1. Route Planning Branch & Bound CIS548 November 15, 2006

2. 2006Kutztown University2 Limo Driver Review of problem Review of problem  Driver has n pickup/dropoff pairs  Limo leaves garage in morning; returns in evening  Goal :: complete pick/drop tasks traveling shortest possible distance Asymmetric TSP Asymmetric TSP  Once pickup is made driver must proceed to dropoff point  Trip to next pickup point always emanates from last dropoff point  So trip from A to B is actually from dA to pB  Therefore distance(A → B) ≠ distance(B → A)  Thus, we have asymmetric TSP One standard technique to obtain optimal solution – Branch & bound One standard technique to obtain optimal solution – Branch & bound

3. 2006Kutztown University3 Branch & Bound One approach – generate tree of all possible paths One approach – generate tree of all possible paths  Example – GenPerms GenPerms  O (Time) = N!  O (Space) = N  Can we do better? Generate tree; prune non-optimal branches Generate tree; prune non-optimal branches For each node generated.. For each node generated..  Calculate an upper & lower bound for reaching goal  If lower bound of node exceeds upper bound for some other node: » Prune the node » Branch below node is pruned as well  If node contains goal » If its value ≤ lower bound of all other nodes  DONE » Otherwise, continue

4. 2006Kutztown University4 B&B – S 3 implementation Use same heuristic as before Use same heuristic as before  Sum of min distances to remaining pickup points  Use f-value for lower bound  Use sum of max distances as upper bound Order nodes by f-value (lower bound) Order nodes by f-value (lower bound) Keep global upper & lower bounds Keep global upper & lower bounds When a node is generated.. When a node is generated..  If its lower bound is ≥ global upper bound, discard it  If its lower (upper) bound ) global one, update global value(s)  Insert new node into linked list of waiting nodes If global upper bound is changed.. If global upper bound is changed..  Prune linked list of waiting nodes  Discard nodes with lower bound ≥ global upper bound

5. 2006Kutztown University5 S 3 with B&B – Discussion What have we gained? What have we gained? What are upper & lower bounds on time? What are upper & lower bounds on time?  With & without B&B? What are upper & lower bounds on space? What are upper & lower bounds on space?  With & without B&B? What are O (Time) and O (Space)? What are O (Time) and O (Space)?  With & without B&B? What are empirical results? What are empirical results? Key parameters – Key parameters –  Time :: number of nodes expanded & produced  Space :: maximum length of nodes-waiting list

6. 2006Kutztown University6 S 3 with B&B – Exercises Construct a Limo Driver instance where the maximum length of the nodes-waiting list differs with & without B&B. Construct a Limo Driver instance where the maximum length of the nodes-waiting list differs with & without B&B. Prove or disprove: the number of nodes expanded is the same with & without B&B. Prove or disprove: the number of nodes expanded is the same with & without B&B.  To disprove :: Construct a Limo Driver instance where the number of nodes expanded differs with & without B&B.  To prove :: Prove that if a node is expanded without B&B it will also be expanded with B&B; and prove the converse.