Direct Sums for Parity Decision Trees

TL;DR

This paper establishes direct sum theorems for randomized parity decision trees, demonstrating that solving multiple instances scales linearly with the number of instances under certain conditions.

Contribution

It provides the first direct sum results in the randomized parity decision tree model, applicable when lower bounds use discrepancy or product distributions.

Findings

Direct sum theorem holds for bounds proved via discrepancy.

Applicable when lower bounds are relative to product distributions.

First such results in the randomized parity decision tree model.

Abstract

Direct sum theorems state that the cost of solving $k$ instances of a problem is at least $\Omega(k)$ times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.