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

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

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Unwanted Matching pattern s pace runner text

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Unwanted Matching pattern s pace runner text

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Unwanted Matching pattern s pace runner text

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**Introducing Word Separator #**

# : word separator - special symbol not in S D = S* # : dictionary of words Text T : an element of D+ (T is a sequence T1T2…Tk of k words in D) e.g., T = This#is#a#pen# S = {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# a b aa#b# a#b# #b# b# # a # # b #

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**Normal and Word Suffix Tries**

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

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**Normal and Word Suffix Trees**

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

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**Sizes of Word Suffix Tries and Trees**

For text T = T1T2…Tk 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 = T1T2…Tk 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 = S* # clearly requires constant space (for fixed S). S #

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # a

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # a

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # a a

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # a a

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # a a #

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # a a #

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # b a a # b

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # b a a # b

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**On-line Construction of Word Suffix Trees**

T = aa#b# a,b # b a # a # b #

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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.]**

a # b a,b S,# Different looking!!!! Just change hair!! Body is the same!!

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**Compact Directed Acyclic Word Graphs**

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

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**Sparse CDAWGs T = aa#b# a # b b a # a # b # Word Suffix Tree**

minimization # Sparse Compact Directed Acyclic Word Graph (SCDAWG) Word Suffix Tree

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**Sparse CDAWGs [cont.] T = a#b#a#bab# a # b a # b b # a # a a b a # b #**

minimization a # a a b a # b # # b b # a a b b # # Word Suffix Tree SCDAWG

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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 Different looking!!!! Just change here Body is the same!!**

# b Different looking!!!! a,b S,# Just change here Body is the same!!

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

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

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

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

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

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

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