On the Complexity of the (Approximate) Nearest Colored Node Problem
Given a graph G=(V,E) where each vertex is assigned a color from the set C={c_1, c_2, .., c_sigma}. In the (approximate) nearest colored node problem, we want to query, given v in V and c in C, for the (approximate) distance dist^(v, c) from v to the nearest node of color c. For any integer 1 <= k <= log n, we present a Color Distance Oracle (also often referred to as Vertex-label Distance Oracle) of stretch 4k-5 using space O(kn sigma^{1/k}) and query time O(log{k}). This improves the query time from O(k) to O(log{k}) over the best known Color Distance Oracle by Chechik [Chechik, 2012].
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 Omega (log (log{sigma} / 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 sigma = Theta(n^{epsilon}) for some constant 0 < epsilon < 1, that our Color Distance Oracle has asymptotically optimal query time in regard to k, and that recent Color Distance Oracles for trees [Tsur, 2018] and planar graphs [Mozes and Skop, 2018] 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 [Chechik, 2014]. Finally, we give new conditional lower bounds proving the hardness of answering queries if edge insertions and deletion are allowed that strictly improve over recent bounds in time and generality.
Nearest Colored Node
Distance Oracles
Cell-probe lower bounds
Theory of computation~Design and analysis of algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Cell probe models and lower bounds
68:1-68:14
Regular Paper
Supported by Basic Algorithms Research Copenhagen (BARC), supported by Thorup’s Investigator Grant from the Villum Foundation under Grant No. 16582.
https://arxiv.org/abs/1807.03721
Maximilian
Probst
Maximilian Probst
BARC, University of Copenhagen, Universitetsparken 5, Copenhagen 2100, Denmark
https://orcid.org/0000-0003-3522-156X
10.4230/LIPIcs.ESA.2018.68
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Maximilian Probst
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