Suffix Trees and Suffix Arrays

1
Suffix Trees and Suffix Arrays

• Modern Information Retrieval
• by R. Baeza-Yates and B. Ribeiro-Neto
• Addison-Wesley, 1999.
• (Chapter 8)

2
Introduction

• Word-based indexing
• Inverted indices are good for search words
• Queries such as phrases are expensive to solve
  using Inverted files
• For word-based applications, inverted files
  perform better
• Suffix trees and suffix arrays
• complex queries

3
Text Suffixes
This is a text. A text has many words. Words are
made from letters.

• text. A text has many words. Words are made from
  letters.
• text has many words. Words are made from letters.
• many words. Words are made from letters.
• words. Words are made from letters.
• Words are made from letters.
• made from letters.
• letters.

4
The Suffix Trie and Suffix Tree
5
PAT Trees and PAT Arrays

• Information Retrieval Data Structures and
  Algorithms
• by W.B. Frakes and R. Baeza-Yates (Eds.)
  Englewood Cliffs, NJ Prentice Hall, 1992.
• (Chapters 5)

6
PAT Trees and PAT Arrays

• Problems of tradition IR models
• Documents and words are assumed.
• Keywords must be extracted from the text
  (indexing).
• Queries are restricted to keywords.
• New indices for text
• A text is regarded as a long string.
• Each position corresponds to a semi-infinite
  string (sistring).
• No structures and no keywords

7
Semi-infinite Strings

• ExampleText Once upon a time, in a far away
  land sistring 1 Once upon a time sistring
  2 nce upon a time sistring 8 on a time, in a
  sistring 11 a time, in a far sistring 22 a
  far away land
• Compare sistrings 22 lt 11 lt 2 lt 8 lt 1

8
PAT Tree

• PAT TreeA Patricia tree constructed over all the
  possible sistrings of a text
• Patricia tree
• a binary digital tree where the individual bits
  of the keys are used to decide on the branching
• A zero bit will cause a branch to the left
  subtree
• A one bit will cause a branch to the right
  subtree
• each internal node indicates which bit of the
  query is used for branching
• absolute bit position
• a count of the number of bits to skip
• each external node points to a sistring
• the integer displacement to original text

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1
Example
2
2
Text 01100100010111 sistring 1 01100100010111
sistring 2 1100100010111 sistring
3 100100010111 sistring 4 00100010111
sistring 5 0100010111 sistring 6 100010111
sistring 7 00010111 sistring 8 0010111 ...
3
4
2
1
1
2
2
3
4
2
3
5
1
external node sistring (integer
displacement) total displacement of the bit
to be inspected
1
1
1
1
0
0
1
1
1
2
2
0
1
3
2
internal node skip counter pointer
10
1
Text 01100100010111 sistring 1 01100100010111
sistring 2 1100100010111 sistring
3 100100010111 sistring 4 00100010111
sistring 5 0100010111 sistring 6 100010111
sistring 7 00010111 sistring 8 0010111 ...
2
2
2
4
3
3
6
7
3
4
5
1
1
2
2
1
2
4
3
3
2
2
6
7
3
5
5
4
1
4
2
3
4
8
6
3
5
1
Search 00101
?3?6?4?bits????
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Indexing Points

• The above example assumes every position in the
  text is indexed.i.e. n external nodes, one for
  each indexed position in the text
• Word and phrase searchessistrings that are at
  the beginning of words are necessary
• Trade-off between size of the index and search
  requirements

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Prefix searching

• ideaevery subtree of the PAT tree has all the
  sistrings with a given prefix.
• Search proportional to the query lengthexhaust
  the prefix or up to external node.

Search for the prefix 10100 and its answer
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Proximity Searching

• Find all places where s1 is at most a fixed
  (given by a user) number of characters away from
  s2. in 4 ation gt insulation, international,
  information
• Algorithm1. Search for s1 and s2.2. Select the
  smaller answer set from these two sets and
  sort by position.3. Traverse the unsorted answer
  set, searching every position in the sorted
  set and checking if the distance between
  positions satisfying the proximity condition.

sorttraverse timem1 logm1 m2logm1 (assume
m1ltm2)
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Range Searching

• Search for all the strings within a certain
  lexicographical range.
• Ex the range of abc ..acc
• abracadabra, acacia ?
• abacus, acrimonious X
• Algorithm
• Search each end of the defining intervals.
• Collect all the sub-trees between (and including)
  them.

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Longest Repetition Searching

• the match between two different positions of a
  text where this match is the longest in the
  entire text, e.g., 0 1 1 0 0 1 0 0 0 1 0 1 1 1

the tallest internal node gives a pair of
sistrings that match for the greatest number of
characters
Text 01100100010111 sistring 1 01100100010111
sistring 2 1100100010111 sistring
3 100100010111 sistring 4 00100010111 sistring
5 0100010111 sistring 6 100010111 sistring
7 00010111 sistring 8 0010111
1
2
2
4
3
3
2
6
7
3
5
5
1
4
8
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Most Significant or Most Frequent Matching

• The most frequently occurring strings within the
  text database
• e.g., the most frequent trigram
• Find the most frequent trigram
• find the largest subtree at a distance 3
  characters from root

1
the tallest internal node gives a pair of
sistrings that match for the greatest number of
characters
2
2
4
3
3
i.e., 1, 2, 3 are the same for sistrings
100100010111 and 100010111
2
6
7
3
5
5
1
4
8
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Building PAT Trees as Patricia Trees (1)

• Bucketing of external nodes
• collect more than one external node
• a bucket replaces any subtree with size less than
  a certain constraint (b)save significant number
  of internal nodes
• the external nodes inside a bucket do not have
  any structure associated with themincrease the
  number of comparisons for each search

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Building PAT Trees as Patricia Trees (2)

• Mapping the tree onto the disk using super-nodes
• Advantage save the number of disk access and
  space
• Every disk page has a single entry point,
  contains as much of the trees as possible, and
• terminates either in external nodes or in
  pointers to other disk pages
• The pointers in internal nodes will address
  either a disk page or another node inside the
  same page
• reduces the storage cost of internal nodes
• Example
• Assume a disk page contains on the order of 1,000
  internal/external nodes
• on the average, each disk page contains about 10
  steps of a root-to-leaf path

19
PAT Trees Represented as Arrays

• External node bucket size, b
• If we keep the external nodes in the bucket in
  the same relative order as they would be in the
  tree
• Indirect binary search vs. sequential search

PAT array
1
7
4
8
5
1
6
3
2
2
2
2
4
3
3
0 1 1 0 0 1 0 0 0 1 0 1 1 1 ...
Text
6
7
3
5
5
1
4
8
20
Searching PAT Trees as Arrays

• Prefix searching and range searchingdoing an
  indirect binary search over the array with the
  results of the comparisons being less than,
  equal, and greater than.
• ExampleSearch for the prefix 100 and its answer
• Most frequent, Longest repetition
• Manber and Baeza-Yates (1991)

PAT array
7
4
8
5
1
6
3
2
0 1 1 0 0 1 0 0 0 1 0 1 1 1 ...
Text
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Comparisons

• Signature files
• Use hashing techniques to produce an index
• Advantage
• storage overhead is small (10-20)
• Disadvantages
• the search time on the index is linear
• some answers may not match the query, thus
  filtering must be done

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Comparisons (Continued)

• Inverted files
• storage overhead (30 100)
• search time for word searches is logarithmic
• PAT arrays
• potential use in other kind of searches
• phrases
• regular expression searching
• approximate string searching
• longest repetitions
• most frequent searching