My presentation all shortestpath

All-Pairs Shortest PathTheory and Algorithms

Carlos Andres Theran SuarezProgram Mathematics and Scientific Computing

University of Puerto Rico Carlos.theran@upr.edu

October – 2011Mayaguez-Puerto Rico

Dr Marko Schütz

Introduction

• In this section we consider de problem of finding shortest path between all pair of vertices in a directed graph .

– With a weight function where

For this goal we are going to use the adjacency-matrix of a graph.

• The input: is a which is adjacency-matrix of a • The output: is the matrix of SPL .

Recall

• Single-Source shortest paths.• Nonnegative edge weight.– Dijkstra’s algorithm: Running time Array

Running time Binary heap Running time Fibonacci heap

• General.– Bellman-Ford: Running time

• UDG.– Breadth for search: Running time

What do you think?

Can we solve all-pair shortest paths by running a single source-paths algorithms?

What do you think?

Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths.• Nonnegative edge weight

Dijkstra’s algorithm: Running time Array

What do you think?

Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths.• Nonnegative edge weight

Dijkstra’s algorithm: Running time Array Running time Binary heap

What do you think?

Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths.• Nonnegative edge weight

Dijkstra’s algorithm: Running time Array Running time Binary heap Running time Binary heap

What do you think?

Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths.• Nonnegative edge weight

Dijkstra’s algorithm: Running time Array Running time Binary heap Running time Binary heap

• General.Bellman-Ford: Running time

What do you think?

Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths.• Nonnegative edge weight

Dijkstra’s algorithm: Running time Array Running time Binary heap Running time Binary heap

• General.Bellman-Ford: Running time In a dense graph

Predecessor Matrix

Let the predecessor Matrix, where

Now we define the predecessor subgraph of G for as .

Where

Predecessor Matrix

Outline

1. Present a dynamic programming algorithms based on matrix multiplication to solve the problem.

2. Dynamic programming algorithms called Floyd-Warshall algorithms.

3. Unlike the others algorithms, Johnson's algorithms used adjacency-list representation of a graph.

Shortest path and matrix multiplication

1. The structure of shortest path.Let suppose that we have a shortest path form vertix to vertex , and suppose that have at most edge.

• If then have weight 0.

• If then where has at most edge, by lemma 24.1 is a shortest path from .

So

2. A recursive solution. Let be the minimum weight of any path from vertex to vertex that contains at most edge.

Shortest path and matrix multiplication (cont.)

3. Computing shortest-path weight bottom upInput . We compute

Shortest path and matrix multiplication (cont.)

• Now we can see the relation to the matrix multiplication.

Let the matrix product of For We have .

If we set;

Shortest path and matrix multiplication (cont.)

Computing the sequence of matrix

Shortest path and matrix multiplication (cont.)

• Improving the running time.

Our goal, is to compute matrices, let go to see that we can compute with only matrix product.

Shortest path and matrix multiplication (cont.)

Shortest path and matrix multiplication (cont.)

Shortest path and matrix multiplication (cont.)

The algorithm consider a intermediate vertices of a shortest path.1. The structure of a shortest path.Intermediate vertex in a any vertex of other than or, so it can be the set Let assume that the vertex of are and a subset for some .

• If of path , then all the vertices intermediate are in the set . Thus, a shortest path from vertex to vertex with all intermediate vertices in the set is also a shortest path form to with all in the set .

The Floyd-Warshall algorithm

• If of path , we break down into is a shortest path from to, so of , thus is a shortest path form to with all in the set . Similarly is a shortest path form to with all in the set .

The Floyd-Warshall algorithm (cont)

3. A recursive solution.

Since for every path, all intermediate vertices are in the set , matriz gives the final answer:

The Floyd-Warshall algorithm (cont)

• input: A matrix • output: A matrix of shortest path weight.

• =

The Floyd-Warshall algorithm (cont)

The Floyd-Warshall algorithm (cont)

4. Constructing a Shortest pathWe compute the predecessor matrix just as the Floyd-warshall algorithm compute the matrices so.

Recursive formulation.

The Floyd-Warshall algorithm (cont)

• It is asymtoticaly better than repeated squaring of matrices or the Floyd-Warshall algoritm.

• It use a subroutine both Dijkstra’s algorithm and Bellman-Ford algorithm.

• Johnson's algorithm use the technique of reweighting.

Johnson's algorithm for sparse graphs.

Reweighting If has a negative weight edge but no negative weight cycle, we compute a new set of nonnegative edge weight that allow as to use Dijkstra’s algorithm.

The new set of edge must satisfy two condition.1. is a shortest path form is a shortest path

form .2. For all edges , the new weight is nonnegative.

Johnson's algorithm for sparse graphs (cont.).

• Lemma Give a weighted, directed graph with weight funtion be any funtion mapping vertices to real numbers. For each edge , define.

Johnson's algorithm for sparse graphs (cont.).

• Producing no negative weight by reweighting

Johnson's algorithm for sparse graphs (cont.).

Johnson's algorithm for sparse graphs (cont.).

Johnson's algorithm for sparse graphs (cont.).