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

4. 1 11 19 28 33 40 46 50 60 This is a text. A text has many words. Words are made from letters. 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 < 11 < 2 < 8 < 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

9. 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 ... 4 3 1 2 1 2 2 4 3 3 2 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 3 4 2 3 7 6 4 3 5 1 1 2 2 1 3 4 2 3 2 2 7 6 5 3 5 4 1 4 3 2 4 8 6 3 5 1 Search 00101 註：3和6要4個bits才能區辨

11. 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

12. 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

13. 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 ==> 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. sort+traverse time:m1 logm1 +m2logm1 (assume m1<m2)

14. 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.

15. 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 00 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 3 4 3 2 7 6 5 3 5 1 4 8

16. “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 3 4 3 i.e., 1, 2, 3 are the same for sistrings 100100010111 and 100010111 2 7 6 5 3 5 1 4 8

17. 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

18. 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 3 4 2 3 0 1 1 0 0 1 0 0 0 1 0 1 1 1 ... Text 7 6 5 3 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 01 0 0 0 1 0 1 1 1 ... Text

21. 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

22. Comparisons (Continued) • Inverted files • storage overhead (30% ~ 100%) • search time for word searches is logarithmic • PAT arrays • potential use in otherkind of searches • phrases • regular expression searching • approximate string searching • longest repetitions • most frequent searching