### Abstract

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plain. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a coverage measure, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time &Ogr;(n ^{3}polylogn) in the general case, and run in time &Ogr;(n ^{3}polylogn) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly &Ogr;(n ^{3/2}polylog n). These algorithms are significant improvements over the general algorithm of Chew et al. that takes time &Ogr;(n ^{4} log ^{2} n). In the second part of this paper we address the problem of matching polygonal chains. We study the well known Fréchet distance, and present the first algorithm for computing the Fréchet distance under general translations. Our methods also yield algorithms for computing a generalization of the Fréchet distance, and we present a simple approximation algorithm for the Fréchet distance and its generalization that runs in time &Ogr;(n ^{2}polylogn).

Original language | English (US) |
---|---|

Title of host publication | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |

Pages | 295-304 |

Number of pages | 10 |

State | Published - 2001 |

Externally published | Yes |

Event | 2001 Operating Section Proceedings, American Gas Association - Dallas, TX, United States Duration: Apr 30 2001 → May 1 2001 |

### Other

Other | 2001 Operating Section Proceedings, American Gas Association |
---|---|

Country | United States |

City | Dallas, TX |

Period | 4/30/01 → 5/1/01 |

### Fingerprint

### Keywords

- Algorithms
- Measurement
- Performance
- Theory
- Verification

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms*(pp. 295-304)

**Pattern matching for sets of segments.** / Efrat, Alon; Indyk, Piotr; Venkatasubramanian, Suresh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms.*pp. 295-304, 2001 Operating Section Proceedings, American Gas Association, Dallas, TX, United States, 4/30/01.

}

TY - GEN

T1 - Pattern matching for sets of segments

AU - Efrat, Alon

AU - Indyk, Piotr

AU - Venkatasubramanian, Suresh

PY - 2001

Y1 - 2001

N2 - In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plain. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a coverage measure, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time &Ogr;(n 3polylogn) in the general case, and run in time &Ogr;(n 3polylogn) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly &Ogr;(n 3/2polylog n). These algorithms are significant improvements over the general algorithm of Chew et al. that takes time &Ogr;(n 4 log 2 n). In the second part of this paper we address the problem of matching polygonal chains. We study the well known Fréchet distance, and present the first algorithm for computing the Fréchet distance under general translations. Our methods also yield algorithms for computing a generalization of the Fréchet distance, and we present a simple approximation algorithm for the Fréchet distance and its generalization that runs in time &Ogr;(n 2polylogn).

AB - In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plain. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a coverage measure, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time &Ogr;(n 3polylogn) in the general case, and run in time &Ogr;(n 3polylogn) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly &Ogr;(n 3/2polylog n). These algorithms are significant improvements over the general algorithm of Chew et al. that takes time &Ogr;(n 4 log 2 n). In the second part of this paper we address the problem of matching polygonal chains. We study the well known Fréchet distance, and present the first algorithm for computing the Fréchet distance under general translations. Our methods also yield algorithms for computing a generalization of the Fréchet distance, and we present a simple approximation algorithm for the Fréchet distance and its generalization that runs in time &Ogr;(n 2polylogn).

KW - Algorithms

KW - Measurement

KW - Performance

KW - Theory

KW - Verification

UR - http://www.scopus.com/inward/record.url?scp=21044444687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21044444687&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:21044444687

SN - 0898714907

SP - 295

EP - 304

BT - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

ER -