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Sparse Compact Directed Acyclic Word Graphs Shunsuke Inenaga (Japan Society for the Promotion of Science & Kyushu University) Masayuki Takeda (Kyushu University & Japan Science and Technology Agency)

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Traditional Pattern Matching Problem Given ： text T in and pattern P in Return ： whether or not P appears in T : alphabet (set of characters) : set of strings A text indexing structure for T enables you to solve the above problem in O(m) time (for fixed ). m: the length of P

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Suffix Trie A trie representing all suffixes of T a a c b $ c b $ c b $ b $ $ T = aacb$ aacb$ acb$ cb$ b$ $

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Unwanted Matching pace runner pattern s text

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Unwanted Matching pace runner pattern s text

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Unwanted Matching pace runner pattern s text

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Introducing Word Separator # # : word separator - special symbol not in D = # : dictionary of words Text T : an element of D + (T is a sequence T 1 T 2 …T k of k words in D) e.g., T = This#is#a#pen# = {A,…,z} D = {...,This#,...,a#,...is#,...pen#,...}

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Word-level Pattern Matching Problem Given: text T in D + and pattern P in D + Return: whether or not P appears at the beginning of any word in T e.g. T = The#space#runner#is#not#your#good#pace#runner# P = pace#runner#

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Word-level Pattern Matching Problem Given: text T in D + and pattern P in D + Return: whether or not P appears at the beginning of any word in T e.g. T = The#space#runner#is#not#your#good#pace#runner# P = pace#runner#

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Word Suffix Trie A trie representing the suffixes of T which begin at a word. T = aa#b# aa#b# a#b# #b# b# # a a # b # b #

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Normal and Word Suffix Tries a a # b # b # a a # b # # b # # b # b # T = aa#b# Normal Suffix Trie Word Suffix Trie

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Normal and Word Suffix Trees a a # b # b # a a # b # # b # # b # b # T = aa#b# Normal Suffix Tree Word Suffix Tree

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Sizes of Word Suffix Tries and Trees For text T = T 1 T 2 …T k of length n, the word suffix trie of T requires O(nk) space, but the word suffix tree of T requires O(k) space!! because the word suffix tree has only k leaves and has only branching internal nodes.

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Construction of Word Suffix Trees Algorithm by Andersson et al. （ 1996 ） for text T = T 1 T 2 …T k of length n, constructs word suffix trees in O(n) expected time with O(k) space. Our algorithm (CPM’06) builds word suffix trees in O(n) time in the worst case, with O(k) space.

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Our Construction Algorithm We modify Ukkonen’s on-line normal suffix tree construction algorithm by using minimum DFA accepting dictionary D We replace the root node of the suffix tree with the final state of the DFA.

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Minimum DFA The minimum DFA accepting D = # clearly requires constant space (for fixed ). #

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On-line Construction of Word Suffix Trees T = aa#b# a,b a #

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T = aa#b# a,b a # On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a # On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a # On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a # b b On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a # b b On-line Construction of Word Suffix Trees

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T = aa#b# a,b # a a # b b # # On-line Construction of Word Suffix Trees

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Pseudo-Code Just change here

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Like Dress-up Doll...

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Like Dress-up Doll... [cont.] Just change hair!! Body is the same!! a a # b # b # a a # b # # b # # b # b # a,b # # Different looking!!!!

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Compact Directed Acyclic Word Graphs a a # b # # b # # b # b # T = aa#b# Suffix Tree a a # b # # b # # b # b # Compact Directed Acyclic Word Graph (CDAWG) minimization

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Sparse CDAWGs a a # b # b # T = aa#b# Word Suffix Tree Sparse Compact Directed Acyclic Word Graph (SCDAWG) minimization a a # b # b #

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Sparse CDAWGs [cont.] a a # b # # T = a#b#a#bab# Word Suffix TreeSCDAWG minimization # b b a a b # # # b b a # b a # b a a # # # b b a # b a a # b b

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SCDAWG Construction SCDAWGs can be constructed by minimizing word suffix trees in O(k) time. using Revuz’s DAG minimization algorithm (1992)

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SCDAWG Construction [cont.] Question : Direct construction for SCDAWGs? Answer : YES! Using minimal DFA accepting dictionary D, we can directly build SCDAWGs in O(n) time and O(k) space. We modify the CDAWG on-line construction algorithm (Inenaga et al. 05) by using the above DFA.

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Pseudo-Code Just change here Body is the same!! a a # b # # b # # b # b # a a # b # b # Different looking!!!! a,b # #

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Some Events Basically on-line construction of SCDAWGs is similar to that of word suffix trees. Except for the two following unique events: Edge merging Node splitting

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Edge Merging T = a#b#a#bab#bc... a # # b a # b b a,b # a # # b

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Edge Merging T = a#b#a#bab#bc... a # # b a # b b a,b # a # # b a a

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Edge Merging T = a#b#a#bab#bc... a # # b a # b b a,b # a # # b a a a

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Edge Merging T = a#b#a#bab#bc... a # # b a # b b a,b # a a

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Node Splitting T = a#b#a#bab#bc... a # # b a # b b a,b # a a b # b #

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Node Splitting T = a#b#a#bab#bc... a # # b a # b b a,b # a a b # b # b b

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Node Splitting T = a#b#a#bab#bc... a # # b a # b b a,b # a a b # b # b b a # # b a a b # b # b b

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Conclusion We introduced new text indexing structure sparse compact directed acyclic word graphs (SCDAWGs) for word-level pattern matching. We presented an on-line algorithm to construct SCDAWGs directly, in O(n) time with O(k) space. The key is the use of minimum DFA accepting dictionary D.

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Related Work “Sparse Directed Acyclic Word Graphs” by Shunsuke Inenaga and Masayuki Takeda Accepted to SPIRE’06

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