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CAS 721 Course Project

CAS 721 Course Project. Implementing Branch and Bound, and Tabu search for combinatorial computing problem By Ho Fai Ko (9700749). Overview. Background of problem Background of algorithms Branch and Bound Tabu Search Implementation detail Analysis of algorithm. Chip Realization Process.

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CAS 721 Course Project

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  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

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