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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

This study considers the soft capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph G=(V,E), which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v's incident edges covered by the copy is up to a given capacity of v. We extend Bhattacharya et al.'s work [SODA'15 and ICALP'15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with O(log n / epsilon) amortized update time, where n is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a non-uniform and unsplittable demand, and (2) the more general capacitated set cover problem.

Hao-Ting Wei, Wing-Kai Hon, Paul Horn, Chung-Shou Liao, and Kunihiko Sadakane. An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{wei_et_al:LIPIcs.APPROX-RANDOM.2018.27, author = {Wei, Hao-Ting and Hon, Wing-Kai and Horn, Paul and Liao, Chung-Shou and Sadakane, Kunihiko}, title = {{An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.27}, URN = {urn:nbn:de:0030-drops-94312}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.27}, annote = {Keywords: approximation algorithm, dynamic algorithm, primal-dual, vertex cover} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Given a string X[1, n] and a position k in [1, n], the Shortest Unique Substring of X covering k, denoted by S_k, is a substring X[i, j] of X which satisfies the following conditions: (i) i leq k leq j, (ii) i is the only position where there is an occurrence of X[i, j], and (iii) j - i is minimized. The best-known algorithm [Hon et al., ISAAC 2015] can find S k for all k in [1, n] in time O(n) using the string X and additional 2n words of working space. Let tau be a given parameter. We present the following new results. For any given k in [1, n], we can compute S_k via a deterministic algorithm in O(n tau^2 log n tau) time using X and additional O(n/tau) words of working space. For every k in [1, n], we can compute S_k via a deterministic algorithm in O(n tau^2 log n/tau) time using X and additional O(n/tau) words and 4n + o(n) bits of working space. For both problems above, we present an O(n tau log^{c+1} n)-time randomized algorithm that uses n/ log c n words in addition to that mentioned above, where c geq 0 is an arbitrary constant. In this case, the reported string is unique and covers k, but with probability at most n^{-O(1)} , may not be the shortest. As a consequence of our techniques, we also obtain similar space-and-time tradeoffs for a related problem of finding Maximal Unique Matches of two strings [Delcher et al., Nucleic Acids Res. 1999].

Arnab Ganguly, Wing-Kai Hon, Rahul Shah, and Sharma V. Thankachan. Space-Time Trade-Offs for the Shortest Unique Substring Problem. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganguly_et_al:LIPIcs.ISAAC.2016.34, author = {Ganguly, Arnab and Hon, Wing-Kai and Shah, Rahul and Thankachan, Sharma V.}, title = {{Space-Time Trade-Offs for the Shortest Unique Substring Problem}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {34:1--34:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.34}, URN = {urn:nbn:de:0030-drops-68041}, doi = {10.4230/LIPIcs.ISAAC.2016.34}, annote = {Keywords: Suffix Tree, Sparsification, Rabin-Karp Fingerprint, Probabilistic z-Fast Trie, Succinct Data-Structures} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

Let S and S' be two strings of the same length.We consider the following two variants of string matching.
* Parameterized Matching: The characters of S and S' are partitioned into static characters and parameterized characters.
The strings are parameterized match iff the static characters match exactly and there exists a one-to-one function which renames the parameterized characters in S to those in S'.
* Order-Preserving Matching: The strings are order-preserving match iff for any two integers i,j in [1,|S|], S[i] <= S[j] iff S'[i] <= S'[j].
Let P be a collection of d patterns {P_1, P_2, ..., P_d} of total length n characters, which are chosen from an alphabet Sigma.
Given a text T, also over Sigma, we consider the dictionary indexing problem under the above definitions of string matching.
Specifically, the task is to index P, such that we can report all positions j where at least one of the patterns P_i in P is a parameterized-match (resp. order-preserving match) with the same-length substring of $T$ starting at j. Previous best-known indexes occupy O(n * log(n)) bits and can report all occ positions in O(|T| * log(|Sigma|) + occ) time. We present space-efficient indexes that occupy O(n * log(|Sigma|+d) * log(n)) bits and reports all occ positions in O(|T| * (log(|Sigma|) + log_{|Sigma|}(n)) + occ) time for parameterized matching and in O(|T| * log(n) + occ) time for order-preserving matching.

Arnab Ganguly, Wing-Kai Hon, Kunihiko Sadakane, Rahul Shah, Sharma V. Thankachan, and Yilin Yang. Space-Efficient Dictionaries for Parameterized and Order-Preserving Pattern Matching. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganguly_et_al:LIPIcs.CPM.2016.2, author = {Ganguly, Arnab and Hon, Wing-Kai and Sadakane, Kunihiko and Shah, Rahul and Thankachan, Sharma V. and Yang, Yilin}, title = {{Space-Efficient Dictionaries for Parameterized and Order-Preserving Pattern Matching}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {2:1--2:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.2}, URN = {urn:nbn:de:0030-drops-60736}, doi = {10.4230/LIPIcs.CPM.2016.2}, annote = {Keywords: Parameterized Matching, Order-preserving Matching, Dictionary Indexing, Aho-Corasick Automaton, Sparsification} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Two equal-length strings S and S' are a parameterized-match (p-match) iff there exists a one-to-one function that renames the characters in S to those in S'. Let P be a collection of d patterns of total length n characters that are chosen from an alphabet Sigma of cardinality sigma. The task is to index P such that we can support the following operations.
* search(T): given a text T, report all occurrences <j,P_i> such that there exists a pattern P_i in P that is a p-match with the substring T[j,j+|P_i|-1].
* ins(P_i)/del(P_i): modify the index when a pattern P_i is inserted/deleted.
We present a linear-space index that occupies O(n*log n) bits and supports (i) search(T) in worst-case O(|T|*log^2 n + occ) time, where occ is the number of occurrences reported, and (ii) ins(P_i) and del(P_i) in amortized O(|P_i|*polylog(n)) time.
Then, we present a succinct index that occupies (1+o(1))n*log sigma + O(d*log n) bits and supports (i) search(T) in worst-case O(|T|*log^2 n + occ) time, and (ii) ins(P_i) and del(P_i) in amortized O(|P_i|*polylog(n)) time. We also present results related to the semi-dynamic variant of the problem, where deletion is not allowed.

Arnab Ganguly, Wing-Kai Hon, and Rahul Shah. A Framework for Dynamic Parameterized Dictionary Matching. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganguly_et_al:LIPIcs.SWAT.2016.10, author = {Ganguly, Arnab and Hon, Wing-Kai and Shah, Rahul}, title = {{A Framework for Dynamic Parameterized Dictionary Matching}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {10:1--10:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.10}, URN = {urn:nbn:de:0030-drops-60256}, doi = {10.4230/LIPIcs.SWAT.2016.10}, annote = {Keywords: Parameterized Dictionary Indexing, Generalized Suffix Tree, Succinct Data Structures, Sparsification} }

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