Structuring labeled trees for optimal succinctness, and beyond

Abstract

Consider an ordered, static tree /spl Tscr/ on t nodes where each node has a label from alphabet set /spl Sigma/. Tree /spl Tscr/ may be of arbitrary degree and of arbitrary shape. Say, we wish to support basic navigational operations such as find the parent of node u, the ith child of u, and any child of it with label /spl alpha/. In a seminal work over fifteen years ago, Jacobson (1989) observed that pointer-based tree representations are wasteful in space and introduced the notion of succinct data structures. He studied the special case of unlabeled trees and presented a succinct data structure of 2t + o(t) bits supporting navigational operations in O(1) time. The space used is asymptotically optimal with the information-theoretic lower bound averaged over all trees. This led to a slew of results on succinct data structures for arrays, trees, strings and multisets. Still, for the fundamental problem of structuring labeled trees succinctly, few results, if any, exist even though labeled trees arise frequently in practice, e.g. in the data as in markup text (XML) or in augmented data structures. We present a novel approach to the problem of succinct manipulation of labeled trees by designing what we call the xbw transform of the tree, in the spirit of the well-known Burrows-Wheeler transform for strings. The xbw transform uses path-sorting and grouping to linearize the labeled tree /spl Tscr/ into two coordinated arrays, one capturing the structure and the other the labels. Using the properties of the xbw transform, we (i) derive the first-known (near-)optimal results for succinct representation of labeled trees with O(1) time for navigation operations, (ii) optimally support the powerful subpath search operation for the first time, and (iii) introduce a notion of tree entropy and present linear time algorithms for compressing a given labeled tree up to its entropy beyond the information-theoretic lower bound averaged over all tree inputs. Our xbw transform is simple and likely to spur new results in the theory of tree compression and indexing, and may have some practical impact in XML data processing.