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The heaviest induced ancestors problem revisited. (English) Zbl 07286746
Navarro, Gonzalo (ed.) et al., 29th annual symposium on combinatorial pattern matching, CPM 2018, July 2–4, 2018, Qingdao, China. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. LIPIcs – Leibniz Int. Proc. Inform. 105, Article 20, 13 p. (2018).
Summary: We revisit the heaviest induced ancestors problem, which has several interesting applications in string matching. Let $$\mathcal{T}_1$$ and $$\mathcal{T}_2$$ be two weighted trees, where the weight $$\mathsf{W}(u)$$ of a node $$u$$ in either of the two trees is more than the weight of $$u$$’s parent. Additionally, the leaves in both trees are labeled and the labeling of the leaves in $$\mathcal{T}_2$$ is a permutation of those in $$\mathcal{T}_1$$. A node $$x\in \mathcal{T}_1$$ and a node $$y\in\mathcal{T}_2$$ are induced, iff their subtree have at least one common leaf label. A heaviest induced ancestor query $$\mathsf{HIA}(u_1,u_2)$$ is: given a node $$u_1\in\mathcal{T}_1$$ and a node $$u_2\in \mathcal{T}_2$$, output the pair $$(u_1^*,u_2^*)$$ of induced nodes with the highest combined weight $$\mathsf{W}(u^*_1)+ \mathsf{W}(u^*_2)$$, such that $$u_1^*$$ is an ancestor of $$u_1$$ and $$u^*_2$$ is an ancestor of $$u_2$$. Let $$n$$ be the number of nodes in both trees combined and $$\varepsilon>0$$ be an arbitrarily small constant. T. Gagie et al. [“Heaviest induced ancestors andlongest common substring”, Preprint, arXiv:1305.3164] introduced this problem and proposed three solutions with the following space-time trade-offs:
an $$O(n \log^2n)$$-word data structure with $$O(\log n\log\log n)$$ query time
an $$O(n\log n)$$-word data structure with $$O(\log^2 n)$$ query time
an $$O(n)$$-word data structure with $$O(\log^{3+\varepsilon}n)$$ query time.
In this paper, we revisit this problem and present new data structures, with improved bounds. Our results are as follows.
an $$O(n\log n)$$-word data structure with $$O(\log n\log\log n)$$ query time
an $$O(n)$$-word data structure with $$O(\frac{\log^2 n}{\log\log n})$$ query time.
As a corollary, we also improve the LZ compressed index of Gagie et al. [loc. cit.] for answering longest common substring (LCS) queries. Additionally, we show that the LCS after one edit problem of size $$n$$ [A. Amir et al., Algorithmica 82, No. 12, 3707–3743 (2020; Zbl 07272778)] can also be reduced to the heaviest induced ancestors problem over two trees of $$n$$ nodes in total. This yields a straightforward improvement over its current solution of $$O(n\log^3 n)$$ space and $$O(\log^3 n)$$ query time.
For the entire collection see [Zbl 1390.68025].

MSC:
 68W32 Algorithms on strings
Full Text:
References:
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