On total communication complexity of collapsing protocols for pointer jumping problem

TL;DR

This paper proves that collapsing protocols for the multiparty pointer jumping problem have a total communication complexity close to the trivial protocol, establishing a tight lower bound and confirming Liang's conjecture.

Contribution

It establishes a tight lower bound of at least n-2 on total communication complexity for collapsing protocols in MPJ_k^n, confirming Liang's conjecture.

Findings

Total complexity is at least n-2 bits.

Confirms that no collapsing protocol can outperform the trivial total complexity.

Closes the gap between known lower and upper bounds for total complexity.

Abstract

This paper focuses on bounding the total communication complexity of collapsing protocols for multiparty pointer jumping problem ($MPJ_k^n$). Brody and Chakrabati in \cite{bc08} proved that in such setting one of the players must communicate at least $n - 0.5\log{n}$ bits. Liang in \cite{liang} has shown protocol matching this lower bound on maximum complexity. His protocol, however, was behaving worse than the trivial one in terms of total complexity (number of bits sent by all players). He conjectured that achieving total complexity better then the trivial one is impossible. In this paper we prove this conjecture. Namely, we show that for a collapsing protocol for $MPJ_k^n$, the total communication complexity is at least $n-2$ which closes the gap between lower and upper bound for total complexity of $MPJ_k^n$ in collapsing setting.