AlgorithmsBellman-Ford and

FloydTwo basic algorithms for path searching in a

graph

Evlogi Hristov

Telerik Corporation

Student at Telerik Academy

Table of Contents1. Relaxing of edges and paths2. Negative Edge Weights3. Negative Cycles4. Bellman-Ford algorithm

Description Pseudo code

5. Floyd-Warshall algorithm Description Pseudo code

2

Relaxing Edges

Relaxing Edges

if dist[v] > dist[u] + w dist[v] = dist[u] + w

-1

2

52 74

2 + 2 = 4

2 + 5 = 7

4 -1 = 33

Edge relaxation : Test whether traveling along a given edge gives a new shortest path to its destination vertex

Path relaxation: Test whether traveling through a given vertex gives a new shortest path connecting two other given vertices

Negative Edges and Cycles

Negative Edges and Cycles

Negative edges & Negative cycles A cycle whose edges sum to a

negative value Cannot produce a correct "shortest

path" answer if a negative cycle is reachable from the source

S 2-3

-2 1

32 5

2

0

0

3

Bellman-FordAlgorithm

Bellman-Ford Based on dynamic programming approach

Shortest paths from a single source vertex to all other vertices in a weighted graph Slower than other algorithms but

more flexible, can work with negative weight edges

Finds negative weight cycles in a graph

Worst case performance O(V·E)

8

Bellman-Ford (2)

9

//Step 1: initialize graphfor each vertex v in vertices if v is source then dist[v] = 0 else dist[v] = infinity

//Step 2: relax edges repeatedlyfor i = 1 to i = vertices-1 for each edge (u, v, w) in edges if dist[v] > dist[u] + w dist[v] = dist[u] + w

//Step 3: check for negative-weight cyclesfor each edge (u, v, w) in edges if dist[v] > dist[u] + w error "Graph contains a negative-weight cycle"

10

Bellman-Ford Algorithm

Live Demo

A-1

1

0

5

B

C D

23E

-3

2

4

∞∞

∞∞

Bellman-Ford (3) Flexibility

Optimizations http://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm

Disadvantages Does not scale well Count to infinity (if node is

unreachabel) Changes of network topology are not

reflected quickly (node-by-node)11

for i = 1 to i = n for each edge (u, v, w) in edges if dist[v] > dist[u] + w dist[v] = dist[u] + w

Floyd-WarshallAlgorithm

Floyd-Warshall Based on dynamic programming approach

All shortest paths through the graph between each pair of vertices Positive and negative edges Only lengths No details about the path

Worst case performance: O(V 3)13

Floyd-Warshall (2)

14

let dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)

for each vertex v dist[v][v] == 0

for each edge (u,v) dist[u][v] = w(u,v) // the weight of the edge (u,v)

for k = 1 to k = |V| for i = 1 to i = |V| for j = 1 to j = |V| if (dist[i][j] > dist[i][k] + dist[k][j]) dist[i][j] = dist[i][k] + dist[k][j]

15

Floyd-Warshall Algorithm

Live Demo

A

-4

11

4

B

D C

12

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Questions?

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Bellman-Ford

http://algoacademy.telerik.com

Links for more information

Negative weights http://www.informit.com/articles/article.

aspx?p=169575&seqNum=8

MIT Lecture and Proof http://videolectures.net/mit6046jf05_de

maine_lec18/

Optimizations fhttp://en.wikipedia.org/wiki/Bellman%E

2%80%93Ford_algorithmy

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