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### traveling salesperson.pptx

• 1. NADAR SARASWATHI COLLEGE OF ARTS AND SCIENCE,THENI. DEPARTMENT OF COMPUTER SCIENCE TRAVELING SALESPERSON BY B.POORANI
• 2. TRAVELING SALESPERSON ๏ด Let G=(V,E) be a directed graph defining an instance of the traveling salesperson problem. ๏ด Let Cij be the cost of edges (i,j),Cij=infinity if (i,j) and let V=N. ๏ด Each leaf node L is a solution node and represents the tour defined by the path from the root to L.
• 3. ๏ดThe progress of the LCBB algorithm on the problem instance. ๏ดThe initial reduced matrix is that upper=โ.The portion of the state space tree that gets generated. ๏ดStarting with the root node as the E-node ,nodes 2, 3,4 and 5 are generated.
• 4. ๏ด The matrix of is obtained from the by (1) setting all entries in row 1 and column 3 to โ. ๏ด (2) setting the element at position (3,1) to โ,and (3) reducing column 1 by subtracting by 11. ๏ด The value of upper is unchanged and node 4 becomes the next E-node . Its children 6,7 and 8 are generated. ๏ด The live nodes at this time are node 2,3,5,6,7 and 8 .
• 5. ๏ด Node 6 has least c value and become the next E-node. ๏ด Nodes 9 and 10 are generated. ๏ด Node 10 is the next E-node . The solution node , node 11 , is generated. ๏ด The tour length for this node is c(11)=28 and upper is updated to 28. ๏ด For the next E-node , node 5,c(5)=31>upper.
• 6. ๏ด To use LCBB to search the traveling salesperson state space tree , we need to define a cost function c(.) and two other function c^(.) and u(.). ๏ด c(A)={length of tour defined by the path from the root to A , if A is a leaf .cost of a minimum-cost leaf in the subtree A , if A is not a leaf.
• 7. ๏ด A different LCBB algorithm can be arrived at by considering a different tree organization for the solution space. ๏ด This organization is reached by regarding a tour as a collection of n edges. If G=(V,E) has e edges , then every tour contains exactly n of the e edges. ๏ด A possible organization for the state space is a binary tree in which a left branch represents the inclusion of a particular edges while the right branch represents the exclusion of the edges.
• 8. ๏ด An example of low LCBB would work on the dynamic binary tree formulation . consider the cost matrix . ๏ด We must decide which edges to use to partition the solution space into two subsets . if edges (i,j) is used , then the left subtree of the root represents all tour including edges (i,j) and the right subtree represents all tour that do not include edges (i,j).
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