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Route Planning Branch & Bound CIS548 November 15, 2006
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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
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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
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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
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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
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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.
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