CSC 2300 Data 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 c vw. 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 G f, which gives the flow that has been attained at the current stage. Initially, all edges in G f have no flow. Also construct a graph G r, called the residual graph. Each edge of G r tells us how much more flow can be added.

Initial Stage Graphs G, G f, and G r :

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