Why Floyd-Warshall Algorithm better for Dense Graphs and not for Sparse Graphs?

Floyd Warshall Algorithm:

The Floyd Warshall Algorithm is an all pair shortest path algorithm unlike Dijkstra and Bellman Ford which are single source shortest path algorithms. This algorithm works for both the directed and undirected weighted graphs. But, it does not work for the graphs with negative cycles (where the sum of the edges in a cycle is negative). It follows Dynamic Programming approach to check every possible path going via every possible node in order to calculate shortest distance between every pair of nodes....

Idea Behind Floyd Warshall Algorithm:

Suppose we have a graph G[][] with V vertices from 1 to N. Now we have to evaluate a shortestPathMatrix[][] where shortestPathMatrix[i][j] represents the shortest path between vertices i and j. Obviously the shortest path between i to j will have some k number of intermediate nodes. The idea behind floyd warshall algorithm is to treat each and every vertex from 1 to N as an intermediate node one by one. The following figure shows the above optimal substructure property in floyd warshall algorithm:...

Floyd Warshall Algorithm Algorithm:

Initialize the solution matrix same as the input graph matrix as a first step. Then update the solution matrix by considering all vertices as an intermediate vertex. The idea is to pick all vertices one by one and updates all shortest paths which include the picked vertex as an intermediate vertex in the shortest path. When we pick vertex number k as an intermediate vertex, we already have considered vertices {0, 1, 2, .. k-1} as intermediate vertices. For every pair (i, j) of the source and destination vertices respectively, there are two possible cases. k is not an intermediate vertex in shortest path from i to j. We keep the value of dist[i][j] as it is. k is an intermediate vertex in shortest path from i to j. We update the value of dist[i][j] as dist[i][k] + dist[k][j], if dist[i][j] > dist[i][k] + dist[k][j]...

Pseudo-Code of Floyd Warshall Algorithm :

For k = 0 to n – 1 For i = 0 to n – 1 For j = 0 to n – 1 Distance[i, j] = min(Distance[i, j], Distance[i, k] + Distance[k, j]) where i = source Node, j = Destination Node, k = Intermediate Node...

Illustration of Floyd Warshall Algorithm :

Suppose we have a graph as shown in the image: Step 1: Initialize the Distance[][] matrix using the input graph such that Distance[i][j]= weight of edge from i to j, also Distance[i][j] = Infinity if there is no edge from i to j. Step 2: Treat node A as an intermediate node and calculate the Distance[][] for every {i,j} node pair using the formula: = Distance[i][j] = minimum (Distance[i][j], (Distance from i to A) + (Distance from A to j ))= Distance[i][j] = minimum (Distance[i][j], Distance[i][A] + Distance[A][j]) Step 3: Treat node B as an intermediate node and calculate the Distance[][] for every {i,j} node pair using the formula: = Distance[i][j] = minimum (Distance[i][j], (Distance from i to B) + (Distance from B to j ))= Distance[i][j] = minimum (Distance[i][j], Distance[i][B] + Distance[B][j]) Step 4: Treat node C as an intermediate node and calculate the Distance[][] for every {i,j} node pair using the formula: = Distance[i][j] = minimum (Distance[i][j], (Distance from i to C) + (Distance from C to j ))= Distance[i][j] = minimum (Distance[i][j], Distance[i][C] + Distance[C][j]) Step 5: Treat node D as an intermediate node and calculate the Distance[][] for every {i,j} node pair using the formula: = Distance[i][j] = minimum (Distance[i][j], (Distance from i to D) + (Distance from D to j ))= Distance[i][j] = minimum (Distance[i][j], Distance[i][D] + Distance[D][j]) Step 6: Treat node E as an intermediate node and calculate the Distance[][] for every {i,j} node pair using the formula: = Distance[i][j] = minimum (Distance[i][j], (Distance from i to E) + (Distance from E to j ))= Distance[i][j] = minimum (Distance[i][j], Distance[i][E] + Distance[E][j]) Step 7: Since all the nodes have been treated as an intermediate node, we can now return the updated Distance[][] matrix as our answer matrix....

Complexity Analysis of Floyd Warshall Algorithm:

Time Complexity: O(V3), where V is the number of vertices in the graph and we run three nested loops each of size VAuxiliary Space: O(V2), to create a 2-D matrix in order to store the shortest distance for each pair of nodes....

Why Floyd-Warshall Algorithm better for Dense Graphs and not for Sparse Graphs?

Dense Graph: A graph in which the number of edges are significantly much higher than the number of vertices.Sparse Graph: A graph in which the number of edges are very much low. No matter how many edges are there in the graph the Floyd Warshall Algorithm runs for O(V3) times therefore it is best suited for Dense graphs. In the case of sparse graphs, Johnson’s Algorithm is more suitable....

Important Interview questions related to Floyd-Warshall:

How to Detect Negative Cycle in a graph using Floyd Warshall Algorithm?How is Floyd-warshall algorithm different from Dijkstra’s algorithm?How is Floyd-warshall algorithm different from Bellman-Ford algorithm?...

Real World Applications of Floyd-Warshall Algorithm:

In computer networking, the algorithm can be used to find the shortest path between all pairs of nodes in a network. This is termed as network routing.Flight Connectivity In the aviation industry to find the shortest path between the airports.GIS(Geographic Information Systems) applications often involve analyzing spatial data, such as road networks, to find the shortest paths between locations.Kleene’s algorithm which is a generalization of floyd warshall, can be used to find regular expression for a regular language....