The nearest colored node in a tree

Gawrychowski, Paweł; Landau, Gad M.; Mozes, Shay; Weimann, Oren

doi:10.1016/j.tcs.2017.08.021

Cited by 7 publications

(13 citation statements)

References 24 publications

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“…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.…”

Section: Introductionmentioning

confidence: 70%

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

Evald,

Fredslund-Hansen,

Wulff-Nilsen

2021

Preprint

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Given an undirected n-vertex planar graph G = (V, E, ω) with non-negative edge weight function ω : E → R and given an assigned label to each vertex, a vertex-labeled distance oracle is a data structure which for any query consisting of a vertex u and a label λ reports the shortest path distance from u to the nearest vertex with label λ. We show that if there is a distance oracle for undirected n-vertex planar graphs with non-negative edge weights using s(n) space and with query time q(n), then there is a vertex-labeled distance oracle with Õ(s(n)) 1 space and Õ(q(n)) query time. Using the state-of-the-art distance oracle of Long and Pettie [12], our construction produces a vertex-labeled distance oracle using n 1+o(1) space and query time Õ(1) at one extreme, Õ(n) space and n o(1) query time at the other extreme, as well as such oracles for the full tradeoff between space and query time obtained in their paper. This is the first non-trivial exact vertex-labeled distance oracle for planar graphs and, to our knowledge, for any interesting graph class other than trees.

show abstract

Section: Introductionmentioning

confidence: 70%

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

Evald,

Fredslund-Hansen,

Wulff-Nilsen

2021

Preprint

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“…We will show below that the only additional operation we now need to support is determining the parent of a given unmarked leaf in the original D ′ -modified suffix tree (before the leaves were chopped). This can be done using the nearest colored ancestor data structure of [19] over the D-modified suffix tree. For a tree of size N , it achieves O(log log N ) time per query after O(N )-time preprocessing.…”

Section: Reducing the Spacementioning

confidence: 99%

“…In [19] it is shown that, in order to answer nearest colored ancestor queries in a tree with N nodes, it is enough to store some arrays of total size O(N ) and predecessor data structures for O(colors) subsets of [1 . .…”

Section: Reducing the Spacementioning

confidence: 99%

Internal Dictionary Matching

Charalampopoulos¹,

Kociumaka²,

Mohamed³

et al. 2019

Preprint

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We introduce data structures answering queries concerning the occurrences of patterns from a given dictionary D in fragments of a given string T of length n. The dictionary is internal in the sense that each pattern in D is given as a fragment of T . This way, D takes space proportional to the number of patterns d = |D| rather than their total length, which could be Θ(n • d).In particular, we consider the following types of queries: reporting and counting all occurrences of patterns from D in a fragment T [i . . j] and reporting distinct patterns from D that occur in T [i . . j]. We show how to construct, in O((n + d) log O(1) n) time, a data structure that answers each of these queries in time O(log O(1) n + |output|).The case of counting patterns is much more involved and needs a combination of a locally consistent parsing with orthogonal range searching. Reporting distinct patterns, on the other hand, uses the structure of maximal repetitions in strings. Finally, we provide tight-up to subpolynomial factors-upper and lower bounds for the case of a dynamic dictionary.

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“…The nearest α-node to x is either (1) the nearest α-descendant of x, (2) the nearest α-node to y, or (3) the nearest α-node to y 2 . Based on this observation, the structure of Gawrychowski et al [12] finds these three candidate nodes, and returns the one that is closest to x. Our structure is based on a slightly different observation: The nearest α-node to x is either (1) the nearest α-descendant of x, (2) the nearest α-descendant of y, or (3) the nearest α-non-descendant of y.…”

Section: Structure For Large Alphabetmentioning

confidence: 99%

“…In the nearest colored node problem the goal is to store a tree with colors on the nodes such that given a node x and a color α, the nearest node to x with color α can be found efficiently. Gawrychowski et al [12] gave a data-structure for this problem that uses O(n log n) bits and answers queries in O(log log n) time, where n is the number of nodes in the tree. Additionally, they considered a dynamic version of the problem in which the colors of the nodes can be changed.…”

Section: Introductionmentioning

confidence: 99%

Succinct data structures for nearest colored node in a tree

Tsur

2018

Information Processing Letters

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The nearest colored node in a tree

Cited by 7 publications

References 24 publications

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

Internal Dictionary Matching

Succinct data structures for nearest colored node in a tree

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