### Abstract

Let A and B be two sets of n objects in ℝ^{d}, and let Match be a (one-to-one) matching between A and B. Let min(Match), max.(Match), and σ(Match) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match, respectively. Bottleneck matching - a matching that minimizes max(Match) - is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly O(n^{1.5}). For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n^{1.5} log n). A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are point-sets in the plane, an O(n^{5} log n)-time algorithm for determining whether for some translated copy the resemblance gets below a given ρ is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost n. This result is used to compute the smallest such ρ in time O(n^{5} log^{2} n), and an efficient approximation scheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find Match*_{U}, a matching that minimizes max(Match) - min(Match). A minimum deviation matching Match*_{D} is a matching that minimizes (1/n)σ(Match) - min(Match). Algorithms for computing Match*_{U} and Match*_{D} in roughly O(n^{10/3}) time are presented. These algorithms are more efficient than the previous O(n^{4})-time algorithms of Martello, Pulleyblank, Toth, and de Werra, and of Gupta and Punnen, who studied these problems for general bipartite graphs.

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

Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Algorithmica |

Volume | 31 |

Issue number | 1 |

State | Published - 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

- Approximation
- Bipartite graph matching
- Bottleneck matching
- Euclidean distance
- Minkowski norm
- Translation

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Safety, Risk, Reliability and Quality
- Applied Mathematics

### Cite this

*Algorithmica*,

*31*(1), 1-28.

**Geometry helps in bottleneck matching and related problems.** / Efrat, Alon; Itai, A.; Katz, M. J.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 31, no. 1, pp. 1-28.

}

TY - JOUR

T1 - Geometry helps in bottleneck matching and related problems

AU - Efrat, Alon

AU - Itai, A.

AU - Katz, M. J.

PY - 2001

Y1 - 2001

N2 - Let A and B be two sets of n objects in ℝd, and let Match be a (one-to-one) matching between A and B. Let min(Match), max.(Match), and σ(Match) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match, respectively. Bottleneck matching - a matching that minimizes max(Match) - is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly O(n1.5). For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n1.5 log n). A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are point-sets in the plane, an O(n5 log n)-time algorithm for determining whether for some translated copy the resemblance gets below a given ρ is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost n. This result is used to compute the smallest such ρ in time O(n5 log2 n), and an efficient approximation scheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find Match*U, a matching that minimizes max(Match) - min(Match). A minimum deviation matching Match*D is a matching that minimizes (1/n)σ(Match) - min(Match). Algorithms for computing Match*U and Match*D in roughly O(n10/3) time are presented. These algorithms are more efficient than the previous O(n4)-time algorithms of Martello, Pulleyblank, Toth, and de Werra, and of Gupta and Punnen, who studied these problems for general bipartite graphs.

AB - Let A and B be two sets of n objects in ℝd, and let Match be a (one-to-one) matching between A and B. Let min(Match), max.(Match), and σ(Match) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match, respectively. Bottleneck matching - a matching that minimizes max(Match) - is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly O(n1.5). For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n1.5 log n). A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are point-sets in the plane, an O(n5 log n)-time algorithm for determining whether for some translated copy the resemblance gets below a given ρ is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost n. This result is used to compute the smallest such ρ in time O(n5 log2 n), and an efficient approximation scheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find Match*U, a matching that minimizes max(Match) - min(Match). A minimum deviation matching Match*D is a matching that minimizes (1/n)σ(Match) - min(Match). Algorithms for computing Match*U and Match*D in roughly O(n10/3) time are presented. These algorithms are more efficient than the previous O(n4)-time algorithms of Martello, Pulleyblank, Toth, and de Werra, and of Gupta and Punnen, who studied these problems for general bipartite graphs.

KW - Approximation

KW - Bipartite graph matching

KW - Bottleneck matching

KW - Euclidean distance

KW - Minkowski norm

KW - Translation

UR - http://www.scopus.com/inward/record.url?scp=0242333224&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242333224&partnerID=8YFLogxK

M3 - Article

VL - 31

SP - 1

EP - 28

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1

ER -