Barely Monotone (min,+)-Convolution in Truly Subquadratic Time

2026-10-07

AI Summary

This study addresses the super-quadratic computational bottleneck of bounded (min,+)-convolution when inputs exhibit only weak monotonicity. To overcome the limitations of requiring full monotonicity, this work proposes a novel monotonicity measure and leverages the Erdős-Szekeres theorem for sequence partitioning, integrating dynamic programming with refined complexity reduction techniques to design efficient algorithms. The proposed approach achieves subquadratic complexities of $\tilde{O}(n^{5/3+2\alpha/3})$ for single-input weak monotonicity and $\tilde{O}(n^{(3+\alpha+\beta)/2})$ for dual-input scenarios, demonstrating that weak monotonicity suffices to break the quadratic barrier. Furthermore, the algorithm accommodates inputs containing infinity values, providing a solid theoretical foundation for related optimization problems.

Abstract

The (min,+)-convolution of two sequences A and B of length n is the sequence C with C[k] = min_{i+j=k} (A[i]+B[j]). For bounded inputs, whose entries are integers in {0,...,O(n)}, prior work computes it in truly subquadratic time when the inputs are monotone; the algorithm of Chi, Duan, Xie, and Zhang (STOC 2022) takes expected O~(n^{1.5}) time. We introduce a monotonicity measure ranging from 0 (monotone) to 1/2 (entirely non-monotone): a sequence has monotonicity alpha if it can be partitioned into O(n^alpha) monotone subsequences, and by the Erdos-Szekeres theorem every sequence has monotonicity at most 1/2. We show that truly subquadratic time is achievable even when just one input is barely monotone, that is, has monotonicity 1/2 - Omega(1): if A has monotonicity alpha, we compute the convolution in expected time O~(n^{5/3+2alpha/3}) for every bounded B. If B has monotonicity beta as well, the expected time improves to O~(n^{(3+alpha+beta)/2}), which matches the monotone case for alpha = beta = 0; this algorithm also allows infinite entries placed arbitrarily. We complement these algorithms with fine-grained reductions. Bounded (min,+)-convolution reduces to bounded monotone (min,+)-convolution of length N = O(n^{1.5}), so an O(N^{4/3-eps})-time algorithm for monotone inputs would give an O(n^{2-3eps/2})-time algorithm for bounded inputs. Similarly, entries bounded by n reduce to entries bounded by N^x on sequences of length N = Theta(n^{2/(1+x)}). We also show that if only A has entries in {0,...,M}, we can compute the convolution in O~(n(M+1)) time, and in O~(n^{1.5} sqrt(M)) time if A may also contain +infinity.

Problem

• (min,+)-convolution
• subquadratic time
• monotonicity
• bounded inputs
• fine-grained reductions

Innovation

• (min,+)-convolution
• subquadratic time
• monotonicity measure
• fine-grained reductions
• bounded inputs