Structuring labeled trees for optimal succinctness, and beyond

Abstract

Consider an ordered, static tree T on t nodes where each node has a label from alphabet set Σ.

Reducing the Space Requirement of LZ-Index

“…In the first approach, we replace the original LZ-index data structures by new ones allowing the same functionalities, but with reduced redundancy. In the second approach, the main idea is to combine the balanced parentheses [14] representation of the LZ78 trie with the xbw transform [3]. With either approach we achieve (2 + ε)uH k (T ) + o(u log σ ) bits of space to index to text T 1...u , for k = o(log σ u) and any constant ε > 0, and the original time to report the occ occurrences of a pattern P 1...m in T is improved to O(m 2 log σ + (m + occ) log n).…”

“…A different idea to reduce the space requirement of LZ-index is to use the xbw transform of Ferragina et al [3] to represent the LZTrie. This succinct representation supports the operations parent(x), child(x, i), and child(x, α), all of them in O(1) time and using 2n log σ + O(n) bits of space.…”

“…-xbw LZTrie: the xbw representation [3] of the LZTrie, where the nodes are lexicographically sorted according to their upward paths in the trie. We store • S α : the array of symbols labeling the arcs of the trie.…”

“…This is the preorder interval of all the nodes that start with P i+1...m . Next we perform a subpath query for P 1...i in xbw, and get the xbw interval [x 1 , x 2 ] of all the nodes that finish with P 1...i (actually we have to perform rank 1 (S last , i) to avoid counting the same node multiple times, see [3]). Then, we search the Range data structure for [x 1 .…”

“…In this part we define new data structures allowing the same navigation, yet reducing the original redundancy. In the second approach we combine the balanced parentheses representation of Munro and Raman [14] of the LZ78 trie with the xbw transform of Ferragina et al [3], whose powerful operations are useful for the LZ-index search algorithm.…”