A GPU-BASED ALGORITHM FOR APPROXIMATELY FINDING THE LARGEST COMMON POINT SET IN THE PLANE UNDER SIMILARITY TRANSFORMATION

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ > 0, find the largest subset B ⊂ P and a similarity transformation T (translation, rotation and scale) such that h(T(B),Q) < δ, where h(.,.) is the directional Hausdorff distance. This problem stems from real world applications, where δ is determined by the practical uncertainty in the position of the points (pixels). We reduce the problem to finding the depth (maximally covered point) of an arrangement of polytopes in transformation space. The depth is the cardinality of B, and the polytopes that cover the deepest point correspond to the points in B. We present an algorithm that approximates the maximum depth with high probability, thus getting a large enough common point set in P and Q. The algorithm is implemented in the GPU framework, thus it is very fast in practice. We present experimental results and compare their runtime with those of an algorithm running on the CPU.

A New Algorithm for Subset Matching Problem Based on Set-String Transformation

In computer engineering, a number of programming tasks involve a special problem, the so-called tree matching problem (Cole & Hariharan, 1997), as a crucial step, such as the design of interpreters for nonprocedural programming languages, automatic implementation of abstract data types, code optimization in compilers, symbolic computation, context searching in structure editors and automatic theorem proving. Recently, it has been shown that this problem can be transformed in linear time to another problem, the so called subset matching problem (Cole & Hariharan, 2002, 2003), which is to find all occurrences of a pattern string p of length m in a text string t of length n, where each pattern and text position is a set of characters drawn from some alphabet S. The pattern is said to occur at text position i if the set p[j] is a subset of the set t[i + j - 1], for all j (1 = j = m). This is a generalization of the ordinary string matching and is of interest since an efficient algorithm for this problem implies an efficient solution to the tree matching problem. In addition, as shown in (Indyk, 1997), this problem can also be used to solve general string matching and counting matching (Muthukrishan, 1997; Muthukrishan & Palem, 1994), and enables us to design efficient algorithms for several geometric pattern matching problems. In this article, we propose a new algorithm on this issue, which needs only O(n + m) time in the case that the size of S is small and O(n + m·n0.5) time on average in general cases.