Abstract
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 \(\varOmega (\sqrt{D(f)})\) for any \(k \ge 2\). 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 \(\tilde{O}(\sqrt{nk}+k)\) bits. Since All-Equality has deterministic complexity \(\varTheta (nk)\), 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 \(\tilde{\varTheta }(k \sqrt{n})\) for it.
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Notes
- 1.
We consider the number-in-hand model, where each player receives a private input, rather than the perhaps more familiar number-on-forehead model, where each player can see the input of all the other players but not its own.
- 2.
Another reasonable definition for randomized protocols is to take the maximum over all inputs of the expected total number of bits sent. For two players this is asymptotically equivalent to the definition above [13]. For \(k > 2\) players, the expectation may be smaller than the maximum by a factor of \(\log (k)\).
- 3.
The theorem in [14] gives a general construction for any distance up to 1 / 2; here we use distance 1 / 6.
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Fischer, O., Oshman, R., Zwick, U. (2016). Public vs. Private Randomness in Simultaneous Multi-party Communication Complexity. In: Suomela, J. (eds) Structural Information and Communication Complexity. SIROCCO 2016. Lecture Notes in Computer Science(), vol 9988. Springer, Cham. https://doi.org/10.1007/978-3-319-48314-6_5
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