# Discrepancy Minimization in Input-Sparsity Time

A recent work of Larsen [Lar23] gave a faster combinatorial alternative to Bansal's SDP algorithm for finding a coloring x∈{-1,1}^n that approximately minimizes the discrepancy disc(A,x) : = A x _∞ of a general real-valued m× n matrix A. Larsen's algorithm runs in O(mn^2) time compared to Bansal's O(mn^4.5)-time algorithm, at the price of a slightly weaker logarithmic approximation ratio in terms of the hereditary discrepancy of A [Ban10]. In this work we present a combinatorial O(nnz(A) + n^3) time algorithm with the same approximation guarantee as Larsen, which is optimal for tall matrices m=poly(n). Using a more intricate analysis and fast matrix-multiplication, we achieve O(nnz(A) + n^2.53) time, which breaks cubic runtime for square matrices, and bypasses the barrier of linear-programming approaches [ES14] for which input-sparsity time is currently out of reach. Our algorithm relies on two main ideas: (i) A new sketching technique for finding a projection matrix with short ℓ_2-basis using implicit leverage-score sampling; (ii) A data structure for faster implementation of the iterative Edge-Walk partial-coloring algorithm of Lovett-Meka, using an alternative analysis that enables “lazy" batch-updates with low-rank corrections. Our result nearly closes the computational gap between real-valued and binary matrices (set-systems), for which input-sparsity time coloring was very recently obtained [JSS23].

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