Labeling Schemes for Nearest Common Ancestors through Minor-Universal Trees

Abstract

Preprocessing a tree for finding the nearest common ancestor of two nodes is a basic tool with multiple applications. Quite a few linear-space constant-time solutions are known and the problem seems to be well-understood. This is however not so clear if we want to design a labeling scheme. In this model, the structure should be distributed: every node receives a distinct binary string, called its label, so that given the labels of two nodes (and no further information about the topology of the tree) we can compute the label of their nearest common ancestor. The goal is to make the labels as short as possible. Alstrup, Gavoille, Kaplan, and Rauhe [Theor. Comput. Syst. 37(3): 441-456 2004] showed that O(log n)-bit labels are enough, with a somewhat large constant. More recently, Alstrup, Halvorsen, and Larsen [SODA 2014] refined this to only 2.772 log n, and provided a lower bound of 1.008 log n.We connect the question of designing a labeling scheme for nearest common ancestors to the existence of a tree, called a minor-universal tree, that contains every tree on n nodes as a topological minor. Even though it is not clear if a labeling scheme must be based on such a notion, we argue that all already existing schemes can be reformulated as such. Further, we show that this notion allows us to easily obtain clean and good bounds on the length of the labels. As the main upper bound, we show that 2.318 log nbit labels are enough. Surprisingly, the notion of a minoruniversal tree for binary trees on n nodes has been already used in a different context by Hrubes et al. [CCC 2010], and Young, Chu, and Wong [J. ACM 46(3):416-435, 1999] introduced a very closely related (but not equivalent) notion of a universal tree. On the lower bound side, we show that any minor-universal tree for trees on n nodes must contain at least Ω(n 2.174 ) nodes. This highlights a natural limitation for all approaches based on defining a minoruniversal tree. We complement the existential results with a generic transformation that allows us, for any labeling scheme for nearest common ancestors based on a minoruniversal tree, to decrease the query time to constant, while increasing the length of the labels only by lower order terms.

Efficient Labeling for Reachability in Digraphs

“…Alstrup, Dahlgaard, and Knudsen [3] proved the optimal result for adjacency in trees, achieving labels of size log n + O (1). Numerous other functions were considered, both in terms of upper and lower bounds: distance [10][11][12], connectivity [16,18], sibling or ancestor relationship [2], nearest common ancestor in trees [5,14], routing [25] and flow [16]. Often more restricted classes of graphs are analysed, most notably planar graphs [7,8], bounded degree graphs [1] and sparse graphs [4,13,19].…”

Shorter Labels for Routing in Trees

“…Upper bound Distance 1/4 log 2 n − O(log n) [9] 1/4 log 2 n + o(log 2 n) [25] Fixed-port routing Ω(log 2 n/ log log n) [21] O(log 2 n/ log log n) [20] (1 + )-approximate distance Ω(log (1/ ) log n) [25] O(log (1/ ) log n) [25] Nearest common ancestor 1.008 log n − O(1) [10] 2.318 log n + O(1) [31] Designer-port routing log n + Ω(log log n) [5] log n + O((log log n) 2 ) log n + O((log log n) 2 ) log n + O((log log n) 2 ) Ancestry log n + Ω(log log n) [5] log n + O(log log n) [23] Siblings/connectivity (unique) log n + Ω(log log n) [5] log n + O(log log n) [5] Adjacency log n (trivial) log n + O(1) [6] Table 1: Summary of the state-of-the-art bounds for labeling schemes in trees.…”

“…Arguably, trees are one of the most important classes of graphs considered in the context of labeling schemes. Functions studied in the literature on labeling schemes in trees include adjacency [6,13,14], ancestry [2,22,23], routing [20,21,44], distance [5,9,27,29,39], and nearest common ancestors [8,10,19,31,40]. See Table 1 for a summary of the state-of-the-art bounds for these problems.…”

Simpler Adjacency Labeling for Planar Graphs with B-Trees

“…In every case, the length of each individual label is much smaller than the size of a centralised structure, often by a factor close to Θ(n), i.e., we are able to evenly distribute the whole adjacency information. Other functions considered in the context of labeling schemes are ancestry in trees [25,31], routing [24,30,50] or connectivity [39]. However, from the point of view of possible applications, the next most natural question is that of distance labelings, where given labels of two vertices we need to output the exact distance between them in a graph.…”