# Improved Bound for Matching in Random-Order Streams

arXiv: https://arxiv.org/abs/2005.00417 · HTML: https://mlsanity.com/paper/2005.00417/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2020.

## Built on by (most cited first, 2 of 2)

- [One Pass Streaming Algorithm for Super Long Token Attention Approximation in Sublinear Space](https://mlsanity.com/paper/2311.14652/) (2023)
- [Stochastic Matching via Local Sparsification](https://mlsanity.com/paper/2605.14195/) (2026)

## Abstract (arXiv)

We study the problem of computing an approximate maximum cardinality matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, the edges of the input graph G = (V,E) are given as a stream e1, ..., em, and the algorithm is allowed to make a single pass over this stream while using O(n textrmpolylog(n)) space (m = |E| and n = |V|). If the order of edges is adversarial, a simple single-pass greedy algorithm yields a 1/2-approximation in O(n) space; achieving a better approximation in adversarial streams remains an elusive open question. A line of recent work shows that one can improve upon the 1/2-approximation if the edges of the stream arrive in a random order. The state of the art for this model is two-fold: Assadi et al. [SODA 2019] show how to compute a 2/3(~.66)-approximate matching, but the space requirement is O(n1.5 textrmpolylog(n)). Very recently, Farhadi et al. [SODA 2020] presented an algorithm with the desired space usage of O(n textrmpolylog(n)), but a worse approximation ratio of 6/11(~.545), or 3/5(=.6) in bipartite graphs. In this paper, we present an algorithm that computes a 2/3(~.66)-approximate matching using only O(n log(n)) space, improving upon both results above. We also note that for adversarial streams, a lower bound of Kapralov [SODA 2013] shows that any algorithm that achieves a 1-1/e(~.63)-approximation requires (n1+Omega(1/loglog(n))) space. Our result for random-order streams is the first to go beyond the adversarial-order lower bound, thus establishing that computing a maximum matching is provably easier in random-order streams.