We revisit several fundamental graph problems in the deterministic two-party communication model. Our main contributions include: - We give a new Õ(n^{3/2})-bit protocol for computing a maximum matching in general graphs. While the same upper bound can be obtained by simulating the classic algorithms of Micali-Vazirani [Silvio Micali and Vijay V. Vazirani, 1980] and Gabow [Harold N. Gabow, 2017], our protocol is conceptually simple and avoids the intricacies of finding a maximal set of shortest augmenting paths. - We give a new Õ(n)-bit protocol for negative-cycle detection and negative-weight single-source shortest paths. Our protocol simplifies that of Blikstad et al. [Joakim Blikstad et al., 2022] by replacing a long chain of reductions with a more direct approach based on vertex potentials. - We give a combinatorial Õ(n)-bit protocol for computing a maximum matching in bipartite graphs, obtained by reinterpreting the near-linear communication protocol of Blikstad et al. [Joakim Blikstad et al., 2022] through a discretized analysis. Together, these results provide simpler protocols for several basic graph problems. We hope they will inspire further advances on the communication complexity of a wide range of graph problems.

We study the problem of computing a minimum \(s-t\) cut in an unweighted, undirected graph via cut queries. In this model, the input graph is accessed through an oracle that, given a subset of vertices \(S \subseteq V\), returns the size of the cut \((S, V\ \unicode{x005C}\ S)\).

Computing the approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream of elements x1,x2,. ..,xn and a query x, a relative-error quantile estimation algorithm can estimate the rank of x with respect to the stream, up to a multiplicative ±∈ · rank(x ) error. Notably, this requires the sketch to obtain more precise estimates for the ranks of elements on the tails of the distribution, as compared to the additive ±en error regime. This is particularly favorable for some practical applications, such as anomaly detection.

Estimating the ε-approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream x_1,..., x_n from a universe \mathcalU with total order, an additive-error quantile sketch \mathcalM allows us to approximate the rank of any query y\in \mathcalU up to additive ε n error. In 2001, Greenwald and Khanna gave a deterministic algorithm (GK sketch) that solves the ε-approximate quantiles estimation problem using O(ε^-1 łog(ε n)) space \citegreenwald2001space ; recently, this algorithm was shown to be optimal by Cormode and Vesleý in 2020 \citecormode2020tight. However, due to the intricacy of the GK sketch and its analysis, over-simplified versions of the algorithm are implemented in practical applications, often without any known theoretical guarantees. In fact, it has remained an open question whether the GK sketch can be simplified while maintaining the optimal space bound. In this paper, we resolve this open question by giving a simplified deterministic algorithm that stores at most (2 + o(1))ε^-1 łog (ε n) elements and solves the additive-error quantile estimation problem; as a side benefit, our algorithm achieves a smaller constant factor than the \frac11 2 ε^-1 łog(ε n) space bound in the original GK sketch~\citegreenwald2001space. Our algorithm features an easier analysis and still achieves the same optimal asymptotic space complexity as the original GK sketch. Lastly, our simplification enables an efficient data structure implementation, with a worst-case runtime of O(łog(1/ε) + łog łog (ε n)) per-element for the ordinary ε-approximate quantile estimation problem. Also, for the related "weighted'' quantile estimation problem, we give efficient data structures for our simplified algorithm which guarantee a worst-case per-element runtime of O(łog(1/ε) + łog łog (ε W_n/w_\textrmmin )), achieving an improvement over the previous upper bound of \citeassadi2023generalizing.