One-dimensional I test and direction vector I test with array references by induction variable

Transcription

1 Int. J. High Performance Computing an Networking, Vol. 3, No. 4, One-imensional I test an irection vector I test with array references by inuction variable Minyi Guo School of Computer Science an Engineering, University of Aizu, Aizu-Wakamatsu City, Fukushima, Japan State Key Lab for Novel Software Technology, Nanjing University, PRC Fax: minyi@u-aizu.ac.jp Weng-Long Chang* Department of Computer Science an Information Engineering, National Kaohsiung University of Applie Sciences, 415 Chien Kung Roa, 807 Kaohsiung, Taiwan changwl@ mail.stut.eu.tw *Corresponing author Jian Lu State Key Lab for Novel Software Technology, Nanjing University, PRC lj@nju.eu.cn Minglu Li Department of Computer Science an Engineering, Shanghai JiaoTong University, Shanghai, China li-ml@cs.stut.eu.cn Abstract: In this paper, theoretical aspects to emonstrate the accuracy of the Interval Test (the I test an the irection vector I test) to be applie for resolving the problem state above is presente. Also, it is prove from the propose theoretical aspects that uner a specific irection vector θ = ( = 1,, = ) there are integer-value solutions for one-imensional arrays with subscripts forme by inuction variable an uner other specific irection vectors there are no integer-value solutions. Experiments with benchmarks, cite from Parallel loop, Vector loop an TRFD (Perfect benchmark), reveal that our framework can properly enhance the precision of ata epenence analysis for one-imensional arrays with subscripts mentione above. Keywors: parallelising/vectorising compilers; ata epenence analysis; loop parallelisation; loop vectorisation; automatic loop transformation. Reference to this paper shoul be mae as follows: Guo, M., Chang, W-L., Lu, J. an Li, M. (2005) One-imensional I test an irection vector I test with array references by inuction variable, Int. J. High Performance Computing an Networking, Vol. 3, No. 4, pp Copyright 2005 Inerscience Enterprises Lt.

2 220 M. GUO, W-L. CHANG, J. LU AND M. LI Biographical notes: Minyi Guo is currently an Associate Professor at the Department of Computer Software, at the University of Aizu, Japan. Guo has serve as programme committee or organising committee member for many international conferences. He is the Eitor-in-Chief of the Journal of Embee Systems. He is also on the eitorial boar of the International Journal of High Performance Computing an Networking, the Journal of Embee Computing, the Journal of Parallel an Distribute Scientific an Engineering Computing an the International Journal of Computer an Applications. Guo s research interests inclue parallel an istribute processing, parallelising compilers, ata parallel languages, ata mining, molecular computing an software engineering. He is a member of the ACM, IEEE, an IEICE. Weng-Long Chang receive his PhD egree in computer science an information engineering from the National Cheng Kung University, Taiwan, Republic of China, in He is currently an Assistant Professor at the Southern Taiwan University of Technology. His research interests inclue molecular computing an languages an compilers for parallel computing. Jian Lu receive his BS egree, MS egree, an PhD egree from Nanjing University of PR China in 1982, 1984, 1988, respectively. Currently, he is a Professor at Department of Computer Science an Technology, Vice Director of the Institute of Computer Software at Nanjing University an Director of State Key Laboratory of Novel Software Technology of P.R. China. His research interests inclue formal methos, mobile agent technology an software architecture. Dr. Minglu Li is a Full Professor an Vice Char of Department of Computer Science an Engineering of Shanghai Jiao Tong University (SJTU). Now, he is also Director of Gri Computing Center of SJTU, a member of Expert Committee of ChinaGri Program of Ministry of Eucation of China, PI of ShanghaiGri of Science an Technology Commission of Shanghai Municipality(STCSM), an Associate Eitor of International Journal of Gri an Utility Computing, an member of eitorial boar of International Journal of Web Services Research, a member of executive committee of Technical Committee on Services Computing of IEEE. 1 INTRODUCTION Achieving a goo ata epenence analysis is a critical, issue in orer to reuce the communication overhea an to exploit parallelism of applications as much as possible. This task becomes even more important in istribute memory systems where, in aition to accomplishing a high parallelism of computation an low communication overhea among the processors, it is essential to evelop a new analysis technique for ata epenence. There are several well-known ata epenence analysis algorithms applicable for one-imensional arrays uner constant bouns or variable bouns: the GCD test (Banerjee, 1988, 1993, 1997), Banerjee s metho (Banerjee, 1988, 1993, 1997), the I test an the irection vector I test (Kong et al., 1991; Niezielski an Psarris, 1999; Psarris et al., 1991, 1993; Psarris an Kyriakopoulos, 1999), the extene I test (Chang an Chu, 1998), the generalise irection vector I test (Chang an Chu, 2001) an the interval reuction test (Huang an Yang, 2000). There are also several well-known ata epenence analysis algorithms applicable for multi-imensional couple arrays uner constant bouns or variable bouns: the generalise GCD test (Banerjee, 1988, 1993, 1997), the Lamba test (Li et al., 1990), the generalise Lamba test (Chang et al., 1999), the multi-imensional I test (Chang et al., 2001), the multi-imensional irection vector I test (Chang et al., 2002), the Power test (Wolfe an Tseng, 1992) an the Omega test (Pugh, 1992). There are several well-known ata epenence analysis algorithms applicable for arrays with linear subscripts with symbolic coefficients, or with non-linear subscripts uner symbolic bouns: the infinity Banerjee test (Petersen, 1993), the Range test (Blume an Eigenmann, 1998), the infinity Lamba test (Chang an Chu, 2000), the access range test (Paek, 1997; Hoeflinger, 1998) an one analysis metho for pointers an inuction variables (Wu, 2001). In this paper, we propose a sophisticate technique of ata epenence analysis. This approach is to test epenence if there are integer-value solutions for one-imensional arrays with subscripts forme by inuction variable. Without irection vectors, there are integer-value solutions for one-imensional arrays with subscripts forme by inuction variable. Furthermore, it is also shown that uner a specific irection vector θ = ( = 1,, = ) there are integer-value solutions for one-imensional arrays with subscripts forme by inuction variable. Shen et al. (1992) ha inicate that in real programmes one-imensional arrays with subscripts forme by inuction variable occur quite frequently. A -neste loop accessing a one-imensional array with subscripts forme by inuction variable is shown in Figure 1.

3 ONE-DIMENSIONAL I TEST AND DIRECTION VECTOR I TEST WITH ARRAY REFERENCES BY INDUCTION VARIABLE 221 Figure 1 An example of a neste loop with inuction variable Inuction variable is a one scalar integer variable, which is use in a loop, to simulate loop s inex variables: it is incremente or ecremente by a constant amount in each iteration. Every inuction variable can be replace by a linear function in loop s inex variables. The transformation, which oes so, is calle inuction variable substitution. Since the variable K in Figure 1 is one inuction variable, it can be replace z (( I1 L1) ( p1= 2( U p L 1)) ( 1 p + + I 1 2 L2) ( p2= 3( U p L 2 p2 + 1) + + ( i L where is the number of common loops an z is one integer variable (Banerjee, 1988, 1993, 1997; Paek, 1997; Hoeflinger, 1998). Therefore, the coe in Figure 1 is transforme into the coe in Figure 2, after finishing the processing of inuction variable substitution for the variable K. (where n = 2). Furthermore, variables X 2k 1 an X 2k (1 k ) are ifferent instances of the same loop iteration variable, I k. Assume that L 2k 1, L 2k, U 2k 1 an U 2k are, respectively, lower bouns an upper bouns for X 2k 1 an X 2k. Because variables X 2k 1 an X 2k (1 k ) are ifferent instances of the same loop iteration variable, I k, lower bouns for X 2k 1 an X 2k are the same an upper bouns for X 2k 1 an X 2k also are the same. The problem of etermining whether there exists epenence for the array A between S 1 an S 2 in Figure 2 can be reuce to that of checking whether one system of a linear equation with n unknown variables has a simultaneous integer solution, which satisfies the constraints for each variable in the system. It is assume that one linear equation in a system is written as: z ( + 1) ( U L + 1) ( U L ( U L + + ( X2 3 X2 2) ( U L + 1) + ( X X )) = 0, where each L k an each U k are an integer variable an are, respectively, one lower boun an one upper boun for the k-the loop for (1 k ). Because z is the greatest common ivisor for all the coefficients in the left-han sie of equation (1), all the coefficients in equation (1) are ivie by z an equation (1) is rewritten as + 1) ( U L + 1) ( U L ( U L + + ( X2 3 X2 2) ( U L + 1) + ( X2 1 X2) = 0. (2) (1) It is postulate that the constraints to each variable in equation (2) are represente as L X an X U, (3) k 2k 1 2k k Figure 2 The transforme loop after inuction variable substitution for the inuction variable K Because epenence between S 1 an S 2 in Figure 2 may arise in ifferent iterations of the common loops, we eal with the loop iteration variables reference in S 1 as being ifferent variables from those reference in S 2 subject to common loop bouns. Therefore, when analysing epenence arising from a statement pair neste in common loops, the problem will involve n unique variable where (1 k ). Let us use an example to make clear the illustrations state above. Consier the neste o-loop in Figure 3. The variable K in Figure 3(a) is incremente by one in each iteration in the neste loop. So it is a one inuction variable. After finishing the processing of inuction variable substitution for the inuction variable K, the result is shown in Figure 3(b). In Figure 3(b), the lower an upper bouns of the first (outer) loop an the secon (inner) loop are, respectively, 1 an 10. Therefore, the bouns of the o-loop are constants. This o-loop executes 100 iterations by consecutively assigning the values 1, 2,, 10 to J an I an executing the boy (the main statement S) exactly once in each iteration. The net effect of the o-loop execution is then the orere execution of the statements:

4 222 M. GUO, W-L. CHANG, J. LU AND M. LI A(1) = B(1) A(2) = B(2) A(100) = B(100). (a) Figure 3 A neste o-loop in Fortran (a) a neste o-loop with inuction variable an (b) the transforme loop after inuction variable substitution for the inuction variable K To ascertain whether two references to the one-imensional array A may refer to the same element of A, it has to be checke if the following linear equation 10 X 1 10 X 2 + X 3 X 4 = 0 has a simultaneous integer solution uner the constant bouns 1 X 1, X 2 10, an 1 X 3, X It is well known that the problem of fining integer-value solutions to a system of linear equations is NP-har. Therefore, in practice, most well-known ata epenence analysis algorithms are use to solve as many particular cases of this problem as possible. The rest of this paper proffers the following: In Section 2, the summary accounts of ata epenence an interval equation are presente. In Section 3, the theoretical aspects of etermining whether there are integer-value solutions for one linear equation (2) with symbolic coefficients uner the bouns of equation (3) are escribe. Experimental results are given in Section 4. Finally, brief conclusions are rawn in Section 5. 2 BACKGROUND The summary accounts of ata epenence an interval equation are introuce briefly in this section. 2.1 Data epenence It is assume that S 1 an S 2 are two statements within the loop in Figure 2. The loop is presume to contain common loops. Statements S 1 an S 2 are postulate to be embee in common loops. An array A is suppose to appear simultaneously within statements S 1 an S 2. If S 2 uses the element of the array A efine first by S 1, then S 2 is true-epenent on S 1. If S 2 efines the element of the array A use first by S 1, then S 2 is anti-epenent on S 1. If S 2 (b) reefines the element of the array A efine first by S 1, then S 2 is output-epenent on S 1. Each iteration of a loop neste is ientifie by an iteration vector, whose elements are the values of the iteration variables for that iteration. For example, the instance of the statement S 1 uring iteration i = ( i1,, i ) is enote S1( i ); the instance of the statement S 2 uring iteration j = ( j1,, j ) is enote S 2 ( j). If (i 1,, i ) is ientical to (j 1,, j ) or (i 1,, i ) precees (j 1,, j ) lexicographically, then S 1 () i is sai to precee S 2 ( j), enote S1( i ) < S2 ( j). Otherwise, S2 ( j) is sai to precee S1( i ), enote S 1 () i > S2 ( j). In the following, Definition 1 efines irection vectors. Definition 1: A vector of the form θ = ( θ1,, θ ) is terme as a irection vector. The irection vector ( θ1,, θ ) is sai to be the irection vector from S1( i ) to S 2 ( j) if for (1 k ) i k θ k j k, i.e., the relation θ k is one element in a set {<,=,>}. 2.1 Interval equation A linear equation with symbolic coefficients uner the bouns of equation (3) will be sai to be integer solvable if the equation has an integer-value solution satisfying the bouns of each variable. Definitions 2 an 3, respectively, efine integer interval an interval equation (Kong et al., 1991; Psarris et al., 1993). Definition 2: Let [α 1,α 2 ] represent the integer intervals from α 1 to α 2, i.e., the set of all integers between α 1 an α 2. Definition 3: Let a 1,, a n 1, a n, L an U be integers. A linear equation ax 1 1+ ax an 1Xn 1 + ax n n = [ LU, ], (4) which is referre to as an interval equation will be use to enote the set of orinary equations consisting of: ax 1 1+ ax an 1Xn 1+ anxn = L ax 1 1+ ax an 1Xn 1+ ax n n = L+ 1 ax + ax + + a X + a X = U n 1 n 1 n n An interval equation (4) will be sai to be integer solvable if one of the equations in the set, which it efines, is integer solvable. The immeiate way to etermine this is to test if an integer in between L an U is ivisible by the GCD of the coefficients of the left-han-sie terms. If L > U in an interval equation (4), then there are no integer solutions for this interval equation. If the expression on the left-han sie of an interval equation (4) is reuce to zero items, in the processing of testing, then the interval equation (4) will be sai to be integer solvable if an only if L 0 U.

5 ONE-DIMENSIONAL I TEST AND DIRECTION VECTOR I TEST WITH ARRAY REFERENCES BY INDUCTION VARIABLE DETERMINING INTEGER-VALUED SOLUTIONS FOR ONE-DIMENSIONAL ARRAYS WITH SYMBOLIC COEFFICIENTS UNDER SYMBOLIC BOUNDS Given the ata epenence problem of one-imensional arrays with symbolic coefficients an symbolic bouns, we want to fin that uner what conitions there are integer-value solutions uner what conitions there are no integer-value solutions. We start to iscuss from the case without irection vectors for convenience of presentation. 3.1 Determine integer-value solutions for one-imensional arrays with symbolic coefficients uner symbolic bouns without irection vector One linear equation (2) with symbolic coefficients is euce from etermining whether there exists epenence for the array A between S 1 an S 2 in Figure 2. One boun (3) is erive from lower an upper bouns of loop inex variables in Figure 2. If there is no integer-value solution for one linear equation (2) with symbolic coefficients uner the limits of equation (3), then there is no epenence. Otherwise, there is epenence for one linear equation (2) with symbolic coefficients uner the limits of equation (3). Theorem 1 is presente to show that there are integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3). Theorem 1: There are integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3). Proof: 1 From the I test in Kong et al. (1991), one linear equation (2) with symbolic coefficients can be rewritten as the following interval equation: + 1) ( U L + 1) ( U L + + ( X2j 1 X2j) (( U ( U L + + ( X2 3 X2 2) ( U L + 1) + ( X X ) = [0,0] Accoring to the I test, the term X 2 in the interval equation (5) is move to the right-han sie to gain the new interval equation: (5) 4 Because (U L + 1) is the greatest common ivisor for all the coefficients in the left-han sie of the interval equation (7), all the coefficients in the interval equation (7) are ivie by (U L + 1) an the new interval equation is obtaine: + 1) ( U L + 1) ( U L ( U L + + ( X2 3 X2 2) ( U L + 1) = [ L U, U L ]. (7) + 1) ( U L + 1) ( U L + + ( X2j 1 X2j) (( U ( U L + + ( X2 3 X2 2) ( U L + 1) L U U L =, = [0,0]. (8) U L + 1 U L Repeat the processing of step (2) (4) until the term in left-han sie of the interval equation is reuce to zero item. Therefore, we will obtain the new interval equation 0 = [L 1 U 1,U 1 L 1 ]. Because U 1 L 1, L 1 U 1 0 U 1 L 1 is true. Thus, it is at once erive that there are integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3). The following example is use to explain the power of Theorem 1. Consier the o-loop in Figure 4(a). The front-en of one parallel compiler can recognise that the variable K is one inuction variable an the subscript of the array A is forme by the inuction variable K. After finishing the processing of inuction variable substitution for the inuction variable K, the result is shown in Figure 4(b). Because the coefficient of the subscript of the array A is an unknown variable (at compile time), the Power test an the Omega test cannot be applie to eal with the problem. However, in light of Theorem 1, it is at once conclue that there exists output epenence for the array A uner without consiering any irection vector. Therefore, it is inicate from the result that the precision of Theorem 1 is superior to that of the Omega test an the Power test. ( X X ) (( U L + 1) ( U L ( X X ) (( U L + 1) ( U L ( X X ) (( U L + 1) 2j 1 2j j+ 1 j+ 1 ( U L + + ( X X ) ( U L + 1) X = [ L, U ]. 2 1 (6) (a) (b) 3 In light of the I test, the term X 2 1 in the interval equation (6) is move to the right-han sie to gain the new interval equation. Figure 4 An example of o-loop in Fortran program (a) a neste o-loop with inuction variable an (b) the transforme loop after inuction variable substitution for the inuction variable K

6 224 M. GUO, W-L. CHANG, J. LU AND M. LI 3.2 Determine integer-value solutions for one-imensional arrays with symbolic coefficients uner symbolic bouns an irection vector From Theorem 1, it is very clear that there are integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3). Next, uner a specific given irection vector an the limits of equation (3), whether there are integer-value solutions for one linear equation (2) with symbolic coefficients will be iscusse. The following theorems are propose to show that uner what a specific irection vector there are no integer-value solutions uner what a specific irection vector there are integer-value solutions. Theorem 2: There are no integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3) an a specific irection vector (<, *,, *), where is the number of common loops an * is any one of {<, =, >}. Proof: 1 From the I test an the irection vector I test in Kong et al. (1991) an Psarris et al. (1993), one linear equation (2) with symbolic coefficients can be rewritten as the following interval equation: + 1) ( U L + 1) ( U L 1+ 1) ( U L + + ( X2 3 X2 2) ( U L + 1) + ( X X ) = [0,0] Repeat the processing of step (2) to step (4) in Theorem 1, until the term in left-han sie of the interval equation is reuce to two items. Therefore, we will obtain the new interval equation: 1 2 (9) ( X X ) = [0,0]. (10) 3 Because the irection vector is < for the terms X 1 an X 2, accoring to the irection vector I test in Psarris et al. (1993), the two terms X 1 an X 2 are move to the right-han sie. Therefore, we obtain the new interval equation 0 = [1, U 1 L 1 ]. Because 1 0 U 1 L 1 is false, it is, therefore, inferre that there are no integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3) an a specific irection vector (<, *,, *). Theorem 3: There are no integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3) an a specific irection vector (<, *,, *), where is the number of common loops an * is any one of {<, =, >}. Proof: Similar to Theorem 2. Theorem 4: There are no integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3) an a specific irection vector (=, θ 2,, θ 1, θ ), where is the number of common loops an θ k is any element of {<, =, >} for 2 k 1 an θ k is any element of {<, >}. Proof: Similar to Theorem 2. Theorem 5: There are integer-value solutions for one linear equation (2) with symbolic coefficients uner the limits of equation (3) an a specific irection vector (=, =,, =), where is the number of common loops. Proof: Similar to Theorem 2. We now use the o-loop in Figure 4 to explain the power of Theorem 2 to Theorem 5. Consier the o-loop in Figure 4(a) an Figure 4(b). Because the coefficient of the subscript of the array A is an unknown variable (at compile time), the Power test an the Omega test cannot be applie to eal with the problem uner any given irection vectors. However, in light of Theorem 5, it is right away inferre that there are integer-value solutions for the array A uner a specific irection vector (=, =). Accoring to Theorem 2 to Theorem 4, it is at once conclue that there are no integer-value solutions for the array A uner other specific irection vectors. Therefore, the front-en of one parallel compiler at once erive that there only exists loop-inepenence output epenence for the o-loop. Because the source statement an sink statement to the epenence relation is the same statement, the epenence relation can be ignore. The o-loop can be execute in parallel moe. Hence, it is inicate from the result that the precision of Theorem 5 is superior to that of the Omega test an the Power test. 3.3 Extening symbolic subscript forme by inuction variable The expression K + C for the array A in Figure 1 to a K + C, where a is an integer variable an is not equal to zero can be easily extene. Since the variable K in Figure 1 is one inuction variable, it can be replace by z (( I L ) ( ( U L + ( I L ) 1 1 p1 = 2 p1 p1 2 2 p2 = 3 U p L 2 p i 2 L ( ( + + ( where is the number of common loops an z is one integer variable (Banerjee, 1997; Paek, 1997; Hoeflinger, 1988).

7 ONE-DIMENSIONAL I TEST AND DIRECTION VECTOR I TEST WITH ARRAY REFERENCES BY INDUCTION VARIABLE 225 Therefore, the extene expression of the array A is transforme into the coe in Figure 5, after finishing the processing of inuction variable substitution. Eigenman et al., 1988). One hunre an sixty four pairs of one-imensional array references were observe to have subscripts forme by inuction variable. The propose theorems are only applie to test those one-imensional arrays with subscripts forme by inuction variable. It is very clear from the result shown in Table 1 that the propose theorems coul properly solve whether there are efinitive results for one-imensional arrays with subscripts forme by inuction variable. Table 1 The result of solving whether there are integer-value solutions for one-imensional arrays with subscripts forme by inuction variable The number The number of The number of no Benchmark of arrays teste integer-value solutions integer-value solutions Parallel loop Vector loop TRFD Figure 5 The result is obtaine after finishing the processing of inuction variable substitution for the extene expression of the array A The problem of etermining whether there exists epenence for the array A between S 1 an S 2 in Figure 5 is actually equal to that of checking whether there are integer-value solutions for one new linear equation (11) uner the constraints of equation (3). It is assume that the new linear equation (11) is written as a z (( X1 X2) (( U2 L2 + 1) ( U L + 1) ( U L ( U L + + ( X2 3 X2 2) ( U L + 1) + ( X X )) = 0, (11) where each L k an each U k are an integer variable an are, respectively, one lower boun an one upper boun for the k-th loop for 1 k. Because a z is not equal to zero, all the coefficients in equation (11) are ivie by a z an equation (11) is exactly equal to equation (2). Therefore, Theorem 5 also can be applie to eal with whether there is epenent relation for the array A between S 1 an S 2 in Figure 5. 4 EXPERIMENTAL RESULTS The propose theorems have been teste an experimente on the coes abstracte from three numerical packages: Parallel Loop, Vector Loop an TRFD (Perfect Benchmark) (Dongarra et al., 1991; Levine et al., 1991; The Omega test an the Power test have been implemente base on Wolfe an Tseng (1992) an Pugh (1992), to compare their effects with that of the propose theorems. The Omega test an the Power test were applie to resolve the same 164 pairs of one-imensional arrays with symbolic coefficients. It is very clear from the result shown in Table 2 that the Omega test an the Power test coul not be applie for etermining whether there are integer-value solutions for one-imensional arrays with symbolic coefficients. Therefore, it is inicate from the results shown in Tables 1 an 2 that the precision of the propose theorems is superior to that of the Omega test an the Power test. Table 2 The result of the Omega an Power tests for solving whether there are integer-value solutions for 164 pairs of one-imensional arrays with symbolic coefficients Depenence metho The Omega test The Power test The number of arrays teste The number of integer-value solution The number of no integer-value solutions : represents that the Omega an Power tests coul not eal with them. 5 CONCLUSIONS The front-en of a parallelising compiler can easily recognise inuction variable that forms subscripts of one-imensional arrays. Inuction variable can be replace by a linear function in form of loop s inex-variable after the front-en of a parallelising compiler finishes the processing of inuction variable substitution. This paper presents theoretical aspects to emonstrate the accuracy of

8 226 M. GUO, W-L. CHANG, J. LU AND M. LI the Interval Test (the I test an the irection vector I test) to be applie for resolving the problem state above. Also, it is prove from the propose theoretical aspects that uner a specific irection vector θ = ( = 1,, = ) there are integer-value solutions for one-imensional arrays with subscripts forme by inuction variable an uner other specific irection vectors there are no integer-value solutions. Experiments with benchmarks, cite from Parallel loop, Vector loop an TRFD (Perfect benchmark), reveal that this framework can properly enhance the precision of ata epenence analysis for one-imensional arrays with subscripts mentione above. The propose theorems can improve the precision of ata epenence analysis for one-imensional arrays with references forme by inuction variable. Depening on the application omains, it is suggeste that the propose theorems be applie together with the front-en of a parallelising compiler to provie ata epenence analysis for one-imensional arrays with references forme by inuction variable. ACKNOWLEDGEMENT This work was partially supporte by the 973 Program of China uner Grant 2002CB an NSFC of Jiangsu Province uner Grant BK REFERENCES Banerjee, U. (1988) Depenence Analysis for Supercomputing, Kluwer Acaemic Publishers, Norwell, Massachusetts. Banerjee, U. (1993) Loop Transformations for Restructuring Compilers: The Founations, Kluwer Acaemic Publishers, Norwell, Massachusetts. Banerjee, U. (1997) Depenence Analysis, Kluwer Acaemic Publishers, Norwell, Massachusetts. Blume, W. an Eigenmann, R. (1998) Nonlinear an symbolic ata epenence testing, IEEE Transaction on Parallel an Distribute Systems, Vol. 9, No. 12, pp Chang, Chu an Wu, J. 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