CAS 721 Course Project

1. CAS 721Course Project Implementing Branch and Bound, and Tabu search for combinatorial computing problem By Ho Fai Ko (9700749)

2. Overview • Background of problem • Background of algorithms • Branch and Bound • Tabu Search • Implementation detail • Analysis of algorithm

3. Chip Realization Process Requirements Audit Specification Design Implementation Design verification Chip Fabrication Manufacturing test Good chips Source: Essentials of Electronic Testing, Kluwer 2000

4. Manufacturing Test • Identify faulty chips before shipping them to customers • During volume production phase for mature process technologies • Improve yield through fault diagnosis • During yield ramp phase for new process technologies Defect density * Technology yield learning rates * Source Intel corporation

5. Introduction of SOC • System-on-a-chip (SOC) integrates multiple functional components (cores) on a single silicon die • Sample SOC* • 8 million-gate design • 2 ARM processors and 8 ADSL cores • Compared to board-level system, SOCs are: • Smaller and faster • More reliable, lower power and cost • Facing more challenges for test *Source: A DFT and Test Generation Methodology for a Large SoC Design， SNUG Europe 2001

6. Testing cores in SOCs • To test a core in a SOC • Test patterns are applied to the circuit-under-test (CUT) • Test responses from the CUT are obtained for failure analysis • Questions to be answered: • How to apply test patterns to CUT? • How to obtain test responses from CUT? • Problems: • Limited number of primary inputs and outputs on chips • Solution: • Using test access mechanism (TAM) to transport test data

7. Background of problem

8. Problem formulation • Given m TAM lines • Given n cores in a SOC • Given connectivity matrix C, such that • C[i][j] = 1 if TAM (i) can cover core (j) • Find the assignment for each TAM line to cover all the cores in the SOC such that • Number of TAM lines needed is sufficiently small • Objective function: min sum c[i][j] for all i and j • Subject to: • c[i][j] <= C[i][j] • Sum[i=1..m] c[i][j] >= 1 • c[i][j] = 0 or 1

9. Branch and Bound (BB) • A general exhaustive search algorithm • Can be implemented as a backtracking algorithm • Depth-first search approach • Can be implemented using a priority queue • Breadth-first search approach • Terminate when lower bound of solution is found, or when all possible solution are checked • Can speed up algorithm by carefully choosing the branch condition

10. Tabu search (TS) • Originally developed by Glover in 1986 • History-based heuristic search technique • Prohibits (Tabu) the reversal of previous moves • Encourage exploration of parts of the solution space that have not been visited previously • Speed and quality of solution depends on: • Choice of initial solution • The size, variability, and adaptability of the Tabu memory • How it works: • Begins by marching to a local optimum • Since recent moved are tabu, it is forced to search in new solution space • Do not guarantee to reach the optimal solution

11. Why choose Tabu search? • Other heuristics take advantage on exploring the relationships between decisions • Simulated Annealing • Genetic Algorithm • Decision on a variable c[i][j] = 0 or 1 does not depends on the decisions for other variables • The only concern for this problem is to not go backward so often when searching in solution space • Thus, by changing the Tabu length, the search direction from Tabu search can be better directed

12. Implementation – Data structure • Double link list • Store location of 1’s in the sparse matrix • Double link to speed up search process • Priority queue • Simple PUSH and POP operation to the queue • For branch and bound • Breadth-first-search technique • For Tabu search • Act as a FIFO queue with variable length

13. Implementation – Problem • Randomly generate a problem • Parameters • m: Number of TAM lines available • n: Number of Cores to be covered in a SOC • K: A TAM line has to cover at least K cores • The smaller K is, the less likely to find a solution • K <= n • To ensure randomness of the problem • If use a single random number generator, the problem is skewed • Use multiple generators to generate problem iteratively

14. Implementation - Problem • Example: n = 10, and K = 5 • Single generator • Choose random number R between K and n • How to randomly distribute the R 1’s in n column? • Multiple generators • Random select a sample size 1 <= S <= n for 1 iteration • Randomly decide if a 1 is assigned to a column • If total 1’s are not enough (i.e. < K), repeat • Iteration | Sample size | Selected size | |C| • 1 | 4 | 3 | 3 • 2 | 2 | 1 | 4 • 3 | 6 | 4 | 8

15. Implementation – BB • Use priority queue and breadth first search technique for branch and bound • The lower bound: • The value of objective function has to be >= n • If the constraint is satisfied: • Sum[i=1..m] c[i][j] >= 1 for all j • Solution is found • The branching condition: • The constraint to the problem: • c[i][j] <= C[i][j] • If constraint is violated, it is a dead end • If the cost of unfinished solution + number of cores that yet to be covered is already higher than the best one found so far, declare it as dead end

16. Implementation - TS • Use a FIFO queue with length specified as Tabu list • Start by trying to find an initial solution that satisfies all constraints without worrying about optimality • Length of Tabu list is changed every 20 iteration • To helps exploring new solution space • For each iteration: • Find the column with most 1’s to change • Check changes for each element in the column for constraint violation • If not violated, check if solution is better • If is better, keep the current best solution, and Tabu the move to avoid reversal moves

17. Program output

18. Analysis • The size of solution space is 2^(mn) • Even with really good branching condition • The size of solution space is 2^(Kn) • Still really large • Thus, execution time increases exponentially • For Branch and bound • Impossible to perform exhaustive search • For Tabu search • Fast • Does not know optimal solution is reached unless it reaches the lower bound of the solution space • Does not guarantee the optimality of solution

19. Summary • Implemented two algorithms for a combinatorial computing problem • Branch and Bound • Exhaustive search • Execution time grows exponentially • Tabu search • Fast • Does not guarantee to reach optimality