A Uniform Optimization Technique for Offset Assignment Problems

1. A Uniform Optimization Technique for Offset Assignment Problems Rainer Leupers, Fabian David University of Dortmund, Germany Dept. of Computer Science 12

2. Overview • Offset assignment problem • Related work • Genetic algorithm approach • Exploitation of modify registers • Results & conclusions

3. Offset assignment problem Context: Code generation for DSPs Given: DSP address generation unit (AGU) with # address registers (ARs): k # modify registers (MRs): m auto-increment range (AIR): r Auto-increment capabilities: AR[i] += d, d <= r AR[i] += MR[j] Other address computations cause extra code ! Problem: Assign program variables to memory addresses and ARs, such that the use of auto-increment address computations is maximized !

4. 0 1 2 3 0 1 2 3 a c b a c d d b Offset assignment: example variables: { a, b, c, d } access sequence: (b, d, a, c, d, a, c, b, a, d, a, c, d) Layout 1 Layout 2 AR = 3 AR - - AR - - AR - - AR += 2 AR - - AR - - AR += 3 AR -= 2 AR ++ AR - - AR - - AR += 2 AR = 1 AR += 2 AR -= 3 AR += 2 AR ++ AR -= 3 AR += 2 AR - - AR - - AR += 3 AR -= 3 AR += 2 AR ++ Simple Offset Assignment: k = 1 m = 0 r = 1 cost: 9 cost: 5

5. Related work Offset assignment for different AGU models: #ARs #MRs AIR [Bartley92] 1 - 1 [Liao95] k - 1 [Leupers96] k m 1 [Wess97] 1 - 2 [Sudarsanam97] k - r this work k m r Further work on address optimization for fixed layout

6. Genetic algorithm approach (1) Chromosomal representation: n variables, k address registers each individual is a permutation of { 1, ..., n + k - 1} example: n = 6, k = 2 switch to next AR offset mapping 0 1 2 3 4 5 2 5 3 1 7 6 4 AR[1] AR[2] Fitness function: F(I) = # transitions (v,w) in access sequence, such that v, w assigned to different ARs, or |off(v) - off(w) | <= r

7. Genetic algorithm approach (2) Mutation: exchange two gene values x y y x Crossover: standard order crossover operation Optimization procedure: form initial population for N generations do: select parent individuals generate offspring mutate offspring emit best individual

8. Exploitation of modify registers (1) [Leupers96]: Modification of Belady‘s optimal page replacement algorithm can be used for optimal exploitation of m MRs for a fixed offset assignment (only postpass optimization) PRA(I) = # address computations that can be saved by exploiting MRs for a given offset assignment modified fitness function: F´(I) = F(I) + PRA(I) => exploitation of MRs included into GA !

9. Exploitation of modify registers (2) AR = 2 AR - - MR = 3 AR += MR AR ++ AR -= 2 AR - - AR ++ AR - - AR += MR AR -= MR MR = 2 AR += MR AR - - AR += 0 AR += 0 AR -= MR AR ++ AR - - AR ++ AR += 0 AR += 3 AR = 4 AR ++ MR = 2 AR -= MR AR -= MR AR ++ AR += MR AR -= MR AR += MR MR = 3 AR -= MR AR += MR AR - - AR - - AR += 0 AR += 0 AR += MR AR - - AR ++ AR - - AR += 0 AR -= MR heuristic OA + PRA genetic algorithm heuristic OA AR = 2 AR - - AR += 3 AR ++ AR -= 2 AR - - AR += 3 AR -= 3 AR += 2 AR - - AR += 0 AR += 0 AR -= 2 AR ++ AR - - AR += 0 AR += 3 v4 v0 v2 v1 v3 0 1 2 3 4 0 1 2 3 4 v4 v0 v2 v1 v3 v3 v2 v1 v0 v4 0 1 2 3 4

10. Results & conclusions • Statistical evaluation: • 32 % improvement over OA heuristic with postpass • MR optimization • 32 % improvement over Wess‘ simulated annealing • technique • Runtime: typically 10 CPU seconds (Pentium II) • Main contributions: • First uniform offset assignment technique, • arbitrary k, m, r values • Significant improvements in code quality over previous • techniques, largely due to better exploitation of MRs