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Search. www.math.sc.chula.ac.th/~jaruloj/C681/6_BB.ppt. Outline. Problem space/ State space Exhaustive search Depth-first search Breadth-first search Backtracking Branch-and-bound. Problem Space. or State Space. Problem Space. General problem statement Given a problem instance P,

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  1. Search www.math.sc.chula.ac.th/~jaruloj/C681/6_BB.ppt

  2. Outline • Problem space/ State space • Exhaustive search • Depth-first search • Breadth-first search • Backtracking • Branch-and-bound Search

  3. Problem Space or State Space

  4. Problem Space • General problem statement • Given a problem instance P, find answer A=(a1, a2, …, an)such that the criteria C(A, P) is satisfied. • Problem space of P is the set of all possible answers A. Search

  5. Example: Shortest path • Given a graph G=(V,E), and nodesu and v, find the shortest path between u and v. • General problem statement • Given a graph G and nodes u and v, find the path (u, n1, n2, …, nk, v), and (u, n1, n2,…, nk, v) is the shortest path between u and v. • Problem space • Set of all possible path between u and v. • {(u, n1, n2, …, nk, v)| ni is in V, for 1ik}. Search

  6. Example: 0/1 Knapsack • Given a set S of n objects, where the object i has value viand weight wi , and a knapsack with weight capacity C, find the maximum of value of objects in S which can be put in the knapsack. • General problem statement • Given viand wi , for 1  i  n , find the set K such that for each i in K, 1  i  n,  vi is maximum while  wi C. iK iK • Problem space • Any subset of S. Search

  7. Example: n Queens • Given an nxn board, find the n squares in the board to put n queens so that no pair can attack. • General problem statement • Find (p1, p2, …, pn) where pi = (xi, yi) is a square on the board, where there is no pair (xi, yi) and (xj, yj) such that xi = xj or yi = yj. • Problem space • A set of any n positions on the board. Search

  8. Exhaustive Search

  9. Exhaustive Search • Generate every possible answer • Test each answer with the constraint to find the correct answer • Inefficient because the number of answers in the problem space can be exponential. • Examples: • Shortest path • n! paths to be considered, where n is the number of nodes. • 0/1 Knapsack • 2n selections, where n is the number of objects. • n Queens • n2!/n! (n2-n)! Search

  10. State-Space Tree • Let (a1, a2, …, an) be a possible answer. • Suppose ai is either 0 or 1, for 1  i nใ (?, …, ?) (0, ?, …, ?) (1, ?, …, ?) (0, 0, ?, …, ?) (0, 1, ?, …, ?) (1, 0,?, …, ?) (1, 1, ?, …, ?) (0,0,0, …, ?) (0,0,1, …, ?) (0,1,0, …, ?) (0,1,1, …, ?) Search

  11. State-Space Tree: Shortest Path (1) 8 3 5 2 5 2 (1,2) (1,3) 1 1 -6 -1 (1,2,3) (1,2,4) (1,3,4) (1,3,5) 2 2 4 (1,2,3,4) (1,2,3,5) (1,2,4,5) (1,3,4,5) (1,2,3,4,5) Search

  12. Generating Possible Paths • Let {1,2,3, …, n} be a set of nodes and E[i][j] is the weight of the edge between node i and j. path(p) last = last node in the path p fornext = 1 ton donp = p ifnext is not in np and E[last][next] != 0 thennp = np || next path(np) elsereturn Search

  13. State-Space Tree : 0/1 Knapsack • Given a set of objects o1, …, o5. { } {1} {2} {3} {4} {5} {1,2} {1,3} {1,4} {1,5} {1,2,3} {1,2,4} {1,2,5} {1,3,4} {1,3,5} {1,4,5} {1,2,3,4} {1,2,3,5} {1,2,4,5} {1,3,4, 5} {1,2,3,4,5} Search

  14. State-Space Tree : n Queen Search

  15. Depth-first Search • Traverse the tree from root until a leaf is reached. • Then, traverse back up to visited the next unvisited node. depthFirst(v) visited[v] = 1 for each node k adjacent to v doif notvisited[k] then depthFirst(k) Search

  16. 0/1 Knapsack: Depth-first Search Global: maxV=0 maxSack={} DFknapsack(sack, unchosen) for each object p in unchosen dounchosen=unchosen-{p} sack=sack U {p} val=evaluate(sack) ifunchosen is empty ► A leaf is reached. thenmaxV=max(maxV, val) if maxV=val thenmaxSack=sack return else DFknapsack(sack, unchosen) return Search

  17. Breadth-first Search • Traverse the tree from root until the nodes of the same depth are all visited. • Then, visited the node in the next level. breadthFirst(v) Q = empty queue enqueue(Q, v) visited[v] = 1 while not empty (Q) dou = dequeue(Q) for each node k adjacent to u do ifnotvisited[k] thenvisited[k] = true enqueue(Q, k) Search

  18. 0/1 Knapsack: Breadth-first Search BFknapsack Q = empty queue maxV=0 sack = { } unchosen = set of all objects enqueue(Q, <sack, unchosen>) while not empty (Q) do <sack, unchosen> = dequeue(Q) if unchosen is not empty then for each object pinunchosen do enqueue(Q,<sackU{p}, unchosen-{p}>) else maxV = max(maxV, evaluate(sack)) if maxV=evaluate(sack) thenmaxSack = sack Search

  19. Backtracking

  20. Backtracking • Reduce the search by cutting down some branches in the tree Search

  21. 0/1 Knapsack: Backtracking {} 0,0 Sack Current weight, current value Node {1} 2,5 {2} 1,4 {3} 3,8 {4} 2,7 {1,2} 3,9 {1,3} 5,13 {1,4} 4,12 {2,3} 4,12 {2,4} 3,11 {3,4} 5,15 {2,3,4} 6,19 {1,2,3} 6, 17 {1,2,4} 5, 16 Capacity = 5 Search

  22. 0/1 Knapsack: Backtracking BTknapsack(sack, unchosen) for each object p in unchosen dounchosen=unchosen-{p} ifp can be put in sackthensack = sack U {p} ► Backtracking occurs when p cannot be put in sack. val=evaluate(sack) ifunchosen is empty ► A leaf is reached. thenmaxV=max(maxV, val) maxSack=sack return else BTknapsack(sack, unchosen) return Search

  23. Branch-and-Bound

  24. Branch-and-bound • Use for optimization problems • An optimal solution is a feasible solution with the optimal value of the objective function. • Search in state-space trees can be pruned by using bound. Search

  25. State-Space Tree with Bound State bound State bound State bound State bound State bound State bound State bound State bound State bound State bound State bound Search

  26. Branch and Bound From a node N: • If the bound of N is not better than the current overall bound, terminate the search from N. • Otherwise, • If N is a leaf node • If its bound is better than the current overall bound, update the overall bound and terminate the search from N. • Otherwise, terminate the search from N. • Otherwise, search each child of N. Search

  27. 0/1 Knapsack: Branch-and-Bound Global: OvBound=0 BBknapsack(sack, unchosen) if bound(sack, unchosen)>OvBound ► Estimated bound can be better than the overall bound. then for each object p in unchosen dounchosen=unchosen-{p} ifp can be put in sackthensack = sack U {p} ► Backtracking occurs when p cannot be put in sack. val=evaluate(sack) ifunchosen is empty ► A leaf is reached. thenmaxV = max(maxV, val) maxSack = sack OvBound = max(evaluate(sack), OvBound) return else► A leaf is not reached. BBknapsack(sack, unchosen) return Search

  28. 0/1 Knapsack: Estimated bound • Current value Search

  29. 0/1 Knapsack: Branch-and-bound {} 20 0,0 Sack estimated bound Current weight, current value Node {1}17 2,5 {2}18 1,4 {3}16 3,8 {4}19 2,7 {1,2}16 3,9 {1,3} 13 5,13 {2,3}15.5 4,12 {2,4}16.3 3,11 {3,4}15 5,15 {1,2,4} 16 5, 16 Overall bound 16 0 Capacity = 5 Search

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