CSC 2300Data Structures & Algorithms

April 17, 2007

Chapter 9. Graph Algorithms

Today

Network Flow Minimum Spanning Tree

Prim’s Algorithm

Network Flow

Given a directed graph G=(V,E) with edge capacities cvw.

Examples: amount of water that flow through a pipe, or amount of traffic on a street between two intersections.

Also given two vertices: source s and sink t. Goal: determine the maximum amount of flow

that can pass from s to t.

Example

A graph and its maximum flow:

Procedure

Start with our graph G. Construct a flow graph Gf, which gives the flow that

has been attained at the current stage. Initially, all edges in Gf have no flow. Also construct a graph Gr, called the residual graph. Each edge of Gr tells us how much more flow can be

added.

Initial Stage

Graphs G, Gf, and Gr:

Stage 2

Two units of flow are added along s, b, d, t:

Stage 3

Two units of flow are added along s, a, c, t:

Stage 4

One unit of flow added along s, a, d, t:

Algorithm terminates with correct solution.

Greedy Algorithm

We have described a greedy algorithm, which does not always work.

Here is an example:

Algorithm terminates with suboptimal solution.

Improve Algorithm

How to make the algorithm work? Allow it to change its mind! For every edge (v,w) with flow fvw in the flow graph,

we will add an edge in the residual graph (w,v) of capacity fvw.

What are we doing? We are allowing the algorithm to undo its decisions

by sending flow back in the opposite direction.

Augmenting Path

Old:

New:

Improved Algorithm

Two units of flow are added along s, b, d, a, c, t:

Algorithm terminates with correct solution.

Algorithm Always Works?

Theorem. If the edge capacities are rational numbers, then the improved algorithm always terminates with a maximal flow.

What are rational numbers?

Running Time

Say that the capacities are all integers and the maximal flow is f.

Each augmenting flow increases the flow value by at least one.

How many stages? Augmenting path can be found by which algorithm? Unweighted shortest path. Total time? O( f |E| ).

Bad Case

This is a classic bad case for augmenting:

The maximum flow is easily seen by inspection. Random augmentations could go along a path that includes (a,b). What is the worst case for number of augmentations? 2,000,000.

Remedy

How to get around problem on previous slide? Choose the augmenting path that gives the largest

increase in flow. Which algorithm? Modify the Dijkstra’s algorithm that solves a weighted

shortest-path problem.

Minimum Spanning Tree

Example

Second Example

A graph and its minimum spanning tree:

Prim’s Algorithm

Outline:

Prim’s Algorithm

Prim’s Algorithm – Tables

Another Algorithm

Can you name another algorithm that is very similar to Prim’s algorithm?

Dijkstra’s algorithm. What is the major difference?

Running Time

Without heaps? O( |V|2 ). Optimal for dense graphs. With heaps? O( |E| log|V| ). Good for sparse graphs.