among a large number of processors, it is crucial to decide both which processors to communicate with and what information to exchange . We can neither talk to all parties as in the two-party model, nor obtain all the information known to each party as in the decision tree model. A natural measure for the least amount of information required is the information that a nondeterministic algorithm needs to exchange in order to decide the value of the function . In this paper we show that when computing a Boolean function, this information can be obtained deterministically with limited overhead . More precisely, we prove that the deterministic and the nondeterministic communication complexity of multiparty Boolean function evaluation are polynomially related. Tight bounds relate the deterministic and the nondeterministic communication complexity in the two-party model . Let C1 be the nondeterministic communication complex-*

doi:10.1137/0221052
fatcat:ya772efzzrbjtehoiamm5qzvgu