# Define Single Source Shortest Path Algorithm

## Dijkstra's Algorithm Single Source Shortest Path Graph Algorithm

Single Source Shortest Path Suppose G be a weighted directed graph where a define single source shortest path algorithm labeled w u, v associated with each edge u, v in E, called weight of edge u, v. These weights represent the cost to traverse the edge. A path from vertex u to vertex v is a sequence of one or more edges. Variant of single-source shortest problems Given a source vertex, in the weighted diagraph, find the shortest path single party erfurt 2015 to singles nidderau other vertices in the digraph.

Given a destination vertex, t, in the weighted digraph, find the shortest path weights from all other vertices in the digraph. Given any two vertices in the define single source shortest path algorithm digraph, find the shortest path from u to v or v to u. Negative-Weight Edges The negative weight cycle is a cycle whose total is negative. No path from starting vertex S to a vertex on the cycle can be a shortest path. Since a path can run around the cycle many, many times and get any negative cost desired.

Relaxation Technique This technique consists of testing whether we can improve the shortest path algoritgm so far if so update the shortest path. Algoorithm relaxation step may or may not decrease the value of the shortest-path estimate. The following code performs a relaxation step on edge u,v. Defnie that we can solve this problem quite easily with BFS traversal algorithm in the special case when all weights are 1.

The greedy approach to this problem is repeatedly selecting the best choice from those available at that time.

## Single Source Shortest Paths

In this case we are trying to find the smallest number of edges that must be traversed in order to get to every vertex in the graph. The file contains an adjacency list representation of an undirected weighted graph with vertices labeled 1 to If the graph is unweighted, we can use a FIFO queue and keep track of the number of edges taken to get to a particular node. It turns out that we can solve this problem efficiently by solving a more general problem, the single-source shortest-path problem: This involves comparisons and takes time F , where is the cost of a single comparison in Floyd's algorithm and F is a constant. Algorithm 1 Create a set sptSet shortest path tree set that keeps track of vertices included in shortest path tree, i. In other words, the shortest path between s and v is: Do we know an algorithm for determining this? If we are interested only in shortest distance from source to a single target, we can break the for loop when the picked minimum distance vertex is equal to target Step 3. This algorithm is used in routing. For example, the 6th row has 6 as the first entry indicating that this row corresponds to the vertex labeled 6. If the input graph is represented using adjacency list, it can be reduced to O E log V with the help of binary heap. The distance value of vertex 6 and 8 becomes finite 15 and 9 respectively. Update the distance values of adjacent vertices of 7. Initialize all distance values as INFINITE. Each set of tasks is given the entire graph and is responsible for computing shortest paths for a single vertex Figure 3. This array contains, for each node v in the graph, the previous node u in the shortest path between the source node s and v. Graph must not contain a negative edge. Clearly, the choice of shortest-path algorithm for a particular problem will involve complex tradeoffs between flexibility, scalability, performance, and implementation complexity. The first parallel Dijkstra algorithm replicates the graph in each of P tasks.