Branch & bound

Branch & bound

• 1.
  Backtracking And BranchAnd Bound
• 2.
  Subset & PermutationProblems • Subset problem of size n.  Nonsystematic search of the space for the answer takes O(p2n ) time, where p is the time needed to evaluate each member of the solution space. • Permutation problem of size n.  Nonsystematic search of the space for the answer takes O(pn!) time, where p is the time needed to evaluate each member of the solution space. • Backtracking and branch and bound perform a systematic search; often taking much less time than taken by a nonsystematic search.
• 3.
  Tree Organization OfSolution Space • Set up a tree structure such that the leaves represent members of the solution space. • For a size n subset problem, this tree structure has 2n leaves. • For a size n permutation problem, this tree structure has n! leaves. • The tree structure is too big to store in memory; it also takes too much time to create the tree structure. • Portions of the tree structure are created by the backtracking and branch and bound algorithms as needed.
• 4.
  Subset Tree Forn = 4 x1=1 x1= 0 x2=1 x2= 0 x2=1 x2= 0 x3=1 x3= 0 x4=1 x4=0 1110 1011 0111 0001
• 5.
  Permutation Tree Forn = 3 x1=1 x1=2 x1= 3 x2= 2 x2= 3 x2= 1 x2= 3 x2= 1 x2= 2 x3=3 x3=2 x3=3 x3=1 x3=2 x3=1 123 132 213 231 312 321
• 6.
  Backtracking • Search thesolution space tree in a depth- first manner. • May be done recursively or use a stack to retain the path from the root to the current node in the tree. • The solution space tree exists only in your mind, not in the computer.
• 7.
  Backtracking Depth-First Search x1=1x1= 0 x2=1 x2= 0 x2=1 x2= 0
• 8.
  Backtracking Depth-First Search x1=1x1= 0 x2=1 x2= 0 x2=1 x2= 0
• 9.
  Backtracking Depth-First Search x1=1x1= 0 x2=1 x2= 0 x2=1 x2= 0
• 10.
  Backtracking Depth-First Search x1=1x1= 0 x2=1 x2= 0 x2=1 x2= 0
• 11.
  Backtracking Depth-First Search x1=1x1= 0 x2=1 x2= 0 x2=1 x2= 0
• 12.
  O(2n ) Subet Sum& Bounding Functions x1=1 x1= 0 x2=1 x2= 0 x2=1 x2= 0 Each forward and backward move takes O(1) time. {10, 5, 2, 1}, c = 14
• 13.
  Backtracking • Space requiredis O(tree height). • With effective bounding functions, large instances can often be solved. • For some problems (e.g., 0/1 knapsack), the answer (or a very good solution) may be found quickly but a lot of additional time is needed to complete the search of the tree. • Run backtracking for as much time as is feasible and use best solution found up to that time.
• 14.
  Branch And Bound •Search the tree using a breadth-first search (FIFO branch and bound). • Search the tree as in a bfs, but replace the FIFO queue with a stack (LIFO branch and bound). • Replace the FIFO queue with a priority queue (least-cost (or max priority) branch and bound). The priority of a node p in the queue is based on an estimate of the likelihood that the answer node is in the subtree whose root is p.
• 15.
  Branch And Bound •Space required is O(number of leaves). • For some problems, solutions are at different levels of the tree (e.g., 16 puzzle). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 32 4 56 13 14 15 12 11 10 9 78
• 16.
  Branch And Bound FIFO branch and bound finds solution closest to root.  Backtracking may never find a solution because tree depth is infinite (unless repeating configurations are eliminated). • Least-cost branch and bound directs the search to parts of the space most likely to contain the answer. So it could perform better than backtracking.