Efficient Many-To-Many Point Matching in One Dimension

Let S and T be two sets of points with total cardinality n. The minimum-cost many-to-many matching problem matches each point in S to at least one point in T and each point in T to at least one point in S, such that sum of the matching costs is minimized. Here we examine the special case where both S and T lie on the line and the cost of matching s ∈S to t ∈T is equal to the distance between s and t. In this context, we provide an algorithm that determines a minimum-cost many-to-many matching in O(n log n) time, improving the previous best time complexity of O(n2) for the same problem.

[1]  M. Fatih Demirci,et al.  Object Recognition as Many-to-Many Feature Matching , 2006, International Journal of Computer Vision.

[2]  Godfried T. Toussaint,et al.  Faster Algorithms for Computing Distances between One-Dimensional Point Sets , 2005 .

[3]  Heikki Mannila,et al.  Distance measures for point sets and their computation , 1997, Acta Informatica.

[4]  Samuel R. Buss,et al.  A Bipartite Matching Approach to Approximate String Comparison and Search , 1995 .

[5]  Richard M. Karp,et al.  The Restriction Scaffold Problem , 2003, J. Comput. Biol..

[6]  Subhash Suri,et al.  Fast matching algorithms for points on a polygon , 1989, 30th Annual Symposium on Foundations of Computer Science.

[7]  Godfried T. Toussaint,et al.  A Faster Algorithm for Computing the Link Distance Between Two Point Sets on the Real Line , 2005 .

[8]  Godfried T. Toussaint,et al.  C G ] 5 J ul 2 00 5 An O ( n log n )-Time Algorithm for the Restricted Scaffold Assignment , 2009 .

[9]  Pravin M. Vaidya.  Geometry helps in matching , 1988, STOC '88.

[10]  Alon Itai,et al.  Geometry Helps in Bottleneck Matching and Related Problems , 2001, Algorithmica.

[11]  Godfried T. Toussaint,et al.  The Geometry of Musical Rhythm , 2013, JCDCG.

[12]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[13]  Godfried T. Toussaint,et al.  An algorithm for computing the restriction scaffold assignment problem in computational biology , 2005, Inf. Process. Lett..

[14]  Daniel E. Loeb.  A New Proof of Monjardet's Median Theorem , 1996, J. Comb. Theory, Ser. A.

[15]  Alexander Schrijver.  Min-max Relations for Directed Graphs , 1982 .

[16]  Godfried T. Toussaint,et al.  A Comparison of Rhythmic Similarity Measures , 2004, ISMIR.

[17]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[18]  L. Polansky,et al.  Temporal Gestalt Perception in Music , 1980 .

[19]  Rainer E. Burkard,et al.  Linear Assignment Problems and Extensions , 1999, Handbook of Combinatorial Optimization.

[20]  N. Tomizawa,et al.  On some techniques useful for solution of transportation network problems , 1971, Networks.

[21]  Richard M. Karp,et al.  Two special cases of the assignment problem , 1975, Discrete Mathematics.

[22]  Michael Droettboom,et al.  ISMIR 2004: International Conference on Music Information Retrieval, October 10-14, 2004, Universitat Pompeu Fabra, Barcelona, Spain , 2004 .

[23]  Judith Keijsper,et al.  An Efficient Algorithm for Minimum-Weight Bibranching , 1998, J. Comb. Theory, Ser. B.

[24]  Michael Keith.  From Polychords to Polya : Adventures in Musical Combinatorics , 1991 .