Succinct data structures for nearest colored node in a tree

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

“…Hence, the tradeoff from [12] translates to vertex-labeled distance oracles, assuming that the planar graph is undirected. To the best of our knowledge, this is the first non-trivial upper bound for vertexlabeled distance oracles in any interesting graph class other than trees [8,15]. A strength of our result is that any future progress on distance oracles in undirected planar graphs immediately translates to vertex-labeled distance oracles.…”

On the complexity of the (approximate) nearest colored node problem

“…We then prove a lower bound in the cell probe model showing that even for unweighted undirected paths any static data structure that uses space S requires at least Ω log log σ log(S/n)+log log n query time to give a distance estimate of stretch O(polylog(n)). This implies for the important case when σ = Θ(n ǫ ) for some constant 0 < ǫ < 1, that our Color Distance Oracle has asymptotically optimal query time in regard to k, and that recent Color Distance Oracles for trees [23] and planar graphs [16] achieve asymptotically optimal query time in regard to n.…”

“…We point out that the technique presented to query the Color Distance Oracle extends to Compact Routing Schemes as described by Thorup and Zwick [21] such that paths can be computed in O(log k) time. By using a recently devised succinct RMQ structure that is only allowed to query intervals where both indices are multiples of log k by Tsur [23], we can adapt our approach to reduce the intervals as described in lemma 2 down to size O(log k) and then test each remaining index in O(log k) overall time. The data structure only requires…”

“…As HSTs can be balanced, we get an additional factor of O(log n) in our data structures. We can then for each color use a succinct nearest marked ancestor structure as presented by Tsur [23] with O(log log σ log w ) query time.…”

“…This improves the query time from O(k) to O(log k) over the best known Color Distance Oracle by Chechik [6].We then prove a lower bound in the cell probe model showing that even for unweighted undirected paths any static data structure that uses space S requires at least Ω log log σ log(S/n)+log log n query time to give a distance estimate of stretch O(polylog(n)). This implies for the important case when σ = Θ(n ǫ ) for some constant 0 < ǫ < 1, that our Color Distance Oracle has asymptotically optimal query time in regard to k, and that recent Color Distance Oracles for trees [23] and planar graphs [16] achieve asymptotically optimal query time in regard to n.We also investigate the setting where the data structure additionally has to support color-reassignments. We present the first Color Distance Oracle that achieves query times matching our lower bound from the static setting for large stretch yielding an exponential improvement over the best known query time [7].…”