History[ edit ] What is the shortest way to travel from Rotterdam to Groningenin general: It is the algorithm for the shortest pathwhich I designed in about twenty minutes. As I said, it was a twenty-minute invention.
Today we will discuss one of the most important graph algorithms: Dijkstra's shortest path algorithm, a greedy algorithm that efficiently finds shortest paths in a graph.
Many more problems than you might at first think can be cast as shortest path problems, making this algorithm a powerful and general tool. For example, Dijkstra's algorithm is a good way to implement a service like MapQuest that finds the shortest way to drive between two points on the map. It can also be used to solve problems like network routing, where the goal is to find the shortest path for data packets to take through a switching network.
It is also used in more general search algorithms for a variety of problems ranging from automated circuit layout to speech recognition. Let's start by defining a data abstraction for weighted, directed graphs so we can express algorithms independently of the implementation of graphs themselves.
In a weighted graph, each of its edges has a nonnegative weight that we can think of as the distance one must travel when going along that edge. Importantly, we assume that given a vertex, we can traverse the outgoing edges in constant time per edge. Some graph implementations do not have these properties, but we can easily write an almost trivial implementation that does: That is, all the edges must be traversed in the forward direction.
The length of a path is the sum of the weights along these edges e1, We call this property "length" even though for some graphs it may represent some other quantity: To implement MapQuest, we need to solve the following shortest-path problem: Given two vertices v and v', what is the shortest path through the graph that goes from v to v'?
That is, the path for which summing up the weights along all the edges from v to v' results in the smallest sum possible. It turns out that we can solve this problem efficiently by solving a more general problem, the single-source shortest-path problem: Given a vertex v, what is the length of the shortest path from v to every vertex v' in the graph?
It is this problem that we will investigate in this lecture. The single-source shortest path problem can also be formulated on an undirected graph; however, it is most easily solved by converting the undirected graph into a directed graph with twice as many edges, and then running the algorithm for directed graphs.
There are other shortest-path problems of interest, such as the all-pairs shortest-path problem: The Floyd-Warshall algorithm is a good way to solve this problem efficiently.
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. This is the same problem as solving the weighted version where all the weights happen to be 1.As a result, the shortest path algorithm is widely used in network routing protocols, most notably IS-IS (Intermediate System to Intermediate System) and Open Shortest Path First (OSPF).
It is also employed as a subroutine in other algorithms such as Johnson’s. The greedy algorithms are sometimes also used to get an approximation for Hard optimization problems. For example, Traveling Salesman Problem is a NP-Hard problem.
A Greedy choice for this problem is to pick the nearest unvisited city from the current city at every step. Consider the following greedy algorithm that constructs A row by row.
Assume that the first i-1 rows have been constructed. Let aj be the number of 1's in the j-th column in the first i-1 rows. Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks.
It was conceived by computer scientist Edsger W.
Dijkstra in and published three years later. Prim’s Algorithm also use Greedy approach to find the minimum spanning tree. In Prim’s Algorithm we grow the spanning tree from a starting position. Unlike an edge in Kruskal's, we add vertex to the growing spanning tree in Prim's. Why does a greedy method based algorithm find the correct shortest distance from the source to all other vertices in a directed graph?
Is it possible to find the shortest path in an undirected graph from the source vertex to any other vertex if the graph contains negative edge.