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Suppose we have a memory storing $0$s and $1$s and we want to estimate the frequency of $1$s by sampling. We want to do this I/O-efficiently, exploiting that each read gives a block of $B$ bits at unit cost; not just one bit. If the input consists of uniform blocks: either all 1s or all 0s, then sampling a whole block at a time does not reduce the number of samples needed for estimation. On the other hand, if bits are randomly permuted, then getting a block of $B$ bits is as good as getting $B$ independent bit samples. However, we do not want to make any such assumptions on the input. Instead, our goal is to have an algorithm with instance-dependent performance guarantees which stops sampling blocks as soon as we know that we have a probabilistically reliable estimate. We prove our algorithms to be instance-optimal among algorithms oblivious to the order of the blocks, which we argue is the strongest form of instance optimality we can hope for. We also present similar results for I/O-efficiently estimating mean with both additive and multiplicative error, estimating histograms, quantiles, as well as the empirical cumulative distribution function. We obtain our above results on I/O-efficient sampling by reducing to corresponding problems in the so-called sequential estimation. In this setting, one samples from an unknown distribution until one can provide an estimate with some desired error probability. We then provide non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to $\ell_\infty$ and the so-called Kolmogorov-Smirnov distance., Comment: To appear at FOCS 2024

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a new combinatorial algorithm for the following: Given any 3-colorable graph with minimum degree $\ds>\sqrt n$, we can, in polynomial time, make progress towards a $k$-coloring for some $k=\sqrt{n/\ds}\cdot n^{o(1)}$. We balance our main result with the best-known semi-definite(SDP) approach which we use for degrees below $n^{0.605073}$. As a result, we show that $\tO(n^{0.19747})$ colors suffice for coloring 3-colorable graphs. This improves on the previous best bound of $\tO(n^{0.19996})$ by Kawarabayashi and Thorup in 2017., Comment: To appear in STOC'24

Correlation Clustering is a classic clustering objective arising in numerous machine learning and data mining applications. Given a graph $G=(V,E)$, the goal is to partition the vertex set into clusters so as to minimize the number of edges between clusters plus the number of edges missing within clusters. The problem is APX-hard and the best known polynomial time approximation factor is 1.73 by Cohen-Addad, Lee, Li, and Newman [FOCS'23]. They use an LP with $|V|^{1/\epsilon^{\Theta(1)}}$ variables for some small $\epsilon$. However, due to the practical relevance of correlation clustering, there has also been great interest in getting more efficient sequential and parallel algorithms. The classic combinatorial \emph{pivot} algorithm of Ailon, Charikar and Newman [JACM'08] provides a 3-approximation in linear time. Like most other algorithms discussed here, this uses randomization. Recently, Behnezhad, Charikar, Ma and Tan [FOCS'22] presented a $3+\epsilon$-approximate solution for solving problem in a constant number of rounds in the Massively Parallel Computation (MPC) setting. Very recently, Cao, Huang, Su [SODA'24] provided a 2.4-approximation in a polylogarithmic number of rounds in the MPC model and in $\tilde{O} (|E|^{1.5})$ time in the classic sequential setting. They asked whether it is possible to get a better than 3-approximation in near-linear time? We resolve this problem with an efficient combinatorial algorithm providing a drastically better approximation factor. It achieves a $\sim 2-2/13 < 1.847$-approximation in sub-linear ($\tilde O(|V|)$) sequential time or in sub-linear ($\tilde O(|V|)$) space in the streaming setting. In the MPC model, we give an algorithm using only a constant number of rounds that achieves a $\sim 2-1/8 < 1.876$-approximation.

An electric car equipped with a battery of a finite capacity travels on a road network with an infrastructure of charging stations. Each charging station has a possibly different cost per unit of energy. Traversing a given road segment requires a specified amount of energy that may be positive, zero or negative. The car can only traverse a road segment if it has enough charge to do so (the charge cannot drop below zero), and it cannot charge its battery beyond its capacity. To travel from one point to another the car needs to choose a \emph{travel plan} consisting of a path in the network and a recharging schedule that specifies how much energy to charge at each charging station on the path, making sure of having enough energy to reach the next charging station or the destination. The cost of the plan is the total charging cost along the chosen path. We reduce the problem of computing plans between every two junctions of the network to two problems: Finding optimal energetic paths when no charging is allowed and finding standard shortest paths. When there are no negative cycles in the network, we obtain an $O(n^3)$-time algorithm for computing all-pairs travel plans, where~$n$ is the number of junctions in the network. We obtain slightly faster algorithms under some further assumptions. We also consider the case in which a bound is placed on the number of rechargings allowed., Comment: 9 pages, 2 figures, SOSA 2024