Gray Codes for the Ideals of Interval Orders

The generation of combinatorial objects in a Gray code manner means that the difference between successive objects is small, e.g., one element for subsets or one transposition for permutations of a set. The existence of such Gray codes is often equivalent to an appropriately defined graph on these objects being Hamiltonian. We show that if the graphGis the covering graph of the lattice of the order ideals of an interval order, thenG2has a Hamiltonian path. This leads to an algorithm to generate the ideals of interval orders in constant time per ideal. We also prove that the subgraph ofG2induced by the ideals of any fixed cardinality also has a Hamiltonian path. This proves a conjecture of Pruesse and Ruskey for interval orders. We also show how these paths can be combined into a layered Hamiltonian path ofG2, yielding a Gray code on the ideals in nondecreasing order of their cardinalities.

[1]  Mihalis Yannakakis,et al.  Scheduling Interval-Ordered Tasks , 1979, SIAM J. Comput..

[2]  Frank Ruskey,et al.  A Gray Code for the Ideals of a Forest Poset , 1993, J. Algorithms.

[3]  Gideon Ehrlich,et al.  Loopless Algorithms for Generating Permutations, Combinations, and Other Combinatorial Configurations , 1973, JACM.

[4]  I. Rival.  Algorithms and Order , 1988 .

[5]  I. Rival.  Graphs and Order , 1985 .

[6]  J. Scott Provan,et al.  Calculating bounds on reachability and connectedness in stochastic networks , 1983, Networks.

[7]  E. Rodney Canfield,et al.  A loop-free algorithm for generating the linear extensions of a poset , 1995 .

[8]  George Steiner,et al.  An algorithm to generate the ideals of a partial order , 1986 .

[9]  Nachum Dershowitz.  A simplified loop-free algorithm for generating permutations , 1975 .

[10]  Frank Ruskey.  Listing and Counting Subtrees of a Tree , 1981, SIAM J. Comput..

[11]  Frank Ruskey,et al.  Generating Linear Extensions Fast , 1994, SIAM J. Comput..

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

[13]  Frank Ruskey,et al.  Gray codes from antimatroids , 1993 .

[14]  Dennis E. White,et al.  Combinatorial Gray codes , 1980, SIAM J. Comput..

[15]  R. Möhring.  Algorithmic Aspects of Comparability Graphs and Interval Graphs , 1985 .

[16]  Rolf H. Möhring,et al.  Computationally Tractable Classes of Ordered Sets , 1989 .

[17]  Teruo Hikita.  Listing and Counting Subtrees of Equal Size of a Binary Tree , 1983, Inf. Process. Lett..

[18]  Michel Morvan,et al.  A Linear Time and Space Algorithm to Regonize Interval Orders , 1993, Discret. Appl. Math..