Exponential Separation of Information and Communication for Boolean Functions

We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity ≤ O(k), and distributional communication complexity ≥ 2 k . This shows that a communication protocol for a partial boolean function cannot always be compressed to its internal information. By a result of Braverman [Bra12], our gap is the largest possible. By a result of Braverman and Rao [BR11], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity of boolean functions cannot hold, answering a long standing open problem. Our techniques build on [GKR14], that proved a similar result for relations with very long outputs (double exponentially long in k). In addition to the stronger result, the current work gives a simpler proof, benefiting from the short output length of boolean functions.

Authors

• Anat Ganor
• Gillat Kol
• Ran Raz

Keywords

• information complexity
• communication complexity
• amortized communication complexity
• direct sum
• communication compression