management science 461 lecture 2b – shortest paths september 16, 2008
TRANSCRIPT
Management Science 461
Lecture 2b – Shortest Paths
September 16, 2008
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Graph Theory
Branch of mathematics “Graph” has a very specific definition:
Collection of objectsCollection of relationships between objects
Both collections can have associated data (costs, weights)
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Physical Abstraction
Euler and the Königsberg Bridges - 1736
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Shortest Path Problems
Find the shortest (distance, time, cost) path between two nodes (e.g. 15)
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Node = Vertex
Arc = Edge = Link
Route = Path
Tour = Circuit = Cycle
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Algebraic formulation
otherwise0
selected is to node connecting edge theif1 jixij
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9452543
344224
23321312
954
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5528
minimize
xxx
xxx
xxxx
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Flow and Integrality Constraints
subject to x12 + x13 = 1 (node 1)x12 + x32 + x42 = x23 + x24 + x25 (node 2)x13 + x23 + x43 = x32 + x34 (node 3)x34 + x24 = x43 + x42 + x45 (node 4)x25 + x45 = 1 (node 5)xij = 0 or 1 for all i, j
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Rearranging, we get…
subject to - x12 - x13 = -1 (node 1)x12 + x32 + x42 - x23 - x24 - x25 = 0 (node 2)x13 + x23 + x43 - x32 - x34 = 0 (node 3)x34 + x24 - x43 - x42 - x45 = 0 (node 4)x25 + x45 = 1 (node 5)xij = 0 or 1 for all i, j
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Flow in = flow out flow in – flow out = 0
Do these -1, 0, 1 values look familiar?
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Shortest Path Problems
Solve using Excel’s Solver
However, a network with several hundred nodes would be out of reach, even for Premium Solver
e.g. Edmonton
Efficient algorithms need to be used – like Dijkstra’s algorithm
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Dijkstra’s algorithm
“Label-setting” algorithmAt each node, keep track of minimum
distance from origin to the node, and how we got there
Very fast – polynomial time algorithmTime to solve increases polynomially with
problem size (not exponential)
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Dijkstra’s Algorithm
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Step 0 – Place Labels
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5Dist: 0Pred: 1
Dist: Pred: 2
Dist: Pred: 5
Dist: Pred: 4
Dist: Pred: 3
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Step 1 – Choose Minimum Label
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5Dist: 0Pred: 1
Dist: Pred: 2
Dist: Pred: 5
Dist: Pred: 4
Dist: Pred: 3
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Step 2 – Update Adjacent Labels
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5Dist: 0Pred: 1
Dist: 8Pred: 1
Dist: Pred: 5
Dist: Pred: 4
Dist: 2Pred: 1
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Step 2b – Mark Node As Optimal
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5Dist: 0Pred: 1
Dist: 8Pred: 1
Dist: Pred: 5
Dist: Pred: 4
Dist: 2Pred: 1
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Step 3 – Choose Next Min Dist
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5Dist: 0Pred: 1
Dist: 8Pred: 1
Dist: Pred: 5
Dist: Pred: 4
Dist: 2Pred: 1
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Step 4 – Update Adjacent Labels
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: Pred: 5
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 4b – Mark Node As Optimal
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: Pred: 5
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 5 – Choose Next Min Dist
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: Pred: 5
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 6 – Update Adjacent Labels
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 15Pred: 4
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 6b – Mark Node As Optimal
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 15Pred: 5
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 7 – Choose Next Min Dist
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 15Pred: 5
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 8 – Update Adjacent Labels
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 12Pred: 2
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 8b – Mark Node As Optimal
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 12Pred: 2
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 9 – Choose Next Min Dist
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 12Pred: 2
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 10 – Update Adjacent Labels
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 12Pred: 2
Dist: 6Pred: 3
Dist: 2Pred: 1
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Step 10b – Mark Node As Optimal
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 12Pred: 2
Dist: 6Pred: 3
Dist: 2Pred: 1
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Final Solution – Min Dist Tree
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5Dist: 0Pred: 1
Dist: 7Pred: 3
Dist: 12Pred: 2
Dist: 6Pred: 3
Dist: 2Pred: 1