Direct Sums for Parity Decision Trees

2024-12-09

Electron. Colloquium Comput. Complex.

AI Summary

This paper establishes a direct-sum theorem for randomized parity decision trees: whether solving $k$ independent instances requires $Omega(k)$ times the complexity of solving a single instance. Addressing the long-standing absence of direct-sum lower bounds in this model, the work provides the first general direct-sum theorem, unifying proofs under two mainstream lower-bound frameworks—sensitivity-based discrepancy methods and product-distribution techniques—and confirming $Omega(k)$-factor complexity blowup. Technically, the proof integrates analysis of randomized decision tree complexity, distribution-sensitive sensitivity characterizations, and error control under product distributions. The result fills a fundamental gap in parity decision tree theory and extends the applicability of direct-sum principles beyond communication complexity to finer-grained algebraic query models. By doing so, it introduces a new paradigm for multi-instance lower-bound analysis in query complexity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Proving direct sum theorems for parity decision trees
• Establishing randomized parity decision tree model results
• Identifying conditions for lower bound validity

Innovation

• Proves direct sum in randomized parity trees
• Uses discrepancy method for lower bounds
• Applies to product distribution lower bounds