01 knapsack using backtracking

Backtracking Technique ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Outline of Backtracking Approach ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 2 3 5 7 11 4 6 8 9 10

N-Queens Problem ,[object Object],[object Object],[object Object],Q Q Q Q 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

N-Queens Problem: DFS ,[object Object]

N-Queens Problem: DFS (Same Col/Row x) ,[object Object],X 1 =1 X 2 =2 3 4 4 3 4 3 4 2 3 2 3 2 2 4 3 2 3 4 1 4 1 3 2 4 1 3 4 1 2 4 3 2 1 2 3 4 (x 1 , x 2 , x 3 , x 4 )=(2, 4, 1, 3) ,[object Object],[object Object],[object Object]

N-Queens Problem: Backtracking ,[object Object],[object Object],[object Object],[object Object],Q (i, col(i)) Q (k, col(k)) Q (i,col(i)) Q (k,col(k))

N-Queens Problem: Backtracking ,[object Object]

N-Queens Problem ,[object Object],[object Object],[object Object],[object Object],#Nodes Checked DFS (n n ) #Nodes Checked DFS (same col/row X) (n!) #Nodes Checked Backtracking 341 19,173,961 9.73 ⅹ10 12 1.20ⅹ 10 16 24 40,320 4.79 ⅹ10 8 8.72ⅹ 10 10 61 15,721 1.01 ⅹ10 7 3.78ⅹ 10 8 4 8 12 14 n

Graph Coloring ,[object Object],[object Object],[object Object],[object Object],V 1 V 2 V 4 V 3 2-coloring X 3-coloring O

Graph Coloring ,[object Object],[object Object],[object Object],start 1 2 3 2 1 3 2 1 3 2 1 3 X X X X X

Hamiltonian Circuits Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Hamiltonian Circuits Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],HC : X (Eg) V 1 V 2 V 6 V 5 V 3 V 7 V 4 V 8 V 1 V 2 V 5 V 3 V 4

Sum-of-Subsets Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],A w 1 w 2 w 3 w 2 w 3 w 3 w 3 O O O O O O O {w 1 ,w 2 }

Sum-of-Subsets Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],ⅹ 12 7 3 9 7 8 13 4 0 0 3 3 0 4 5 0 0 4 5 5 6 0 0 0 0 ⅹ ⅹ : 0+5+6=11<13 ⅹ ⅹ : 9+6=15>13 ⅹ ⅹ w1=3 w2=4 w3=5 w4=6 0 4 3 7

0-1 Knapsack Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],x 1 =1 x 2 =1 x 3 =1 x 4 =1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 x 2 =1

Branch-and-Bound ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Breadth First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],BFS of a graph BFS of a tree 1 2 3 4 5 6 7 8 9 1 2 5 6 7 8 9 3 4 10 11 12 13 14 15

0-1 Knapsack Problem ,[object Object],[object Object],0 0 115 40 2 115 0 0 82 70 7 115 40 2 98 120 17 0 70 7 80 80 12 80 70 7 70 90 12 98 40 2 50 100 17 0 90 12 90 30 5 82 80 15 82 30 5 40 0 0 60 p1=40 , w1=2 p2=30 , w2=5 p3=50 , w3=10 p4=10 , w4=5

0-1 Knapsack Problem ,[object Object],[object Object],[object Object],0 0 115 40 2 115 0 0 82 70 7 115 40 2 98 120 17 0 70 7 80 80 12 80 70 7 70 90 12 98 40 2 50 100 17 0 90 12 90 30 5 82 80 15 82 30 5 40 0 0 60 p1=40 , w1=2 p2=30 , w2=5 p3=50 , w3=10 p4=10 , w4=5

0-1 Knapsack Problem ,[object Object],0 0 115 40 2 115 0 0 82 70 7 115 40 2 98 120 17 0 70 7 80 90 12 98 40 2 50 100 17 0 p1=40 , w1=2 p2=30 , w2=5 p3=50 , w3=10 p4=10 , w4=5 90 12 90

01 knapsack using backtracking

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01 knapsack using backtracking