Shortest Paths

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Discrete Mathematics and Its Applications (5th Edition)

Kenneth H. Rosen

Chapter 9.6

Based on slides from Chuck Allison, Michael T. Goodrich, and Roberto Tamassia

By Longin Jan Latecki

Weighted Graphs

Graphs that have a number assigned to each edge are called weighted graphs.

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Weighted Graphs

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Weighted Graphs

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Weighted Graphs

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FLIGHT TIMES

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Weighted Graphs

A weighted graph is a graph in which each edge (u, v) has a weight w(u, v). Each weight is a real number.

Weights can represent distance, cost, time, capacity, etc.

The length of a path in a weighted graph is the sum of the weights on the edges.

Dijkstra’s Algorithm finds the shortest path between two vertices.

Dijkstra's Algorithm

Dijkstra Animation

Demo

Problem: shortest path from a to z

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Theorems

Dijkstra’s algorithm finds the length of a shortest path between two vertices in a connected simple undirected weighted graph G=(V,E).

The time required by Dijkstra's algorithm is O(|V|2).

It will be reduced to O(|E|log|V|) if heap is used to keep {vV\Si : L(v) < }, where Si is the set S after iteration i.

The Traveling Salesman Problem

The traveling salesman problem is one of the classical problems in computer science.

A traveling salesman wants to visit a number of cities and then return to his starting point. Of course he wants to save time and energy, so he wants to determine the shortest cycle for his trip.

We can represent the cities and the distances between them by a weighted, complete, undirected graph.

The problem then is to find the shortest cycle (of minimum total weight that visits each vertex exactly one).

Finding the shortest cycle is different than Dijkstra’s shortest path.It is much harder too, no polynomial time algorithm exists!

The Traveling Salesman Problem

Importance:• Variety of scheduling application can be solved as a

traveling salesmen problem.

• Examples:

• Ordering drill position on a drill press.

• School bus routing.

• The problem has theoretical importance because it represents a class of difficult problems known as NP-hard problems.

THE FEDERAL EMERGENCY MANAGEMENT AGENCY

A visit must be made to four local offices of FEMA, going out from and returning to the same main office in Northridge, Southern California.

FEMA traveling salesman Network representation

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FEMA - Traveling Salesman

• Solution approaches

– Enumeration of all possible cycles.• This results in (m-1)! cycles to enumerate for a graph with m

nodes. • Only small problems can be solved with this approach.

Possible cyclesCycle Total Cost

1. H-O1-O2-O3-O4-H 210 2. H-O1-O2-O4-O3-H 195 3. H-O1-O3-O2-O3-H 240 4. H-O1-O3-O4-O2-H 200 5. H-O1-O4-O2-O3-H 225 6. H-O1-O4-O3-O2-H 200 7. H-O2-O3-O1-O4-H 265 8. H-O2-O1-O3-O4-H 235 9. H-O2-O4-O1-O3-H 25010. H-O2-O1-O4-O3-H 22011. H-O3-O1-O2-O4-H 26012. H-O3-O1-O2-O4-H 260

Minimum

For this problem we have

(5-1)! / 2 = 12 cycles. Symmetrical problemsneed to enumerate only (m-1)! / 2 cycles.

FEMA – full enumeration

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FEMA – optimal solution

The Traveling Salesman Problem

Unfortunately, no algorithm solving the traveling salesman problem with polynomial worst-case time complexity has been devised yet.

This means that for large numbers of vertices, solving the traveling salesman problem is impractical.

In these cases, we can use efficient approximation algorithms that determine a path whose length may be slightly larger than the traveling salesman’s path, but