Abstract
We prove that some exact geometric pattern matching problems reduce in linear time to k -SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of \(k\ge 3\) points within a set of n points in the plane, and for searching for an affine image of a set of \(k\ge d+2\) points within a set of n points in d-space. As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height.
Highlights
The k-SUM problem is a fixed-parameter version of the NP-complete SUBSET SUM problem
We give examples of computational geometry problems that reduce to 3-SUM or k-SUM
Our results are motivated by the nontrivial improved upper bounds on the complexity of 3-SUM and k-SUM proven in the recent years
Summary
The k-SUM problem is a fixed-parameter version of the NP-complete SUBSET SUM problem. There exists a subquadratic algorithm to detect equilateral triangles in a point set This contrasts with our current knowledge on the related 3-SUM-hard problem of finding three collinear points, known as GENERAL POSITION TESTING. Aronov, Ezra, and Sharir [8] study the following problem: Given three sets A, B, C of n points in the plane, decide whether there exists (a, b, c) ∈ A × B × C that simultaneously satisfies two real polynomial equations. They provide a subquadratic upper bound on the algebraic decision tree complexity of this problem. The last section is dedicated to the proof of Corollaries 4 and 5
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