Space Efficient Suffix Array Construction using Induced Sorting LMS Substrings

International Journal of Information Sciences and Techniques (IJIST) Vol.3, No.4, July 2013
DOI : 10.5121/ijist.2013.3402 11
Space Efficient Suffix Array Construction using
Induced Sorting LMS Substrings
Rajesh. Yelchuri1
, Nagamalleswara Rao.N2
Department of Computer Science and Engineering, R.V.R & J.C College of Engg.
Chowdavaram, Guntur, Andhra Pradesh -522119,India
1
rajesh.yelchuri@gmail.com
2
nnmr_m@yahoo.com
ABSTRACT
This paper presents, an space efficient algorithm for linear time suffix array construction. The algorithm
uses the techniques of divide-and-conquer, and recursion. What differentiates the proposed algorithm from
the variable-length leftmost S-type (LMS) substrings is the efficient usage of the memory to construct the
suffix array. The modified induced sorting algorithm for the variable-length LMS substrings uses efficient
usage of the memory space than the existing variable length left most S-type(LMS) substrings algorithm
KEYWORDS
Divide and Conquer, Suffix Array.
1. INTRODUCTION
This document describes, the concept of suffix arrays was introduced by Manber and Myers in
SODA’90 [4] and SICOMP’93 [3] as a space efficient alternative to suffix trees. It has been well
recognized as a fundamental data structure, useful for a broad range of applications, for e.g.,
string search, data indexing, searching for patterns in DNA or protein sequences, data
compression and also in Burrows-Wheeler transformation. For an n-character string, denoted by
STR, its suffix array, denoted by SAR(STR), is an array of indices pointing to all the suffixes of
STR, sorted according to their ascending(or descending) lexicographical order. The suffix array
of STR itself requires only n[log n]-bit space. However, different suffix array construction
algorithms may require different space and time complexities. During the past decade, a many
researches have been developing suffix array construction algorithms that are both time and space
efficient, for which we suggest a detailed survey from Puglisi [5]. Time and space efficient suffix
array construction algorithms has become popular because of their wide usage. Construction of
suffix arrays are needed for large scale applications, e.g., biological genome database and web
searching and, where the size of a huge data set is measured in billions of characters [6], [7], [8],
[9], [10].Time and space efficient linear time algorithms are crucial for large-scale applications to
have predictable worst-case performance. The three known algorithms are KSP [1],KA [12],
[13],KS [11], [2] all are reported in 2003.
2. BASIC NOTATIONS
In this section we bring out some basic terminology, used in the presentation of the algorithm. Let
STR be a string of n characters in an array [0..n-1], and ∑(STR) be the alphabet of STR. To
denote a substring in STR where i and j ranges from 0 to n-1,i<j, we denote it as STR[i..j]. For
simplicity assume, STR is supposed to be terminated by a character called as sentinel and

12
represented by $, which is the unique lexicographically smallest character in STR. Let
suffix(STR, i) be the suffix in STR starting at STR[i] and running to the end of the character
array i.e. to the sentinel.
A suffix suffix(STR,i) is called as S-type or L-type, if suffix(STR, i) < suffix(STR,i+1) or
suffix(STR, i) > suffix(STR,i+1), respectively. The last suffix suffix(STR,n-1) consisting of only
the single character $ (the sentinel) which is predefined as S-type. We can classify a character
STR[i] to be S-type or L-type. To store the type of every character/ suffix, we introduce an n-bit
Boolean array b, where b[i] records the type of character STR[i] as well as suffix(STR, i): 1 for S-
type and 0 for L-type. From the S-type and L-type descriptions, we observed the following
properties:
Property 1:STR[i] is S-type, if STR[i] < STR[i+1] or
STR[i]=STR[i+1] and suffix(STR,i+1) is S-type.
Property 2:STR[i] is L-type, if STR[i] > STR[i+1] or
STR[i]=STR[i+1] and suffix(STR,i+1) is L-type.
By reading STR once from right to left, we can store the type of each character/suffix into type
array ‘b’ in O(n) time.
As defined earlier, SAR(STR) (the notation of SAR is used for it when there is no confusion in
the context), i.e., the suffix array of STR, stores the indices of all the suffixes of STR according to
their lexicographical order. We observe that the pointers for all the suffixes beginning with a
same character must span successively. Let us call a sub array in SAR for all the suffixes with the
same first character as a bucket, where the head and the tail of a bucket refer to the first and the
last items of the bucket. There must be no tie between any two suffixes sharing the identical
character but of different types i.e., in the same bucket, all the suffixes of the same type are
grouped together and the S-type suffixes are to the right of the L-type suffixes [12], [13].
Therefore, each bucket can be divided into two sub-buckets with respect to the types of suffixes
inside i.e. the L and S-type buckets, where the S-type bucket is on the right of the L-type bucket.
3. Existing Algorithm: INDUCED SORTING VARIABLE LENGTH LMS
SUBSTRINGS
A. Algorithm Framework
The framework of existing linear time suffix array sorting algorithm SAR-IS[15] that samples
and sorts the variable-length LMS-substrings, is given in section III-C. Lines 1 to 4 give the
reduced problem, which is then again recursively solved by the lines 5-8, and finally from the
solution of the reduced problem, Line 9 induces the final solution for the original problem.
B. Basic Definitions
We start by introducing the terms of leftmost S-type (LMS) character, suffix, and substring as
follows:
Definition 1:(LMS Character/Suffix) A character STR[i], iЄ[1,n-1] is called LMS, if STR[i] is S-
type and STR[i-1] is L-type. A suffix suffix(STR,i) is called LMS, if STR[i] is an LMS character.

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Definition 2: (LMS-Substring) An LMS-substring is (i) a substring STR[i..j] with both STR[i]
and STR[j] being LMS characters and there exists no other LMS character in the substring, for i
≠ j; or (ii) the sentinel itself. If we treat the LMS-substrings as elementary blocks of the string, we
can effortlessly sort all the LMS substrings, then by using the order index of each LMS substring
as its name, and replace all of the LMS-substrings in STR by their names. Therefore, the string
STR can be represented by a shortened string, denoted by R1, thus the problem size can be further
minimized to fast up solving the problem in divide-and-conquer manner
Definition 3: (Order of Substring) To find out the order of any two LMS-substrings, first
compare their corresponding characters from left to right. For each pair of characters, compare
their lexicographical values first and then their types, if the two characters are of the same
lexicographical value, where the S-type is taken as highest priority than the L-type. From this
definition ,we see that two LMS-substrings can be of the same order index, i.e., the same name, if
they have same, in terms of the lengths, and the characters, and the types. Assigning the S-type
character a higher priority is based on a property directly from the definitions of L-type and S-
type suffixes in [12]: suffix(STR, i)> suffix(STR, j), if (1) STR[i] > STR[j], or (2)
STR[i]=STR[j], suffix(STR, i)and suffix(STR, j) are S-type and L-type, respectively. To sort all
the LMS-substrings, no excess physical space is essential for storing them. We simply maintain a
pointer array, denoted by P1, which contains the pointers for all the LMS-substrings in STR and
can be made by scanning STR or by reading the Boolean array b once from right to left in O(n)
time.
Definition 4: (Pointer Array P1) is an array which has the pointers for all the LMS substrings in
STR with their original positional order being conserved. If we have all the LMS substrings
sorted in the buckets in their lexicographical order, where all the LMS substrings in a bucket are
identical, now we name each and every item of the pointer array P1 by the index of its bucket to
result in a revived string R1. We say the two equal size substrings STR[i..j] and STR[i′..j′] are
identical, if and only if STR[ i + k]=STR[i′ +k] and b[i +k]=b[i′ +k], for k Є [0,j-i].
C. Algorithm
SAR-IS(STR,SAR)
STR- is input string;
SAR-output of suffix array of STR;
b:array[0..n-1] of Boolean;
P1,R1:array[0....n1] of integer; n1=||R1||
BKT:array[0..||∑(STR)||-1] of integer;
Step 1. Scan STR once to classify all the characters as L-Type or S-Type into b;
Step 2. Scan b once to find all LMS –substrings in STR into P1;
Step 3. Induced sort all the LMS-substrings using P1 and BKT;
Step 4. Name each LMS-substring in STR by its bucket index to get a new shortened string R1;
Step 5. if each character in R1 is unique then
Step 6. Directly compute SAR1 from R1;
Step 7. else
Step 8. SAR-IS(R1,SAR1); //Recursive call

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Step 9. Induce SAR from R1;
Step 10. Return
The above mentioned algorithm is the existing one.
4. Proposed Algorithm
In SA-IS, the additional working space is mainly composed of the bucket counter array ‘BKT’
and the type array ‘t’ at each recursion level. Our proposed algorithm differs from the existing
one in two cases. They are
1. We use the MSB bit of the suffix array to store the type of the character(S-type or L-
type) thereby avoiding the space needed for the type array ‘t’ suggested in the existing
algorithm.
2. We reuse the unused space in SAR for the bucket array BKT.
We have observed that the input STR has been reduced to at least n/3 at the initial level (level-0)
for the standard suffix array datasets .So, we can use of the unused space of SAR for the variable
BKT in deeper levels rather than creating memory using malloc. As, in the existing algorithm -1
is used as initialization (default) value for suffix array SAR. In the proposed algorithm we use
0X7FFFFFFF as initialization value for suffix array SAR as the MSB bit is used to classify the S-
type or L-type characters. Here we assume a 32-bit machine and the integer occupies 4-bytes.
The variable Buf_ptr is used which records the start address of the unused space of SAR at initial
level(i.e level-0) so that we can reuse this space in the next levels (i.e. from 1st Level) for the
bucket array (See Fig 1). We can also make use of this space for the L or S-type arrays if the
space is still available.
As we can see the space of SAR0 is reused for the level-1 because the size of the problem gets
decreased as the level progresses.
4.1 Algorithm
SAR-IS (STR, SAR)
STR- is input string;
SAR-output of suffix array of STR;
P1, R1: array [0...n1] of integer; n1=||R1||
BKT: array [0...||∑ (STR) ||-1] of integer; //uses unused space in subsequent iterations
Buf_ptr : pointer to unused space in SAR
Step 1. Scan STR once to classify all the characters as L-Type or S-Type into MSB bits of SAR;
Step 2. Scan MSB’s of SAR once to find all LMS substrings in STR into P1;
Step 3. if level Not Equal to 0 then
BKT=buf_ptr;//assign the start address of unused buffer
Step 4. Induced sort all the LMS-substrings using P1 and BKT;

15
Step 5. Name each LMS-substring in STR by its bucket index to get a new shortened string R1;
Step 6. If level Equal to 0 then assign the start address of
unused space of SAR to buf_ptr.
Step 7. if Each character in R1 is unique then
Step 8. Directly compute SAR1 from R1;
Step 9. else SAR-IS(R1,SAR1); //Recursive call
Step 10.Once again scan STR to classify all the characters as L-Type or S-Type into MSB bits
of SAR;
Step 11.Induce SAR from SAR1;
Step 12.return
Fig 1.Example for the re usage of the buffer SAR
The re usage of the buffer is illustrated in Fig 1.The notation L 0, L 1, L 2 stands for Level-0,
Level-1, Level-2.
4.2 Experimental Results
The algorithm was implemented in VC++ using the Microsoft Visual Studio under Windows XP
platform. The Table II and Fig 2 give the overview of the space consumed by the existing and the
proposed algorithms. The data sets in Table I used in our experiment are downloaded from
Canterbury [14] and Manzini-Ferragina[16].
Dataset ||∑||,Characters
bible.txt 63,4047392
chr22.dna 4,34553758
e.coli 4,4638690
howto 197,39422105
world192.txt 94.2473400
sprot34.dat 66,109617186
etext99 146,105277340
rfc 120,116,421,901
rctail196 93,114,711,151
linux-2.4.5.tar 256,21,508,430
w3c2 256,104,201,579
alphabet 26,100000
random 26,100000
TABLE I Datasets used in the Experiment
L 2
L 1
L 0 B0
SAR0
R1 SAR1 BKT1
R2 SA R 2 BKT2
R1

16
1
2
4
8
16
32
64
128
256
512
1024
Existing Algorithm
Proposed Algorithm
Dataset Space(in Mega Bytes)
Existing
Algorithm
Proposed
Algorithm
bible.txt 21.81 20.10
chr22.dna 179.10 165.85
e.coli 25.25 22.9
howto 204.47 189.11
world192.txt 13.61 12.58
sprot34.dat 556.57 524.48
etext99 544.14 503.74
rfc 590.53 556.99
rctail196 577.29 548.81
linux-2.4.5.tar 130.82 103.53
w3c2 521.11 498.60
alphabet 1.35 1.23
random 1.48 1.23
TABLE II Space Consumed by the Existing and Proposed Algorithm
Fig 2. Logarithmic graph (base 2) showing the comparison between Existing and Proposed Algorithm
The datasets that are in Table I are downloaded from the benchmark repositories for SACAs,
which includes Canterbury [14], Manzini-Ferragina[16].These datasets have constant alphabets
with sizes less than or equal to 256 and one byte is taken for each character.

17
4.3 Conclusions
The proposed algorithm makes the algorithm space efficient by using the MSB bit of SAR to
classify L-type and S-type characters and reuses the space of SAR for the bucket array at each
level there by reducing nearly 25% of the space needed when compared to the existing
algorithm. The results for the various data sets are shown in the Table II.
REFERENCES
[1] D.K. Kim, J.S. Sim, H. Park, and K. Park, “Linear-Time Construction of Suffix Arrays,” Proc.
Ann. Symp Combinatorial Pattern Matching (CPM ’03), pp. 186-199. 2003.
[2] J. Karkkainen, P. Sanders, and S. Burkhardt, “Linear Work Suffix Array Construction,” J. ACM,
no. 6, pp. 918-936, Nov. 2006.
[3] U. Manber and G. Myers, “Suffix Arrays: A New Method for On-Line String Searches,” SIAM J.
Computing, vol. 22, no. 5, pp. 935-948, 1993.
[4] U. Manber and G. Myers, “Suffix Arrays: A New Method for On-Line String Searches,” Proc.
First Ann. ACM-SIAM Symp. Discrete Algorithms (SODA ’90), pp. 319-327, 1990.
[5] S.J. Puglisi, W.F. Smyth, and A.H. Turpin, “A Taxonomy of Suffix Array Construction
Algorithms,” ACM Computing Surveys, vol. 39, no. 2, pp. 1-31, 2007.
[6] R. Grossi and J.S. Vitter, “Compressed Suffix Arrays and Suffix Trees with Applications to Text
Indexing and String Matching,” Proc. Symp. Theory of Computing (STOC ’00), pp. 397-406,
2000.
[7] T.W. Lam, K. Sadakane, W.K. Sung, and S.M. Yiu, “A Space and Time Efficient Algorithm for
Constructing Compressed Suffix Arrays,” Proc. Int’l Conf. Computing and Combinatorics, pp.
401-410, 2002.
[8] G. Manzini and P. Ferragina, “Engineering a Lightweight Suffix Array Construction Algorithm,”
Algorithmica, vol. 40, no. 1, pp. 33- 50, Sept. 2004.
[9] S. Kurtz, “Reducing the Space Requirement of Suffix Trees,” Software Practice and Experience,
vol. 29, pp. 1149-1171, 1999.
[10] W.K. Hon, K. Sadakane, and W.K. Sung, “Breaking a Time-and-Space Barrier for Constructing
Full-Text Indices,” Proc. 44th Ann. IEEE Symp. Foundations of Computer Science (FOCS ’03),
pp. 251-260, 2003.
[11] J. Karkkainen and P. Sanders, “Simple Linear Work Suffix Array Construction,” Proc. 30th Int’l
Conf. Automata, Languages, and Programming (ICALP ’03), pp. 943-955, 2003.
[12] P. Ko and S. Aluru, “Space Efficient Linear Time Construction of Suffix Arrays,” Proc. Ann.
Symp. Combinatorial Pattern Matching(CPM ’03), pp. 200-210. 2003.
[13] P. Ko and S. Aluru, “Space-Efficient Linear Time Construction of Suffix Arrays,” J. Discrete
Algorithms, vol. 3, nos. 2-4, pp. 143-156, 2005
[14] The Canterbury Corpus website. [Online]. Available: http://corpus.canterbury.ac.nz/.
[15] GeNong, Sen Zhang, Wai Hong Chan, “Two Efficient Algorithms for Linear Time Suffix Array
Construction”, IEE Transactions on Computers, vol. 60, pp.1471-1484,Oct.2011.

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[16] Light weight corpus datasets [Online].Available:
http://people.unipmn.it/manzini/lightweight/corpus

Space Efficient Suffix Array Construction using Induced Sorting LMS Substrings

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Space Efficient Suffix Array Construction using Induced Sorting LMS Substrings