1. Prabhas Chongstitvatana 1 Backtracking Eight queens problem 1 try all possible C 64 8 = 4,426,165,368 2 never put more than one queen on a given row, vector representation : each row specify which column (3,1,6,2,8,6,4,7)

2. Prabhas Chongstitvatana 2 (3,1,6,2,8,6,4,7) 1 2 3 4 5 6 7 8 X X X X X X X X

3. Prabhas Chongstitvatana 3 Queen1 for i 1 = 1 to 8 do for i 2 = 1 to 8 do.... for i 3 = 1 to 8 do sol = [i 1, i 2,... i 8 ] if solution ( sol ) then write sol stop write “ there is no solution ” Num. Of positions = 8 8 = 16,777,216 (first soln after 1,299,852 )

4. Prabhas Chongstitvatana 4 3 Never put queen on the same row (different numbers on soln vector) Queen2 sol = initial-permutation while sol != final-permutation and not solution(sol) do sol = next-permutation if solution(sol) then write sol else write “ there is no solution ”

5. Prabhas Chongstitvatana 5 Permutation T[1.. n] is a global array initialize to [1,2,.. n] initial call perm(1) Perm(i) if i = n then use T else for j = i to n do exchange T[i] and T[j] perm(i+1) exchange T[i] and T[j] Number of positions 8! = 40,320 (first soln after 2830)

6. Prabhas Chongstitvatana 6 8-queen as tree search a vector V[1..k] of integers between 1 and 8 is k-promising, if none of the k queens threatens any of the others. A vector V is k-promising if, for every pair of integers i and j between 1 and k with i != j, we have V[i] - V[j] is-not-in {i-j, 0, j-i}. Solutions to the 8-queen correspond to vectors that are 8-promising.

7. Prabhas Chongstitvatana 7 Let N be the set of k-promising vectors, k: 0.. 8. Let G = (N,A) be the directed graph such that (U,V) is-in A iff there exists an integer k, k:0..8, such that U is k-promising V is (k+1)-promising, and U[i] = V[i] for every i in [1..k]... k = 0 k = 8 Number of node < 8! (node 2057, first soln after 114 )

8. Prabhas Chongstitvatana 8 General Template for backtracking Backtrack ( v[1..k] )// v is k-promising vector if solution ( v ) then write v else for each (k+1)-promising vector w such that w[1..k] = v[1..k] do backtrack( w[1.. k+1] )

9. Prabhas Chongstitvatana 9 Branch and bound The assignment problem, n agents are to be assigned n tasks, each agent having exactly one task. If agent i, i:1..n, is assigned task j, j: 1..n, then the cost of performing this task is c ij. Given the cost matrix, the problem is to assign agents to tasks so as to minimize the total cost of executing the n tasks.

10. Prabhas Chongstitvatana 10 Upper bound on the answer : a:1, b:2, c:3, d:4 = 11+15+19+28 = 73 Lower bound (sum smallest elements) 11+12+13+22 = 58 answer [58.. 73] 1 2 3 4 a 11 12 18 40 b 14 15 13 22 c 11 17 19 23 d 17 14 20 28

11. Prabhas Chongstitvatana 11 Explore a tree whose nodes correspond to partial assignment. Use lower bound to guide the search. a:1 a:2 a:3 a:460 58 65 78 * a:1 ; 11+14+13+22 = 60

12. Prabhas Chongstitvatana 12 1 2 3 4 a 11 12 18 40 b 14 15 13 22 c 11 17 19 23 d 17 14 20 28

13. Prabhas Chongstitvatana 13 a:1 a:2 a:3 a:4 60 65 78 * a:2,b:1 a:2,b:3 a:2,b:4 68 59 64

14. Prabhas Chongstitvatana 14 a:1 a:2 a:3 a:4 60 65 * 78 * a:2,b:1 a:2,b:3 a:2,b:4 68 * 64 * a:2,b:3,c:1,d:4 a:2,b:3,c:4,d:1 64 65 *

15. Prabhas Chongstitvatana 15 a:2 a:3 a:4 a:1 65 * 78 * a:2,b:1 a:2,b:3 a:2,b:4 68 * 64 * a:2,b:3,c:4,d:1 a:2,b:3,c:1,d:4 64 a:1,b:4 a:1,b:3 a:1,b:2 a:1,b:3,c:4,d:2 61 a:1,b:3,c:2,d:4 69* 68* 66* 65*