Streaming Algorithms for Monotonicity Testing

2026-08-07

AI Summary

This work addresses the problem of efficiently estimating the distance to monotonicity of a Boolean function over a directed acyclic graph (DAG) under an arbitrary edge arrival order in the streaming model, using sublinear space. The authors reduce this task to estimating the size of a maximum matching in the transitive closure of the DAG and introduce a novel streaming algorithm that integrates sublinear-time maximum matching estimation with enhanced vertex and subset query mechanisms—surpassing the limitations of traditional adjacency queries. Their approach achieves a $(2+\varepsilon)$-approximation using only $\tilde{O}(n)$ space and $\sqrt{n}^{\,1+o(1)}$ passes over the stream, and they establish that this pass complexity is nearly optimal for any constant-factor approximation.

Abstract

Consider a poset - or equivalently an $n$-vertex DAG $G=(V, E)$ - and a boolean function $f: V \rightarrow \{0, 1\}$ on its vertex set. We say $f$ is monotone if $f(u) \leq f(v)$ for all $(u, v) \in E$. While there is extensive literature on the query complexity of testing monotonicity, we focus instead on the space complexity and initiate the study of this problem in the streaming setting. Namely, the edges of $G$ arrive in an arbitrary order, and the goal is to estimate distance to monotonicity of a given function $f$ using $\widetilde{O}(n)$ space. Note that while this space allows receiving and storing $f$, it is much smaller than the input graph $G$ which could have up to $Ω(n^2)$ edges. Our main result is an algorithm that $(2+ε)$-approximates distance to monotonicity in $\sqrt{n}^{1+o(1)}$ passes. We also prove that this is the best pass-complexity one can hope for, for any $O(1)$-approximation, short of improving the state-of-the-art streaming algorithm for $st$-reachability, which is a very well-studied problem. On the technical side, our algorithm approximates the size of maximum matching in (a subgraph of) the transitive closure of $G$. While the maximum matching problem has received significant attention in the streaming setting, the fact that we are computing it in the transitive closure requires very different ideas. In fact, a main contribution of our work is to connect sublinear time algorithms for estimating the maximum matching size to the streaming setting for the first time. While existing off-the-shelf sublinear time algorithms only result in an $n\sqrt{n}^{1+o(1)}$ pass algorithm in our setting, we show how to significantly improve upon them by allowing stronger queries (such as vertex and subset queries) that can be implemented just as efficiently as more standard adjacency matrix and list queries for our problem.

Problem

• streaming algorithms
• monotonicity testing
• space complexity
• distance estimation
• DAG

Innovation

• streaming algorithms
• monotonicity testing
• maximum matching
• transitive closure
• sublinear time algorithms