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Dependence Analysis Kathy Yelick Bebop group meeting, 8/3/01

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Outline Motivation What the compiler does: –What is data dependence? –How is it represented? –How is it used?

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Motivation: Optimization Many compiler optimizations are based on the idea of either: –Reordering statements (or smaller units) –Executing them in parallel In the paper on transforming loops to recursive functions, they are reordering the iterations of matrix multiply. Goal: do this without changing the semantics of the program.

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References Chau-Wen Tseng’s lecture notes: –www.cs.umd.edu/class/spring1999/cmsc732/www.cs.umd.edu/class/spring1999/cmsc732/ Gao Guang’s lecture note (U.Del.) Others

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Definition and Notation Data dependence: –Given two program statements a and b, b depends on a if: b follows a (roughly) they share a memory location and one of them writes to it. –Written: b a –Example: a: x = y + 1; b: z = x * 3; Because b depends on a, the two statements cannot be reordered, nor can they be run in parallel

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Classification A dependence, a b, is one of the following: –true of flow dependence: a writes a location that b later reads (read-after write or RAW) –anti-dependence a reads a location that b later writes (write-after-read or WAR) –output dependence a writes a location that b later writes (write-after-write or WAW)

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More Dependences The following is also a dependence –An input dependence a b occurs when a reads a location that b later reads (read-after-read or RAR) But we usually are not interested in these, because they don’t constraint order or parallelism Examples: trueantioutputinput a = = aa = = a a = = a

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Example Example: S1: A = 0 S2: B = A S3: C = A + D S4: D = 2 S1 S2 S3 S4 S 2 S 1 :flow dep S 3 S 1 :flow dep S 4 S 3 :flow dep S 1 0 S 3 : output-dep S 2 -1 S 3 : anti-dep S1 S2 S3 S4

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How to Compute Dependences? Data dependence relations can be found by comparing the IN and OUT sets of each node. –The IN set of a statement, IN(S), is the set of variables (or, more precisely, the set of memory locations, usually referred to by variable names) that may be used (or read) by this statement. –The OUT set of a statement, OUT(S), is the set of memory locations that may be modified (written or stored) by the statement.

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Computing and representing the memory locations can be very difficult if the program contains aliases. Fortunately, the IN/OUT sets can be computed conservatively Assuming that S 2 is reachable from S 1, the following shows how to intersect these sets to test for data dependence: OUT(S 1 ) IN(S 2 )S 1 S 2 flow dependence IN(S 1 ) OUT(S 2 )S 1 -1 S 2 anti-dependence OUT(S 1 ) OUT(S 2 )S 1 0 S 2 output dependence Computing Dependences in Practice

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Dependences in Loops A loop-independent dependence exists even if there were no loop for (j = 0; j < 100; j++) { a[j] = = a[j] } A loop-carried dependence is induced by the iterations of a loop. The source and sink occur on different iterations. for (j = 0; j < 100; j++) { a[j] = = a[j-1] }

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Data Dependences in Loops Find the dependence relations due to the array X in the program below: (1) for I = 2 to 9 do (2) X[I] = Y[I] + Z[I] (3) A[I] = X[I-1] + 1 (4)end for Approach: In a simple loop, we can unroll the loop and see which statement instances depend on which others (each array element is a variable): (2) X[2]=Y[2]+Z[2] X[3] =Y[3]+Z[3] X[4]=Y[4]+Z[4] (3) A[2]=X[1]+1 A[3] =X[2]+1 A[4]=X[3]+1 I = 2 I = 3 I = 4

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In the example, there is a loop-carried, lexically forward flow dependence relation. (The (1) annotation in the figure below will be explained shortly.) Data Dependences in Loops S2S2 S3S3 (1) - Loop-carried vs loop-independent - Lexical-forward vs lexical backward

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Iteration Space Example do I = 1, 5 do J = I, 6... enddo Written out in lexicographic order the iteration space is: –(1,1), (1,2),…, (1,6),(2,2),(2,3),… Equivalent to sequential execution order J I

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Basic Concepts Let R n be the set of all real n-vectors, (n >1) A lexicographic order < u on these vectors is a relation: i < u j on vectorsi = { i 1 … i n } j = { j 1 … j n } iff i 1 = j 1, j 1 = j 2 … and i u < j u The leading element of a vector is the first non- zero element A negative vector has: leading element < 0 A positive vector has: leading element > 0

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Given 2 n-vectors i,j i = (i 1, … i n ) j = (j 1, … j n ) Their distance vector = (j 1 - i 1, j 2 - i 2, …) –Represents the number of iterations between accesses to the same location Their direction vector S = (sig (j 1 - i 1 ), sig(j 2 - i 2 ), …) –Represents the direction in iteration space Given distance vector ==> one can derive direction vector but not vise versa. Distance/Direction Vectors

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It is often convenient to deal with in- completely specified direction vectors Example 1: {(0, 0, 0, 1), (0, -1, 0, 1), (0, 0, 1, 1), (0, -1, 1, 1)} ==> {(0, 0, 0, 1)} Example 2: {(0, -1, 0, -1), (0, 0, 0, -1), (0, 1, 0, -1)} ==> {(0, *, 0, -1)}

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Distance/Direction Vectors Let i, j denote two vectors in R n and s their direction vector. Then i < j iff s has one of the following n forms: (1, *, *, …, *) (0, 1, *, …, *) (0, 0, 1, *, …, *) (0, 0, …, 0, 1). More precisely, i < u j for a u in 1 u n, iff s has the form with a leading 1 after (u - 1) zeros, I.e., s is positive. Notation (0, 1, -1) (=, >, <)

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do i = 3,100 S:A(2i) = B(i) + 2 T:C(i) = D(i) + 2A(2i +1) + A(2i - 4) + A(i) 1.A(2i), D(i) 2. A(2i), A(2i + 1) 3. A(2i), A(2i - 4) An Example no dependence ?

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Note: the direction is from S to T (i 2 > i 1 ) 2i 1 = 2i 2 - 4, so i 2 - i 1 = 2 i.e. a flow dependence is caused for example: all iteration pairs : (i 2 = i 1 + 2, 3 i 98) for each pair: direction vector = (1) distance vector = (2) i 1 = 3 i 2 = 5 Constant Example: A(2i) and A (2i - 4)

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For A(2i), A(i) this causes a flow dependence of T on S, the set of associated iteration pairs is {(i, j) | j = 2i, 3 i 50)} for i, j in this set direction vector = (1) distance vector = 2i - i = i Summary: T is flow-dependent on S with direction vector = (1) distance vector between (3) to (50) Example: A(2i) and A (2i - 4)

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Testing for Parallel Loops A dependence D = (d 1,…,d k ) is carried at level i, if d i is the first nonzero element of the distance/direction vector A loop l i is parallel if there does not exist a dependence D j carried at level i. If the loop is parallel, it may also be reordered.

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