Balanced parentheses strike back

Abstract

An ordinal tree is an arbitrary rooted tree where the children of each node are ordered. Succinct representations for ordinal trees with efficient query support have been extensively studied. The best previously known result is due to Geary, Raman, and Raman [SODA 2004, pages 1-10]. The number of bits required by their representation for an n-node ordinal tree T is 2n + o(n), whose first-order term is information-theoretically optimal. Their representation supports a large set of O(1)-time queries on T . Based upon a balanced string of 2n parentheses, we give an improved 2n + o(n)-bit representation for T . Our improvement is two fold: Firstly, the set of O(1)-time queries supported by our representation is a proper superset of that supported by the representation of Geary, Raman, and Raman. Secondly, it is also much easier for our representation to support new queries by simply adding new auxiliary strings.

Fully-Functional Succinct Trees

“…For example, formerly the extra data structure for level-ancestor has required O(n log log n/ √ log n) bits [29], or O(n(log log n) 2 / log n) bits 3 [21], and that for child has required O(n/(log log n) 2 ) bits [22]. The previous representation with maximum functionality [9] supports all the operations in Table 1, except insert and delete, in constant time using 2n + O(n log log log n/ log log n)-bit space.…”

“…For example, the first succinct representation of BP [26] supported only findclose, findopen, and enclose (and other easy operations) and each operation used different data structures. Later, many further operations such as lmost-leaf [27], lca [35], degree [7], child and childrank [22], and level-ancestor [29], were added to this representation by using other types of data structures for each. There exists another elegant data structure for BP supporting findclose, findopen, and enclose [16].…”

“…Therefore the tree occupies Θ(n log n) bits, which causes a space problem for manipulating large trees. Much research has been devoted to reducing the space to represent static trees [20,26,27,29,16,17,5,10,7,8,22,19,2,18,35,21,9] and dynamic trees [28,34,6,1], achieving so-called succinct data structures for trees.…”

“…The advantage of DFUDS, when it was created, was that it supported a more complete set of operations in constant time, most notably going to the i-th child of a node. Later, this was also achieved using BP representation, yet requiring complicated additional data structures with nonnegligible lower-order terms in their space usage [22]. Another type of succinct ordinal trees, based on tree covering [17,19,9], has also achieved constant-time support of known operations, yet inheriting the problem of nonnegligible lower-order terms in size.…”

Succinct Trees in Practice

“…While the constanttime complexities achieved for the sophisticated queries are remarkable, the Ω(n log n)-bit space complexity is not justified in terms of Information Theory: there are only C n ∼ 4 n /n 3/2 different general trees of n nodes, and thus log C n = 2n − Θ(log n) bits are sufficient to distinguish any one of them. This huge space gap has motivated a large body of research [16,19,20,21,11,4,10,7,15,12,25,17,18,8,27] achieving 2n + o(n) bits of space and constant time for an impressive set of operations. Table 1 lists those we consider in this paper.…”