OPTIMAL PARALLEL SUPERPRIMITIVITY TESTING FOR SQUARE ARRAYS

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1 Parallel Processing Letters, c World Scientific Publishing Company OPTIMAL PARALLEL SUPERPRIMITIVITY TESTING FOR SQUARE ARRAYS COSTAS S. ILIOPOULOS Department of Computer Science, King s College London, Strand London, England and School of Computing, Curtin University Perth, WA, Australia csi@dcs.kcl.ac.uk and MAUREEN KORDA Department of Computer Science, King s College London, Strand London, England mo@dcs.kcl.ac.uk Received January 1996 Revised May 1996 Communicated by G. Beauquier ABSTRACT We present an optimal O(log log n) time algorithm on the CRCW PRAM which tests whether a square array, A, of size n n, is superprimitive. If A is not superprimitive, the algorithm returns the quasiperiod, i.e., the smallest square array that covers A. Keywords: String Algorithms, Parallel Algorithms, PRAM Algorithms, Complexity 1. Introduction The problem of identifying the repetitive structures in a string is common in computer science. A typical regularity of a string is its period. We say that a string x has a period u, when x is a prefix of the string u k for some integer k > 1, where u k is formed by concatenating k copies of u. The period of a string can be generalised to the notions of seed and cover. A cover of x is a substring w such that x can be reconstructed from superpositions and concatenations of w. A seed of x is a substring w such that a superstring of x can be constructed from superpositions and concatenations of w. For example, abc is a period of abcabcabca, abca is a cover of abcabcaabca, and abca is a seed of abcabcaabc. A Partially supported by SERC grants GR/F 00898, GR/L and GR/J 17844, NATO grant CRG , ESPRIT BRA grant 7131 for ALCOM II, and MRC grant G Supported by a Medical Research Council Studentship.

2 variant of the covering problem (see [11]) defined above, was shown to have applications to DNA sequencing by hybridization using oligonucleotide probes. The shortest cover, w of a string x, w < x, is said to be its quasiperiod. If there is no such w, then x is said to be superprimitive. A linear time sequential algorithm was given in [2] for superprimitivity testing which returns the quasiperiod if it exists. Breslauer [3] presented a linear time on-line algorithm for the same problem. Moore and Smyth [14] presented a linear time algorithm for computing all the covers of a given string. Using the PRAM model of computation, Breslauer [4] gave an optimal O(α(n) log log n)-time algorithm for computing the quasiperiod, where α(n) is the inverse Ackermann function; Breslauer [4] also produced a non-optimal algorithm that requires O(log log n) time and O(nlog n/log log n) processors. Iliopoulos and Park [13] gave an optimal O(log log n)-time algorithm for the same problem. The notion of covers was extended to two dimensions in [10]. Given an n n array, A, a square subarray M of A is called a cover of A, if every element A(i,j) is contained within some occurrence of M in A. A sequential linear time algorithm was presented in [10] for the all-covers problem. The shortest square subarray Q of A that covers A is said to be the quasiperiod of A. If there is no subarray that covers A, then we say that A is superprimitive. This paper considers the parallel complexity of finding the quasiperiod of A and presents an optimal, O(log log n)-time algorithm for superprimitivity testing using the common CRCW PRAM model. While the algorithm in [10] uses the Aho- Corasick automaton, the algorithm presented here uses the notion of witnesses in an elimination procedure and is therefore alphabet independent. The paper is organized as follows: in the next section we present some definitions and results used in the sequel. In Section 3 we present an O(log log n) parallel algorithm for computing all the square borders of a square array. In Section 4 we present a constant time algorithm for performing a bout between two borders. In Section 5 we present an O(log log n)-time algorithm for superprimitivity testing. 2. Preliminaries A string is a sequence of zero or more symbols from an alphabet Σ. The set of all strings over the alphabet Σ is denoted by Σ. A string x of length n is represented by x 1 x n, where x i Σ for 1 i n. A string w is a substring of x if x = uwv for u,v Σ. A string w is a prefix of x if x = wu for u Σ. Similarly, w is a suffix of x if x = uw for u Σ. The string xy is a concatenation of two strings x and y. The concatenations of k copies of x is denoted by x k. A non-empty string u is a period of x if x is a prefix of u k for some integer k > 1, or equivalently if x is a prefix of ux. The period of a string x is the shortest period of x. A two dimensional string is an n m array of nm symbols drawn from an alphabet Σ. The n n square array A can be represented by A[1 n,1 n]; the size n of A is denoted by A. An m m array M is a subarray of A if the upper left corner of M can be aligned with an element A(i,j), 1 i,j n m + 1 and M[1 m,1 m] = A[i i + m 1,j j + m 1]. In this case, the

3 subarray M is said to occur at A(i,j). A subarray M is said to be a prefix of A if M occurs at A(1,1). Similarly, a subarray M is a suffix of A if M occurs at A(n m + 1,n m + 1). A substring w of x is called a cover of x, if x can be constructed by concatenations and superpositions of w. A subarray M of A is called a cover of A, if every element A(i,j) is contained within some occurrence of M. The shortest non-empty square subarray Q of A that covers A is said to be the quasiperiod of A. If there is no subarray that covers A, then we say that A is superprimitive. A string b is a border of x if b is a prefix and a suffix of x. The empty string and x itself are trivial borders of x. A m m subarray M is a border of A if M occurs at positions A(1,1), A(n m + 1,1), A(1,n m + 1) and A(n m + 1,n m+1) (see Figure 1). The empty array and the array A itself are trivial borders of A. M M A M M Figure 1: M is a border of the square array A Fact 1.. A string u is a period of x = ub if and only if b is a non-trivial border of x. Proof. It follows immediately from the definitions of period and border.. Fact 2. A cover of string x is also its border. A cover of array A is also its border. Proof. A cover of a string x occurs as both a prefix and a suffix (in order to cover positions 1 and n) and therefore it is a border of x. A cover of an array A occurs at positions A(1,1), A(n m + 1,1), A(1,n m + 1) and A(n m + 1,n m + 1) ( in order to cover positions A(1,1), A(1,n), A(n,1) and A(n,n)) and therefore is a border of A.. By Fact 2, all borders of A are candidates for the quasiperiod. Let B u and B v be two borders of A such that B u < B v, then the following facts are analogous to those for strings: Fact 3. If B u covers B v, then B v is not the quasiperiod of A. Proof. Assume that B v is the quasiperiod of A; by the definition of the quasiperiodicity B v covers A. However, B u covers B v, which in turn implies that B u also covers A. Therefore B v cannot be the quasiperiod of A, a contradiction, since B u is a shorter cover than B v of A.. Fact 4. If B u B v /2, then B u covers B v. Proof. Let B u and B v be p p and q q matrices respectively. From the

4 definition of a border, B u must occur at all corners of B v. Specifically, B u occurs at locations (1,1), (q p+1,1), (1,q p+1), and (q p+1,q p+1). If p = q/2, then B v occurs at (1,1), (q/2+1,1), (1,q/2+1), (q/2+1,q/2+1) and each occurrence covers q/2 q/2 locations. Clearly, B v is covered. If p > q/2 the occurrences of B u overlap and therefore cover B v.. Fact 5. The quasiperiod Q of an array A covers all the borders of A. Proof. Let B be a border of A. If Q B /2, then Q covers B by Fact 4. Now let Q > B /2. One can see that Q occurs at all four corners of a border B of A; this follows from the fact that Q is also a border of A. From the fact that Q covers A, it follows that Q covers positions A(j, Q +1),...,A(j, B Q ) for j = 1,... Q or equivalently B(j, Q + 1),...,B(j, B Q ) for j = 1,..., Q ; therefore Q covers the first Q rows of B. Similarly we can show that Q is covering the last Q rows, the first Q columns as well as the last Q columns. The rest of the positions in B that are not covered by the above occurrences of Q, are also within another occurrence of Q, since all positions of A are within an occurrence of B; these occurrences are clearly within B. Thus Q covers B.. Our algorithm uses the above Facts to eliminate candidates during the competition which is described in Section 5. The parallel algorithm presented in this paper is based on the common CRCW PRAM (Concurrent Read Concurrent Write Parallel Random Access Machine). The common CRCW PRAM is a shared memory model of parallel computation which consists of a collection of identical processors and a shared memory. Each processor is a RAM, working synchronously and communicating via the shared memory. The memory is accessed by concurrent writes and concurrent reads. Concurrent writes are assumed to write the same value. We measure the complexity of an algorithm on this model by the pair (t,p) where t denotes the time and p the number of processors used by the algorithm. The product t p is the total number of operations required by the algorithm. A parallel algorithm for a problem is said to be optimal if its total number of operations is asymptotically the same as that required by the fastest sequential algorithm for the problem. An optimal algorithm is said to be work-time optimal if its time is the best possible (i.e., matches a lower bound). Our algorithm for superprimitivity testing makes use of the following known algorithms: 1 Given a string of length n, the algorithm in [1] (which will be referred to as the All-Period algorithm) computes all the periods of the string in optimal O(log log n) time. 2 Given a 2-dimensional string of length n n, the algorithm in [6] [7] [8] (which will be referred to as the Witness algorithm) computes the witnesses of the string in optimal O(log log n) time. 3 Given a 2-dimensional m m pattern, a 2-dimensional n n text, and the witnesses of the pattern, the string matching algorithm Constant- Match in [9] [7] finds all occurrences of the pattern in the text in constant time with n 2 processors.

5 4 The integer minimum algorithm in [12] (which will be referred to as the Integer-Min algorithm) computes the minimum of at most n integers whose values are in [1..n] in optimal constant time. The following theorem by Brent [5] is useful in analyzing parallel algorithms, since it allows us to count only the time and the total number of operations. Theorem 1. If a parallel computation can be performed in time t using q operations, then it can be performed in time t + (q t)/p using p processors. For example, if a parallel algorithm runs in O(log log n) time using O(n) operations, then it can be performed in the same O(log log n) time with n/log log n processors. Theorem 1 requires the assignment of processors to their tasks, which can be easily done in our algorithm. 3. Computing all Borders Here we describe a common-crcw PRAM algorithm which computes all the square borders of an n n array A in O(log log n) time and linear work. The algorithm makes use of two n n matrices C and R such that : { 1, A[1...i,j] is a border of A[1...n,j]; C(i,j) = 0, otherwise. { 1, A[i,1...j] is a border of A[i,1...n]; R(i,j) = 0, otherwise. Step 1. Compute C(i,j) and R(i,j), 1 i,j n. This can be done in O(log log n) time using n 2 /log log n processors. We use the All-Periods algorithm of [1]. Step 2. Compute M(i,j) = C(i,j) + R(i,j), 1 i,j n. This can easily be done in constant time using n 2 processors. Step 3. We now consider all positions A(i,i) of A, with M(i,i) = 2; the prefix A[1..i,1..i] of A is a candidate for a border. If M(i,i) = 0, then neither the prefix of the i-th row of length i is a border of that row nor the prefix of the i-th column of length i is a border of that column; therefore A[1..i,1..i] cannot be a border of A. If M(i,i) = 1, then either the prefix of the i-th row of length i is not a border of that row or the prefix of the i-th column of length i is not a border of that column; therefore A[1..i,1..i] cannot be a border of A. If M(i,i) = 2, then both the i-th row and i-th column have borders of length i. Now, for all positions M(i,i) = 2, we know that if M(0,i) = M(1,i)... = M(i 1,i) 1, then rows r 0,r 1,...,r i of A have borders of length i. Similarly the columns c 0,c 1,...,c i have borders of length i, if positions M(i,0) = M(i,1)... = M(i,i 1) 1. Using this criterion we will check whether the prefix A[1..i,1..i] of A occurs on the top right (Step 3.1), bottom left (Step 3.2), and bottom right (Step 3.3) corners of A. Step 3.1 Check whether C(i,j) = 1, for all j = 1,...,i. If this is the case, then A[1..i,1..i] occurs at A(n i + 1,1); otherwise A[1..i,1..i] is not a border.

6 This can easily be done in constant time using n 2 processors (see pseudo-code below). Step 3.2 Check whether R(j,i) = 1, for all j = 1,...,i. If this is the case then A[1..i,1..i] occurs at A(1,n i + 1); otherwise A[1..i,1..i] is not a border. This can easily be done in constant time using n 2 processors (see pseudo-code below). Step 3.3 Check whether R(j,i) = 1 for all j = n i,...,n. If this the case, then A[1..i,1..i] occurs at A(n i+1,n i+1); otherwise A[1..i,1..i] is not a border. This can easily be done in constant time using n 2 processors (see pseudo-code below). The pseudo-code for the All-Borders algorithm is as follows: begin Compute C(i,j), 1 i,j n ; Compute R(i,j), 1 i,j n ; comment Use the All-Periods algorithm for finding all the periods of a string. M(i,j) 0 ; forall i,j pardo if R((i,j) = 1 and C(i,j) = 1 then Processor p ij writes 2 to M(i,j) ; if R((i,j) = 1 xor C(i,j) = 1 then Processor p ij writes 1 to M(i,j) ; odpar forall processors p ij pardo if C(i,j) = 0 and j < i then Processor p ij writes 1 to M(i,i) ; if R(j,i) = 0 and j < i then Processor p ij writes 1 to M(i,i) ; if R(j,i) = 0 and n j < i then Processor p ij writes 1 to M(i,i) ; odpar if M(i,i) = 2 then return A[1...i,1...i]) ; end From the above the following theorem follows: Theorem 3.1. There exists an algorithm to compute all the borders of an n n square array A in O(log log n) time using O(n 2 /log log n) processors.. 4. Bouts The algorithm makes use of Fact 2 that all borders of x are candidates for the quasiperiod. We employ an elimination procedure called the bout: For two borders B u and B v, B u < B v, of A, check if B u covers B v. If B u covers B v, then B v is eliminated since B v cannot be the quasiperiod by Fact 3. Otherwise

7 B u is eliminated since B u cannot be the quasiperiod by Fact 5. Testing whether B u covers B v can be done by firstly dividing B v into disjoint consecutive square blocks of size B u B u and secondly by checking whether each row of every block is covered by an occurrence of B u. We need the following definitions in order to establish the lemma below that forms the basis of our elimination procedure. Let α be the leftmost position above the i-th row such that B u occurs within the block (see Figure 2). Let ζ be the position such that B u occurs and its top right corner δ is the rightmost position within the block above the i-th row (see Figure 2). Also let µ be the position such that B u occurs and its bottom right corner γ is the rightmost position within the block below the i-th row (see Figure 2). Finally let λ be the position such that B u occurs and its bottom left corner β is the leftmost position within the block below the i-th row. µ λ α α ζ δ a b c d e f a b f β γ h Figure 2: Covering a sub-array Lemma 4.1. An array B u covers B v if for every row of every block of B v at least one of the following pairs of points (as defined above) exists {(α,γ),(α,δ),(β,δ), (β,γ)}. Proof. We consider the i-th row of a block of B v. It is clear that if any of the above pairs of points exists then the i-th row is covered (see Figure 2). Assume that α exists. This implies that there is at least one occurrence of B u in the block (above the i-th row) and α is the position of the leftmost occurrence of B u above the i-th row and inside the block. This implies that the segment [b,f] of the i-th row is covered (see Figure 2). In order for the segment [a,b] to be covered then B u must occur to the left of α. But a covering occurrence above α implies the existence of γ and a covering occurrence below α implies the existence of δ. Assume that α does not exist. In this case there is no occurrence of B u in the block (above the i-th row). Let λ be the leftmost occurrence of B u to the right of the first column of the block (see Figure 2). If λ does not exist, then the row is not covered. The existence of λ implies the existence β and that the segment [c,f] of the i-th row is covered by B u. The segment [a,c] can be only

8 covered by an occurrence of B u to the left of the block, which in turn implies the existence of either µ or ζ and therefore the existence of either γ or δ.. Given two borders B u and B v, we use Lemma 4.1 as the basis of the following procedure, which determines whether B u covers B v, to eliminate one of them. Step 1. If B u B v /2, then B u covers B v by Fact 4. Otherwise we use the Constant-Match algorithm to find all occurrences of B u in B v. Step 2. We divide B v into disjoint consecutive square blocks of size B u B u. We assign B u 2 processors to each block. Step 3. For every column of each block we compute the top-most occurrence of B u ; this can be done in constant time using the Integer-Min algorithm and B u processors for each column. We mark all positions of the column below the topmost occurrence. One can see that b (as in Figure 2) is the leftmost mark on the i-th row of the block; we compute the position b (if it exists) in constant time, again using the Integer-Min algorithm. If b exists, then it implies that α exists. Testing the existence of β,γ,δ is done in a similar manner for every row of every block. Step 4. If none of the four pairs of points given by Lemma 4.1 exists in a row of some block, then B u does not cover B v. Otherwise by Lemma 4.1, B v covers B u. Theorem 4.2. Given two borders B u and B v and the witnesses for B u, one can test whether B u covers B v in constant time using n 2 processors.. 5. Superprimitivity Testing The steps of the algorithm are as follows: Step 1. Compute all borders of A using the All-Borders algorithm in O(log log n) time using n 2 /log log n processors. For 1 i < log n, let R i be the range of borders such that {t : 2 i B t < 2 i+1 }. Let B i be the smallest border in the range R i. Note that there are at most log n such borders and that only these can be candidates for the quasiperiod by Facts 3 and 4. In each interval R i the smallest border can be found in constant time using the Integer-Min algorithm. Step 2. Compute the witnesses for each B i selected in Step 1 using the Witness algorithm. This is done in O(log log 2 i ) time and O(2 i ) work for i = 1,...,log n. Therefore the total number of operations is bounded by: log n i=0 log n 2 2i < ( i=0 2 i ) 2 = O(n 2 ) Step 3. First we consider all borders B i such that B i n/log log n performing bouts in a binary tree fashion using the constant time Bout algorithm. The number of operations during the bout is dominated by the pattern matching phase which requires B i 2 operations. There are O(log log n) levels of bouts.

9 The number of candidates for the quasiperiod will be halved at each level and the number of operations required is bounded by: B i n log log n B i 2 B i n log log n (2 i+1 ) 2 < B i n log log n 2 i+1 2 n = O(( log log n )2 ) Since there are log log n levels of bouts, the time for this step is O(log log n) and the total number of operations is O((n/log log n) 2 ). We let b be the only border which survives this step. Step 4. We must now perform bouts sequentially between the remaining borders and b. The borders which have not yet competed in a competition are all B i such that n/log log n < B i n. There are at most log log log n of them. The total sequential time to complete the competition is therefore O(log log log n). The total number of operations for this step is bounded by: b 2 + B i>n/ log log n log log log n B i 2 (n/log log n) 2 + (n/2 i ) 2 = O(n 2 ) Let B be the border which survives this round. If B = A then A is superprimitive. Otherwise b is the quasiperiod of A. 6. Conclusions We have presented an optimal O(log log n)-time algorithm for superprimitivity testing of a square array. The problem of designing an optimal parallel algorithm to compute all the covers of a square array is an open one. 7. References i=0 [1] A.Apostolico, D.Breslauer and Z.Galil, Optimal parallel algorithms for periods, palindromes and squares, Proc. of the 19th International Colloquium on Automota, Languages and Programming, Lecture Notes in Computer Science, 623, (1992) [2] A.Apostolico, M.Farach and C.S.Iliopoulos, Optimal superprimitivity testing for strings, Inform. Process. Lett. 39, (1991) [3] D.Breslauer, An on-line string superprimitivity test, Inform. Process. Lett. 44 (1992) [4] D.Breslauer, Testing string superprimitivity in parallel, Inform. Process. Lett, to appear. [5] R.P. Brent, The parallel evaluation of general arithmetic expressions, Journal of Assoc. Comput. Mach., 21, (1974) [6] M.Crochemore, L.Gasieniec, Z.Galil, K.Park and W.Rytter. Constant time deterministic sample computation and its applications, SIAM Journal of Computing to appear.

10 [7] R.Cole, M.Crochemore, Z.Galil, L.Gasieniec, R.Hariharan, S.Muthukrishnan, K.Park and W.Rytter, Optimally fast parallel algorithms for preprocessing and pattern matching in one and two dimensions, 34th IEEE Symposium on Foundations of Computer Science, Palo Alto, USA, [8] R.Cole, Z.Galil, R.Hariharan, S.Muthukrishnan and K.Park, Optimal parallel witness computation, manuscript, [9] M.Crochemore, L.Gasieniec, R.Hariharan, S.Muthukrishnan and W.Rytter, An optimal constant time algorithm for 2D pattern matching, manuscript, [10] M.Crochemore, C.S.Iliopoulos and M.Korda, Two-dimensional prefix matching and covering for square matrices, submitted, [11] A.M. Duval and W.F. Smyth, Covering a circular string with substrings of fixed length, Int. Journal of Foundations of Computer Science, to appear. [12] F.Fich, R.Ragde and A.Widgerson, Relations between concurrent-write models of parallel computation, SIAM Journal of Computing, 17, (1988) [13] C.S.Iliopoulos and K.Park, An optimal O(log log n) time algorithm for parallel superprimitivity testing, Journal of the Korea Information Science Society, 21(8), (1994) [14] D.W.G.Moore and W.F.Smyth, Computing the covers of a string in linear time, Proc. 5th ACM-SIAM Symp. Discrete Algorithms, (1994)

Computing repetitive structures in indeterminate strings

Computing repetitive structures in indeterminate strings Pavlos Antoniou 1, Costas S. Iliopoulos 1, Inuka Jayasekera 1, Wojciech Rytter 2,3 1 Dept. of Computer Science, King s College London, London WC2R