Public vs. private randomness in simultaneous multi-party communication complexity

Orr Fischer*, Rotem Oshman, Uri Zwick

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


In simultaneous number-in-hand multi-party communication protocols, we have k players, who each receive a private input, and wish to compute a joint function of their inputs. The players simultaneously each send a single message to a referee, who then outputs the value of the function. The cost of the protocol is the total number of bits sent to the referee. For two players, it is known that giving the players a public (shared) random string is much more useful than private randomness: public-coin protocols can be unboundedly better than deterministic protocols, while private-coin protocols can only give a quadratic improvement on deterministic protocols. We extend the two-player gap to multiple players, and show that the private-coin communication complexity of a k-player function f is at least Ω(√D(f)) for any k ≥2. Perhaps surprisingly, this bound is tight: although one might expect the gap between private-coin and deterministic protocols to grow with the number of players, we show that the All-Equality function, where each player receives n bits of input and the players must determine if their inputs are all the same, can be solved by a private-coin protocol with Õ (√ nk + k) bits. Since All-Equality has deterministic complexity Θ (nk), this shows that sometimes the gap scales only as the square root of the number of players, and consequently the number of bits each player needs to send actually decreases as the number of players increases. We also consider the Exists-Equality function, where we ask whether there is a pair of players that received the same input, and prove a nearly-tight bound of Θ (k √ n) for it.

Original languageEnglish
Title of host publicationStructural Information and Communication Complexity - 23rd International Colloquium, SIROCCO 2016, Revised Selected Papers
EditorsJukka Suomela
PublisherSpringer Verlag
Number of pages15
ISBN (Print)9783319483139
StatePublished - 2016
Event23rd International Colloquium on Structural Information and Communication Complexity, SIROCCO 2016 - Helsinki, Finland
Duration: 19 Jul 201621 Jul 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9988 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference23rd International Colloquium on Structural Information and Communication Complexity, SIROCCO 2016


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