Average case analysis for tree labelling schemes

We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. Gavoille et al. proved that for any such distance labelling scheme, the maximum label length is at least 18log^2n-O(logn) bits, where n is the number of vertices in the input tree T. They also gave a separator-based labelling scheme that has the optimal label length @Q(logn@?log(H"n(T))), where H"n(T) is the height of T. We present two distance labelling schemes, namely, the backbone-based scheme and rake-based scheme, which also achieve the optimal label length. The two schemes always perform at least as well as the separator scheme. Furthermore, the rake-based scheme has a much smaller expected label length under certain tree distributions. With these new schemes, we also can find the least common ancestor of any two vertices based on their labels only.

[1]  Bruce A. Reed,et al.  The height of a random binary search tree , 2003, JACM.

[2]  Hosam M. Mahmoud,et al.  Evolution of random search trees , 1991, Wiley-Interscience series in discrete mathematics and optimization.

[3]  Luc Devroye,et al.  A note on the height of binary search trees , 1986, JACM.

[4]  Ran Raz,et al.  Distance labeling in graphs , 2001, SODA '01.

[5]  Mikkel Thorup,et al.  Approximate distance oracles , 2001, JACM.

[6]  Tandy J. Warnow,et al.  A Few Logs Suffice to Build (almost) All Trees: Part II , 1999, Theor. Comput. Sci..

[7]  Stephen Alstrup,et al.  Small induced-universal graphs and compact implicit graph representations , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[8]  Moni Naor,et al.  Implicit representation of graphs , 1992, STOC '88.

[9]  Philip Bille,et al.  Labeling schemes for small distances in trees , 2003, SODA '03.

[10]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[11]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[12]  Ming-Yang Kao.  Tree Contractions and Evolutionary Trees , 1998, SIAM J. Comput..

[13]  Philippe Flajolet,et al.  The Average Height of Binary Trees and Other Simple Trees , 1982, J. Comput. Syst. Sci..

[14]  Stephen Alstrup,et al.  Nearest common ancestors: a survey and a new distributed algorithm , 2002, SPAA.

[15]  Abraham Lempel,et al.  On the Complexity of Finite Sequences , 1976, IEEE Trans. Inf. Theory.

[16]  David Peleg,et al.  Compact and localized distributed data structures , 2003, Distributed Computing.

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

[18]  David Peleg,et al.  Approximate Distance Labeling Schemes , 2001, ESA.

[19]  Melvin A. Breuer,et al.  Coding the vertexes of a graph , 1966, IEEE Trans. Inf. Theory.

[20]  David Peleg,et al.  Proximity-Preserving Labeling Schemes and Their Applications , 1999, WG.

[21]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[22]  Melvin A. Breuer,et al.  An unexpected result in coding the vertices of a graph , 1967 .

[23]  Wojciech Szpankowski,et al.  Height in a digital search tree and the longest phrase of the Lempel-Ziv scheme , 2000, SODA '00.