15 puzzle problem using branch and bound

15 puzzle problem using branch and bound

• 1.
  15 PUZZLE PROBLEM USINGBRANCH AND BOUND CREATED BY:- ABHISHEK KUMAR SINGH
• 2.
  • Branch andBound • The search for an answer node can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an answer node. It is similar to backtracking technique but uses BFS-like search. • There are basically three types of nodes involved in Branch and Bound • 1. Live node is a node that has been generated but whose children have not yet been generated. • 2. E-node is a live node whose children are currently being explored. In other words, an E-node is a node currently being expanded. • 3. Dead node is a generated node that is not to be expanded or explored any further. All children of a dead node have already been expanded.
• 3.
  • Cost function: •Each node X in the search tree is associated with a cost. The cost function is useful for determining the next E-node. The next E-node is the one with least cost. The cost function is defined as, • C(X) = g(X) + h(X) where g(X) = cost of reaching the current node from the root h(X) = cost of reaching an answer node from X • Ideal Cost function for 15-puzzle Algorithm : • We assume that moving one tile in any direction will have 1 unit cost. Keeping that in mind, we define cost function for 15- puzzle algorithm as below : • c(x) = f(x) + h(x) where f(x) is the length of the path from root to x (the number of moves so far) and h(x) is the number of non-blank tiles not in their goal position (the number of mis- - placed tiles). There are at least h(x) moves to transform state x to a goal state
• 4.
  1111 2@ 34 5 6 8 9 10 7 11 13 14 15 12 1111 2@ 33 4 5 6 7 8 9 10 11 12 13 14 15 Initial arrangement Goalarrangement
• 5.
  1 2 34 5 6 8 9 10 7 11 13 14 15 12 up right left down 1 2 4 5 6 3 8 9 10 7 11 13 14 15 12 1 2 3 4 5 6 8 9 10 7 11 13 14 15 12 1 2 3 4 5 6 7 8 9 10 11 13 14 15 12 1 2 3 4 5 6 8 9 10 7 11 13 14 15 12 C=1+4 C=1+4 C=1+2 C=1+4
• 6.
  1 2 4 56 3 8 9 10 7 11 13 14 15 12 1 2 3 4 5 6 8 9 10 7 11 13 14 15 12 1 2 3 4 5 6 7 8 9 10 11 13 14 15 12 1 2 3 4 5 6 8 9 10 7 11 13 14 15 12 1 2 3 4 5 6 7 8 9 10 11 13 14 15 12 1 2 3 4 5 6 7 8 9 10 15 11 13 14 12 1 2 3 4 5 6 7 8 9 10 11 13 14 15 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 13 14 15 12 C=2+1 C=2+3 C=3+0C=3+2 C=2+3
• 7.
  • ALGORITHM 15-PUZZLEPROBLEM • struct list_node • { • list_node *next; • list_node *parent; • float cost; • }; • algorithm LCSearch(list_node *t) • { • if (*t is an answer node) • { • print(*t); • return; • } • E =t; • Initialize the list of live nodes to be empty;
• 8.
  • while (true){ • for each child x of E • { • if x is an answer node { • print the path from x to t; • return; } • Add (x); • x->parent = E; } • if there are no more live nodes • { • print ("No answer node"); • return; • } • E = Least(); • } • } •