The LCA Problem Revisited

Bender, Michael A.; Farach-Colton, Martı́n

doi:10.1007/10719839_9

Cited by 537 publications

(558 citation statements)

References 3 publications

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“…Finally, we are ready to prove (2). Take any a, a ∈ A with lca T 1 (a, a ) = u 1 and lca T 2 (a, a ) = u 2 as in the preceding paragraph.…”

Section: Lemma 13mentioning

confidence: 87%

“…An R* consensus tree of T 1 and T 2 is a tree τ with Λ(τ ) = L which satisfies r(τ ) ⊆ R maj and which maximizes the number of internal nodes. 2 The next two lemmas describe some useful properties of the strong clusters of R maj .…”

Section: R* Consensus Treesmentioning

confidence: 99%

“…To compute count r,r , count r,f , and count f,r , we could preprocess T 1 and T 2 in O(n) time so that lowest common ancestor queries in T 1 and T 2 could be answered in O(1) time [2,12]. Then, for any given a, b ∈ L, a brute-force solution would easily obtain all of count r,r (a, b), count r,f (a, b), and count f,r (a, b) ,r (a, b), count r,f (a, b), and count f,r (a, b) for all a, b ∈ L. In the rest of this section, we show how to obtain these values more efficiently.…”

Section: Constructing the R* Consensus Treementioning

confidence: 99%

See 2 more Smart Citations

Constructing the R* Consensus Tree of Two Trees in Subcubic Time

Jansson

Sung

2012

Algorithmica

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The previously fastest algorithms for computing the R* consensus tree of two given (rooted) phylogenetic trees with a leaf label set of cardinality n run in Θ(n 3 ) time (Bryant and Berry in Adv. Appl. Math. 27(4): 705-732, 2001; Kannan et al. in SIAM J. Comput. 27(6):1695-1724, 1998. In this manuscript, we describe a new O(n 2 √ log n)-time algorithm to solve the problem. This is a significant improvement because the R* consensus tree is defined in terms of a set R maj which may contain Ω(n 3 ) elements, so any direct approach that explicitly constructs R maj requires Ω(n 3 ) time.

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“…Finally, we are ready to prove (2). Take any a, a ∈ A with lca T 1 (a, a ) = u 1 and lca T 2 (a, a ) = u 2 as in the preceding paragraph.…”

Section: Lemma 13mentioning

confidence: 87%

Section: R* Consensus Treesmentioning

confidence: 99%

Section: Constructing the R* Consensus Treementioning

confidence: 99%

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Constructing the R* Consensus Tree of Two Trees in Subcubic Time

Jansson

Sung

2012

Algorithmica

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“…There exists a linear space RMQ data structure that supports queries in constant time, see e.g., [4]. An RMQ query on the LCP array computes the length of the longest common prefix of two suffixes GSA[r 1 ] and GSA[r 2 ], denoted LCP (r 1 , r 2 ).…”

Section: Countingmentioning

confidence: 99%

Cross-Document Pattern Matching

Kucherov

Nekrich

Starikovskaya

2012

Combinatorial Pattern Matching

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Abstract. We study a new variant of the string matching problem called cross-document string matching, which is the problem of indexing a collection of documents to support an efficient search for a pattern in a selected document, where the pattern itself is a substring of another document. Several variants of this problem are considered, and efficient linear-space solutions are proposed with query time bounds that either do not depend at all on the pattern size or depend on it in a very limited way (doubly logarithmic). As a side result, we propose an improved solution to the weighted level ancestor problem.

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“…Subsequently, simpler algorithms with better constant factors have been proposed in [53,13,28,58,14]. On a pointer machine, [36] show a lower bound of Ω(log log n) on the query time, which matches the upper bound of [60].…”

Section: Distance Queries In Treesmentioning

confidence: 56%

Labeling Schemes with Queries

Korman

Kutten

Structural Information and Communication Complexity

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We study the question of "how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels". Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, the value f (u, v) can be inferred by merely inspecting the labels of u and v.This paper introduces a natural generalization: the notion of f -labeling schemes with queries, in which the value f (u, v) can be inferred by inspecting not only the labels of u and v but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one query) enables us to reduce the label size of many labeling schemes significantly. In particular, we show that to support the distance function on n-node trees as well as the flow function on n-node general graphs, O(log n + log W )-bit labels are sufficient and necessary, where W is the maximum (integral) capacity of an edge. We note that it was shown that any labeling scheme (without queries) supporting either the flow function on general graphs or the distance function on trees, must have label size Ω(log 2 n + log n log W ). Using a single query, we also show a routing labeling scheme in the fixed-port model using O(log n)-bit labels, while the lower bound on the label size of any such routing labeling scheme (without queries) is Ω(log 2 n/log log n). We note that all our labeling schemes with queries have asymptotically optimal label size.We then show several extensions. In particular, we show that our labeling schemes with queries on trees can be extended to the dynamic scenario by some adaptations of known model translation methods. Second, we show that the study of the queries model can help with the traditional model too. That is, using ideas from our routing labeling scheme with one query, we show how to construct a 3-approximation routing scheme without queries in the fixed-port model with Θ(log n)-bit labels. Finally, we turn to a non-distributed environment and first translate known results on finding NCA to supporting distance queries. We then use the above to show that one can preprocess a general weighted graph using almost linear space so that flow queries can be answered in almost constant time.

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The LCA Problem Revisited

Cited by 537 publications

References 3 publications

Constructing the R* Consensus Tree of Two Trees in Subcubic Time

Constructing the R* Consensus Tree of Two Trees in Subcubic Time

Cross-Document Pattern Matching

Labeling Schemes with Queries

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