The interval-merging problem

Abstract A closed interval is an ordered pair of real numbers [ x ,  y ], with x  ⩽  y . The interval [ x ,  y ] represents the set { i  ∈  R ∣ x  ⩽  i  ⩽  y }. Given a set of closed intervals I = { [ a 1 , b 1 ] , [ a 2 , b 2 ] , … , [ a k , b k ] } , the Interval-Merging Problem is to find a minimum-cardinality set of intervals M ( I ) = { [ x 1 , y 1 ] , [ x 2 , y 2 ] , … , [ x j , y j ] } , j  ⩽  k , such that the real numbers represented by I = ⋃ i = 1 k [ a i , b i ] equal those represented by M ( I ) = ⋃ i = 1 j [ x i , y i ] . In this paper, we show the problem can be solved in O( d  log  d ) sequential time, and in O(log  d ) parallel time using O( d ) processors on an EREW PRAM, where d is the number of the endpoints of I . Moreover, if the input is given as a set of sorted endpoints, then the problem can be solved in O( d ) sequential time, and in O(log  d ) parallel time using O( d /log  d ) processors on an EREW PRAM.

[1]  Afonso Ferreira,et al.  Parallel computation on interval graphs: algorithms and experiments , 2002, Concurr. Comput. Pract. Exp..

[2]  Kangbok Lee,et al.  Continuous slab caster scheduling and interval graphs , 2004 .

[3]  Li Sheng,et al.  Tagged Probe Interval Graphs , 2001, J. Comb. Optim..

[4]  Roded Sharan,et al.  A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs , 2001, SIAM J. Comput..

[5]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[6]  Min-Sheng Lin,et al.  A linear-time algorithm for computing K-terminal reliability on proper interval graphs , 2002, IEEE Trans. Reliab..

[7]  Tak Wah Lam,et al.  Finding connected components in O(log n loglog n) time on the EREW PRAM , 1993, SODA '93.

[8]  Sara Nicoloso,et al.  Sum coloring and interval graphs: a tight upper bound for the minimum number of colors , 2004, Discret. Math..

[9]  Ke Wang,et al.  Interestingness-Based Interval Merger for Numeric Association Rules , 1998, KDD.

[10]  V. A. Bojarshinov.  Edge and total coloring of interval graphs , 2001, Discret. Appl. Math..

[11]  Madhumangal Pal,et al.  An Optimal Algorithm to Solve 2-Neighbourhood Covering Problem on Interval Graphs , 2002, Int. J. Comput. Math..

[12]  Ruay-Shiung Chang,et al.  Jump Number Maximization for proper Interval Graphs and Series-Parallel Graphs , 1999, Inf. Sci..

[13]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[14]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[15]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[16]  Thomas Zeitlhofer,et al.  List-coloring of interval graphs with application to register assignment for heterogeneous register-set architectures , 2003, Signal Process..

[17]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[18]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[19]  WILLIAM C. K. YEN,et al.  An Optimal Algorithm for Solving the Searchlight Guarding Problem on Weighted Interval Graphs , 1997, Inf. Sci..

[20]  Sandy Irani,et al.  Scheduling with Conflicts on Bipartite and Interval Graphs , 2003, J. Sched..

[21]  Dieter Kratsch,et al.  Approximating Bandwidth by Mixing Layouts of Interval Graphs , 2002, SIAM J. Discret. Math..

[22]  Madhumangal Pal,et al.  An optimal parallel algorithm to construct a tree 3-spanner on interval graphs , 2005, Int. J. Comput. Math..

[23]  D. West.  Introduction to Graph Theory , 1995 .

[24]  Richard Cole,et al.  Parallel merge sort , 1988, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[25]  Alex Alves Freitas,et al.  Speeding up Knowledge Discovery in Large Relational Databases by Means of a New Discretization Algorithm , 1996, BNCOD.

[26]  Donald B. Johnson,et al.  Connected Components in O (log^3/2 n) Parallel Time for the CREW PRAM , 1997, J. Comput. Syst. Sci..