### Abstract

This paper considers the following problem which we call the largest common point set problem (LCP): given two point sets P and Q in the Euclidean plane, find a subset of P with the maximum cardinality which is congruent to some subset of Q. We introduce a combinatorial-geometric quantity λ(P, Q), which we call the inner product of the distance-multiplicity vectors of P and Q, show its relevance to the complexity of various algorithms for LCP, and give a non-trivial upper bound on λ(P, Q). We generalize this notion to higher dimensions, give some upper bounds on the quantity, and apply them to algorithms for LCP in higher dimensions. Along the way, we prove a new upper bound on the number of congruent triangles in a point set in the four-dimensional space.

Original language | English |
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Pages | 314-323 |

Number of pages | 10 |

DOIs | |

Publication status | Published - 1997 Jan 1 |

Event | Proceedings of the 1997 13th Annual Symposium on Computational Geometry - Nice, Fr Duration: 1997 Jun 4 → 1997 Jun 6 |

### Other

Other | Proceedings of the 1997 13th Annual Symposium on Computational Geometry |
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City | Nice, Fr |

Period | 97/6/4 → 97/6/6 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

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## Cite this

*Distribution of distances and triangles in a point set and algorithms for computing the largest common point sets*. 314-323. Paper presented at Proceedings of the 1997 13th Annual Symposium on Computational Geometry, Nice, Fr, . https://doi.org/10.1145/262839.262989