Maintenance of a Percing Set for Intervals with Applications

We show how to efficiently maintain a minimum piercing set for a set S of intervals on the line, under insertions and deletions to/from S. A linear-size dynamic data structure is presented, which enables us to compute a new minimum piercing set following an insertion or deletion in time O(c(S) log |S|), where c(S) is the size of the new minimum piercing set. We also show how to maintain a piercing set for S of size at most (1 + Ɛ)c(S), for 0 < Ɛ ≤ 1, in O(log |S|/Ɛ) amortized time per update. We then apply these results to obtain efficient (sometimes improved) solutions to the following three problems: (i) the shooter location problem, (ii) computing a minimum piercing set for arcs on a circle, and (iii) dynamically maintaining a box cover for a d-dimensional point set.

[1]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[2]  Wen-Lian Hsu,et al.  Linear Time Algorithms on Circular-Arc Graphs , 1991, Inf. Process. Lett..

[3]  Robert J. Fowler,et al.  Optimal Packing and Covering in the Plane are NP-Complete , 1981, Inf. Process. Lett..

[4]  Frank Nielsen,et al.  Fast stabbing of boxes in high dimensions , 1996, Theor. Comput. Sci..

[5]  Kazuo Nakajima,et al.  An Optimal Algorithm for Finding a Maximum Independent Set of a Circular-Arc Graph , 1988, SIAM J. Comput..

[6]  Binhai Zhu,et al.  Shooter location problems revisited , 1997, Canadian Conference on Computational Geometry.

[7]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

[8]  Victor J. Milenkovic,et al.  Limited Gaps , 1994, CCCG.

[9]  C. Pandu Rangan,et al.  Optimal Parallel Algorithms on Circular-Arc Graphs , 1989, FSTTCS.

[10]  Peter L. Hammer,et al.  Stability in Circular Arc Graphs , 1988, J. Algorithms.

[11]  Joseph Y.-T. Leung,et al.  Efficient algorithms for interval graphs and circular-arc graphs , 1982, Networks.

[12]  Subhas C. Nandy,et al.  Shooter Location Problem , 1996, CCCG.

[13]  Subhas C. Nandy,et al.  Generalized Shooter Location Problem , 1999, COCOON.

[14]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[15]  Majid Sarrafzadeh,et al.  Minimum Cuts for Circular-Arc Graphs , 1990, SIAM J. Comput..

[16]  Alan A. Bertossi,et al.  Sequential implementation of the two algorithms for the minimum weighted dominating set and minimum weighted total dominating set , 2001 .