1. Suffix Array: Data structures and applications
Zhifeng Liu
School of Computer Science,
Univ. of Waterloo
Dec. 06, 2004

2. Outline Introduction Suffix array and enhanced suffix array
An example - Is P a substring of S?
Conclusions
References

3. Introduction Why suffix array?
Suffix tree’s drawbacks :
Space consumption: 20n (n=|S|, string length) [Kur99]
Memory Locality: Loss of efficiency
Suffix array (PAT array):
Manber & Myers[Man93]
also Gonnet & Baeza-Yates[Gon93]

4. Suffix array Definition & an example
Informal Definition:
same information as a suffix tree but more compact.
Suffixes in an alphabetic order
Example: the suffix array for banana# is: # a#
ana#
anana#
banana#
na#
nana#
From Prof. Brown’s Assign 2’s Handout

5. Suffix array isn’t perfect either
Less space: 4n but
direct constructing time: O(nlogn)
Linear constructing time via suffix tree but sacrifices space
Binary search for a substring P takes O(mlogn) (m=|P|)
So enhanced suffix array!

6. Enhanced Suffix Array =suffix array+ additional tables
suftab
lcptab
Ssuftab[i] 2
aaacatat$ 1 3
aacatat$
acaaacatat$ 4
acatat$ 6
atat$ 5 8
at$
caaacatat$ 7
catat$
tat$ 9 t$ 10 $
0-[0,10]
1-[0,5]
2-[6,7]
1-[8,9]
2-[0..1]
3-[2..3]
2-[4..5]
Fig1 The enhanced array for S=acaaacatat$ and its lcp-interval tree
Adapted from [Abo04]

7. Enhanced Suffix Array(2)
0-[0,10]
1-[0,5]
2-[6,7]
1-[8,9]
2-[0..1]
3-[2..3]
2-[4..5] 10 ca a t $ 1 5 7 9 2 3 4 6 8
a..$
c..$
t..$
Fig2 The lcp-interval tree vs suffix tree for S=acaaacatat$

8. Enhanced Suffix Array(3) -the more tables, the more likely a suffix tree?
0-[0,10]
1-[0,5]
2-[6,7]
1-[8,9]
2-[0..1]
3-[2..3]
2-[4..5]
ChildTab: Up, down and next fields to record the parent-child,sibling relationships.
The lcp-interval tree is like a suffix tree. However, it is virtual but can simulate suffix tree traversal efficiently.
Fig3 ChildTab records the linked relationship in the lcp-interval tree

9. Enhanced suffix array replaces suffix tree
Every algorithms using suffix tree can be systematically replaced by (enhanced) suffix array in the same time complexity
Bottom-up traversal of suffix tree ->suffix array with lcptab and lcp-interval tree
Top-down traversal of suffix tree->suffix array with childtab
Answer Decision Query

10. Answer Decision Queries
Algorithm Answering decision queries
c := 0
queryFound := true
(i, j ) := getInterval(0,n,P[c])
while (i, j ) <>⊥ and c<m and queryFound = True
if i <> j then
l := getlcp(i, j )
min := min{l, m} queryFound := S[suftab[i]+ c..suftab[i]+min − 1] = P[c..min− 1]
c := min
(i, j ) := getInterval(i, j,P[c])
else queryFound := S[suftab[i]+ c..suftab[i]+ m− 1] = P[c..m− 1]
if queryFound then
Report [i, j] as a occurrence of P
else print(P is not found in S)

11. Answer Decision Queries (cont’d)
Ssuftab[i]
aaacatat$
aacatat$
acaaacatat$
acatat$
atat$
at$
caaacatat$
catat$
tat$ t$ $
P=cb
P=caaa
Longest common string

12. Additional tables eat too much space?
There are tricks to reduce space requirements.
If string length n=|S| <232,each integer index needs 4 bytes.
suftab needs 4n; lcptab also needs 4n?
No! Usually only a few entries in lcptab >255. So
Store each entry in lcptab with 1 byte and allocate another table for long-lcp-values
Space saved, time efficiency reserved though the worst-case time complexity may be affected

13. Conclusions
Suffix array: there is always a tension between space and speed. Research tries to release the tension;
Suffix array can replace with suffix tree;
Suffix array is practical:
Faster and easier to implement

14. References
[Abo04] Mohamed Ibrahim Abouelhoda , Stefan Kurtz , Enno Ohlebusch, Replacing suffix trees with enhanced suffix arrays, Journal of Discrete Algorithms, Volume 2,  Issue 1  (March 2004) p.54-86
[Abo02A] Mohamed Ibrahim Abouelhoda , Stefan Kurtz , Enno Ohlebusch, The Enhanced Suffix Array and Its Applications to Genome Analysis, Proceedings of the Second International Workshop on Algorithms in Bioinformatics, September 17-21,2002, p
[Abo02B] Mohamed Ibrahim Abouelhoda , Enno Ohlebusch , Stefan Kurtz, Optimal Exact Strring Matching Based on Suffix Arrays, Proceedings of the 9th International Symposium on String Processing and Information Retrieval, p.31-43, September 11-13, 2002
[Gon92] Gaston H. Gonnet , Ricardo A. Baeza-Yates , Tim Snider, New indices for text: PAT Trees and PAT arrays, Information retrieval: data structures and algorithms, Prentice-Hall, Inc., Upper Saddle River, NJ, 1992
[Kur99] S. Kurtz, Reducing the space requirement of suffix trees, Software—Practice and Experience 29 (13) (1999) 1149–1171.
[Man93] Udi Manber , Gene Myers, Suffix arrays: a new method for on-line string searches, SIAM Journal on Computing, v.22 n.5, p , Oct. 1993