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)

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

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 )

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 ”

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)

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.

Prabhas Chongstitvatana 7 Let N be the set of k-promising vectors, k: 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 )

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] )

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.

Prabhas Chongstitvatana 10 Upper bound on the answer : a:1, b:2, c:3, d:4 = = 73 Lower bound (sum smallest elements) = 58 answer [ ] a b c d

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: * a:1 ; = 60

Prabhas Chongstitvatana a b c d

Prabhas Chongstitvatana 13 a:1 a:2 a:3 a: * a:2,b:1 a:2,b:3 a:2,b:

Prabhas Chongstitvatana 14 a:1 a:2 a:3 a: * 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: *

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*