A. Darte. On the complexity of loop fusion. In Proceedings of the  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the memory layout before, i.e. which have been of stride eight, are now of stride one. The stride one array access is illustrated on the right side of Figure 4.3. Consequently, all words in the cache line are now used within executions of consecutive iterations. 4.3. 2 Loop Fusion Loop fusion [Dar99] is a transformation which takes two adjacent loops that have the same iteration space traversal and combines their bodies into a single loop, i.e. loops with the same loop bounds. The loop fusion sometimes also called jamming is the opposite operation of loop distribution or loop fission ....

A. Darte. On the Complexity of Loop Fusion. In Proceedings of the International Conference on Parallel Architectures and Compilation Techniques (PACT'99), pages 149--157, Newport Beach, California, USA, October 1999.

....and communications. Real life DSP applications run in open loop with a time unit in the order of magnitude of the millisecond, a volume of data in the order of magnitude of the mega byte and consist of thousands of elementary tasks. The mapping problem has been proved to be NP complete [10, 21] and is usually decomposed into sub problems which are solved separately by dedicated algorithms [5] making global optimization impossible. Work based on Integer Programming with Boolean variables led to a combinatorial explosion [21] A lot of work has been done to optimise local criteria such as ....

A. Darte. On the complexity of loop fusion. Parallel Coomputing, 26(9):1175--1193, August 2000.

.... approaches consider basically unit execution time for each iteration and zero communication for each communication step (UET model) Previous attempts of scheduling uniform dependence loops include the free scheduling method introduced in [10] tiling transformation [9, 17, 20] and loop fusion [12, 5]. The problem of scheduling uniform dependence loops is a very special case of scheduling Directed Acyclic Graphs (DAGs) The general DAG scheduling problem is known to be NP complete (see Ullman [19] so many researchers have tackled special cases of the above problem [8, 10] hoping to come up ....

Alain Darte. On the complexity of loop fusion. Parallel Computing, 26(9):1175--1193, 2000.

....information for each node, what is the best topological order that minimizes the number of color changes among consecutive nodes The problem assumes a PRH with capacity 1 (a system consisting of only one fully reconfigurable FPGA) if transfered to reconfigurable computing domain. Researchers in [19, 8] have studied this problem for applying loop fusion to code generation in compiler optimization area. By reduction from vertex cover problem, it has been shown that the general formulation of the problem is NPHard. 4 Experimental Results This section describes the experiments carried out to ....

Alain Darte. "On the complexity of loop fusion". Parallel Computing, 26(9):1175--1193, 2000.

....has been done on improving locality and parallelism by loop fusion. Kennedy and McKinley [13] presented an algorithm for fusing a collection of loops to minimize parallel loop synchronization and maximize parallelism. They proved that finding loop fusions that maximizes locality is NP hard. Darte [5] discusses the complexity of maximal fusion of parallel loops. A fast algorithm was presented by Kennedy in [12] that allows accurate modeling of data sharing as well as the use of fusion enabling transformations. Ding [6] illustrates the use of loop fusion in reducing storage requirements through ....

A. Darte. On the complexity of loop fusion. In International Conference on Parallel Architectures and Compilation Techniques, Newport Beach, CA, Oct. 1999.

....McKinley [10] presented an algorithm for fusing a collection of loops to minimize the parallel loop synchronization overhead and maximize parallelism. They proved that finding loop fusions that maximize locality is NP hard. Two polynomial time algorithms for improving locality were given. Darte [5] discusses the complexity of maximal fusion of parallel loops. Recently, the arrays presented in Fig. 2(a) 115.2MB N A 35.7 sec. b# 15.4MB N A 4.9 sec. 7.7MB N A 2.8 sec. 57.6MB N A 18.3 sec. T1(b,c,d, f ) T1(b,c,d, f ) 1.728GB 0 0 57.6MB 17.8 sec. 18.5 sec. Table 2: Loop ....

A. Darte. On the complexity of loop fusion. In Proc. International Conference on Parallel Architectures and Compilation Techniques (PACT'99), Newport Beach, CA, October 1999.

....single loop nest would fail to respect the dependences required element V[i] j 1] is used by iteration (i; j) of S2 but is generated in iteration (i; j 1) of S1. We show how the validity of loop fusion is tested in Section 4.3. However, it turns out that these loops can be fused. The trick [18] is to renumber S2.IterationSpace by shifting it by 1 in both i and j. This aligns iteration (i 1; j 1) of S1 with iteration (i; j) of S2. Now no dependence violation occurs. 4 Using Component Dependence Metadata This section illustrates how component dependence metadata can be used to solve ....

Alain Darte. On the complexity of loop fusion. In Proceedings of the

....single loop nest would fail to respect the dependences required element V[i] j 1] is used by iteration (i; j) of S2 but is generated in iteration (i; j 1) of S1. We show how the validity of loop fusion is tested in Section 5.3. However, it turns out that these loops can be fused. The trick [16] is to renumber S2.IterationSpace by shifting it by 1 in both i and j. This aligns iteration (i 1; j 1) of S1 with iteration (i; j) of S2. Now no dependence violation occurs. 5 class Region2 public int i lower, i upper, j lower, j upper; Constructor Region2(int i l,int i u, int ....

Alain Darte. On the complexity of loop fusion. In Proceedings of the 1999 International Conference on Parallel Architectures and Compilation Techniques (PACT '99), pages 149--157, Newport Beach, California, October 12--16, 1999. IEEE Computer Society Press.

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A. Darte. On the complexity of loop fusion. In PACT'99, pages 149-157, Newport Beach, CA, Oct. 1999.

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Alain Darte. On the complexity of loop fusion. Parallel Computing, 26(9):1175--1193, July 2000.

....developing a heuristic based on a maxflow mincut algorithm. Then, several authors (see the Related Work section) contributed to loop fusion optimizations, but with slightly di#erent objectives, focusing on loop fusion for locality [17] weighted loop fusion [19] maximal fusion (number of loops) [4], loop fusion for memory reduction [25, 15] etc. All these approaches keep in mind array contraction but they do not optimize directly for it. They target variants of data locality (for example, number of fused dependences) and, in favorable cases (but not always) they can achieve array ....

....array contraction are in practice not very large, solutions based on integer linear programming (ILP) for example can be practical, especially if the number of variables and constraints remain small. The goal of this section is to give such an ILP formulation, extending the techniques developed in [19, 2, 4], also for loop fusion but with di#erent objectives. We start with loop fusion first. For each statement u, we introduce an integer variable #(u) that represents the loop (i.e. cluster) number in which u will be placed in the fusion partition. If e = u, v) is a precedence or contractable arc, ....

Alain Darte. On the complexity of loop fusion. Parallel Computing, 26(9):1175--1193, July 2000.

.... without changing the iteration domain) is the basic tool for Allen, Callahan, and Kennedy s algorithm [1] This transformation is su cient for parallelism detection in many practical cases, and its simplicity makes possible its study with other objectives: for example, partial loop distribution [7] can be used to derive parallel loops including loop independent dependences, fusion techniques can derive codes with better locality [23] loop distribution can be extended for complex ow programs [17] etc. Our goal here is to make a similar study when shifting statements, the main question ....

Alain Darte. On the complexity of loop fusion. In International Conference on Parallel Architectures and Compilation Techniques (PACT'99), Newport Beach, CA, October 1999.

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A. Darte. On the complexity of loop fusion. In Proceedings of the

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A. Darte. On the complexity of loop fusion. Parallel Computing, 26(9):1175--1193, 2000.

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Alain Darte. "On the complexity of loop fusion". Parallel Computing, 26(9):1175--1193, 2000.

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Alain Darte. On the complexity of loop fusion. In Proceedings of International Conference on Parallel Architecture and Compilation Techniques, pages 149--157, Newport Beach, California, October 1999.

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Alain Darte. On the complexity of loop fusion. Parallel Computing, 26(9):1175{ 1193, 2000.

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A. Darte. On the complexity of loop fusion. In Proceedings of the

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Alain Darte. "On the complexity of loop fusion". Parallel Computing, 26(9):1175--1193, 2000.

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Alain Darte. On the complexity of loop fusion. In Proceedings of International Conference on Parallel Architecture and Compilation, pages 149--157, Newport Beach, CA, Oct 1999.

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A. Darte. On the Complexity of Loop Fusion. In Proc. of the Int. Conference on Parallel Architectures and Compilation Techniques, pages 149-157, Newport Beach, California, USA, 1999.

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A. Darte. On the complexity of loop fusion. In International Conference on Parallel Architectures and Compilation Techniques, Newport Beach, CA, Oct. 1999.

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A. Darte. On the complexity of loop fusion. In IEEE PACT, pages 149--157, 1999.

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