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

433

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