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arXiv · 2610.10503

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

MohammadTaghi Hajiaghayi·Danny Mittal·Saeed Seddighin
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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.

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MohammadTaghi Hajiaghayi, Danny Mittal, Saeed Seddighin. 2026-10-07. Barely Monotone (min,+)-Convolution in Truly Subquadratic Time. https://arxiv.org/abs/2610.10503

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