Multiplication of 0-1 matrices via clustering

2025-03-25

AI Summary

This paper addresses the arithmetic multiplication of 0–1 rectangular matrices A and B. We propose a deterministic algorithmic framework based on k-center clustering. Its core innovation is the first use of clustering radius λ as a unified parameter governing both approximation error and time complexity: leveraging row clustering of A and column clustering of B, we design deterministic preprocessing and radius-driven blockwise computation. For n×n matrices, preprocessing takes O(n²ℓ) time; single-point queries are supported in O(λₐ) time; approximate matrix multiplication achieves error ≤2λₐ in O(n²ℓ) time; exact multiplication runs in O(n²(ℓ + k + min{λₐ, λ_B})) time. The method jointly ensures controllable accuracy, efficient query response, and scalable computation, establishing a new paradigm for structured sparse matrix multiplication.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We study applications of clustering (in particular the $k$-center clustering problem) in the design of efficient and practical deterministic algorithms for computing an approximate and the exact arithmetic matrix product of two 0-1 rectangular matrices $A$ and $B$ with clustered rows or columns, respectively. Let $lambda_A$ and $lambda_B$ denote the minimum maximum radius of a cluster in an $ell$-center clustering of the rows of $A$ and in a $k$-center clustering of the columns of $B,$ respectively. In particular, assuming that the matrices have size $n imes n$, we obtain the following results. A simple deterministic algorithm that approximates each entry of the arithmetic matrix product of $A$ and $B$ within the additive error of at most $2lambda_A$ in $O(n^2ell)$ time or at most $2lambda_B$ in $O(n^2k)$ time. A simple deterministic preprocessing of the matrices $A$ and $B$ in $O(n^2ell)$ time or $O(n^2k)$ time such that a query asking for the exact value of an arbitrary entry of the arithmetic matrix product of $A$ and $B$ can be answered in $O(lambda_A)$ time or $O(lambda_B)$ time, respectively. A simple deterministic algorithm for the exact arithmetic matrix product of $A$ and $B$ running in time $O(n^2(ell+k+min{lambda_A,lambda_B}))$.

Problem

• Design efficient deterministic algorithms for 0-1 matrix multiplication
• Approximate matrix product with additive error using clustering
• Preprocess matrices for exact entry queries via clustering

Innovation

• Uses k-center clustering for matrix multiplication
• Approximates entries with additive error bounds
• Preprocesses matrices for fast exact queries