Parallel Algorithms for Computing Prüfer-Like Codes for Labeled Trees

In 1918 Prüfer showed a one-to-one correspondence between labeled trees on n nodes and sequences of (n 2) node labels, obtained by iteratively deleting the leaf with the smallest label in the current tree and appending the label of the adjacent node to the code. In 1953 Neville proposed two additional methods for encoding a labeled tree of order n into a sequence of (n 2) labels. Recently, yet another tree-code was proposed by the authors. As a generalization, we define a tree-code to be Prüfer-like, if it is a sequence of (n 2) node labels computed by iteratively deleting the leaves of the tree in some deterministic order. The four aforementioned Prüfer-like codes can be computed sequentially in O(n) time. Two EREW PRAM O(log n)-time algorithms for computing the Prüfer code have been proposed in the literature. Neither of these algorithms is workoptimal, since they require O(n) processors. In this paper we propose three different EREW PRAM O(log n)-time algorithms for computing the tree-codes proposed by Neville, and by the authors. The algorithm for Neville's third encoding requires O(n / log n) processors and therefore is work-optimal. To the best of our knowledge, no other work-optimal algorithm for computing Prüfer-like tree-code has been published. Index Terms -Prüfer codes, tree-codes, labeled trees, parallel algorithms. • The authors are with the School of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 32816-2362. Email: {deo, pmicikev}@cs.ucf.edu. • This research was supported in part by Honeywell Space Systems.