发表机构
University of Maryland（马里兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出单调性度量，证明即使输入仅勉强单调，也能在真正次二次时间内计算 (min,+)-卷积，并给出匹配的细粒度归约。
AI 中文摘要
两个长度为 n 的序列 A 和 B 的 (min,+)-卷积是序列 C，其中 C[k] = min_{i+j=k} (A[i]+B[j])。对于有界输入，即条目为 {0,...,O(n)} 中的整数，先前的工作在输入单调时以真正次二次时间计算；Chi、Duan、Xie 和 Zhang (STOC 2022) 的算法期望运行时间为 O~(n^{1.5})。我们引入一个单调性度量，范围从 0（单调）到 1/2（完全非单调）：一个序列具有单调性 alpha，如果它可以被划分为 O(n^alpha) 个单调子序列，并且根据 Erdos-Szekeres 定理，每个序列的单调性至多为 1/2。我们证明，即使只有一个输入是勉强单调的，即具有单调性 1/2 - Omega(1)，真正次二次时间也是可实现的：如果 A 具有单调性 alpha，我们对于每个有界 B 计算卷积的期望时间为 O~(n^{5/3+2alpha/3})。如果 B 也具有单调性 beta，期望时间改进为 O~(n^{(3+alpha+beta)/2})，这与 alpha = beta = 0 时的单调情况匹配；该算法还允许任意放置的无限条目。我们用细粒度归约补充这些算法。有界 (min,+)-卷积归约到长度为 N = O(n^{1.5}) 的有界单调 (min,+)-卷积，因此单调输入的 O(N^{4/3-eps})-时间算法将给出有界输入的 O(n^{2-3eps/2})-时间算法。类似地，条目以 n 为界归约到长度为 N = Theta(n^{2/(1+x)}) 的序列上以 N^x 为界的条目。我们还表明，如果只有 A 的条目在 {0,...,M} 中，我们可以在 O~(n(M+1)) 时间内计算卷积，如果 A 也可能包含 +infinity，则在 O~(n^{1.5} sqrt(M)) 时间内计算。
英文摘要
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.
