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Shortest Paths.
Shortest Paths.
Dijkstra's Algorithm.
Dijkstra's Algorithm.
In 1959, Edsger Dijkstra, a Dutch computer scientist, presented an algorithm that may be used with a weighted graph. The graph must embrace a non-negative value on each of its edges in order to be considered directed or undirected. This algorithm was given the name "Dijkstra's Algorithm" by the inventor.
An algorithm that is used for finding the shortest distance, or path, from starting node to target node in a weighted graph is known as Dijkstra’s Algorithm.
This algorithm makes a tree of the shortest path from the starting node, the source, to all other nodes (points) in the graph.
Dijkstra's algorithm makes use of weights of the edges for finding the path that minimizes the total distance (weight) among the source node and all other nodes. This algorithm is also known as the single-source shortest path algorithm.
But... WAIT! Let's first learn about Graphs.
In basic terms, graphs are data structures that are used to represent connections between a few items. These elements are referred to as nodes (or vertices) and are typically real-time objects, people, or things. Additionally, two nodes can only be joined if an edge exists between them.
Generally, graphs are suited to real-world applications, such as graphs can be used to illustrate a transportation system/network, where nodes represent facilities that transfer or obtain products and edges show routes or subways that connect nodes.
Graphs can be divided into two parts;
Undirected Graphs: For every couple of associated nodes, if an individual could move from one node to another in both directions, then the graph is termed as an undirected graph.
Directed Graphs: For every couple of associated graphs, if an individual could move from one node to another in a specific (single) direction, then the graph is known as the directed graph. In this case, arrows are implemented rather than simple lines in order to represent directed edges.
Weighted Graphs
Weighted Graphs
The weight graphs are the graphs where edges of the graph have “a weight” or “cost” and also where weight could reflect distance, time, money or anything that displays the “association” amid a couple of nodes it links. These weights are an essential element under Dijkstra's Algorithm.
Step by step process of Algorithm Implementation
The very first step is to mark all nodes as unvisited,
Mark the picked starting node with a current distance of 0 and the rest nodes with infinity,
Now, fix the starting node as the current node,
For the current node, analyse all of its unvisited neighbours and measure their distances by adding the current distance of the current node to the weight of the edge that connects the neighbour node and current node,
Compare the recently measured distance with the current distance assigned to the neighbouring node and make it as the new current distance of the neighbouring node,
After that, consider all of the unvisited neighbours of the current node, mark the current node as visited,
If the destination node has been marked visited then stop, an algorithm has ended, and
Else, choose the unvisited node that is marked with the least distance, fix it as the new current node, and repeat the process again from step 4.
Example 1:
We will calculate the shortest path between node C and the other nodes in the graph.
During the execution of the algorithm, each node will be marked with its minimum distance to node C as we have selected node C.
In this case, the minimum distance is 0 for node C. Also, for the rest of the nodes, as we don’t know this distance, they will be marked as infinity (∞), except node C (currently marked as red dot).
Now the neighbors of node C will be checked, i.e, node A, B, and D. We start with B, here we will add the minimum distance of current node (0) with the weight of the edge (7) that linked the node C to node B and get 0+ 7= 7.
Now, this value will be compared with the minimum distance of B (infinity), the least value is the one that remains the minimum distance of B, like in this case, 7 is less than infinity, and marks the least value to node B.
Now, the same process is checked with neighbour A. We add 0 with 1 (weight of edge that connects node C to A), and get 1. Again, 1 is compared with the minimum distance of A (infinity), and marks the lowest value. .
The same is repeated with node D, and marked 2 as lowest value at D.
Since, all the neighbors of node C have checked, so node C is marked as visited with a green check mark.
Now, we will select the new current node such that the node must be unvisited with the lowest minimum distance, or the node with the least number and no check mark. Here, node A is the unvisited with minimum distance 1, marked as current node with red dot.
We repeat the algorithm, checking the neighbor of the current node while ignoring the visited node, so only node B will be checked.
For node B, we add 1 with 3 (weight of the edge connecting node A to B) and obtain 4. This value, 4, will be compared with the minimum distance of B, 7, and mark the lowest value at B as 4.
After this, node A marked as visited with a green check mark. The current node is selected as node D, it is unvisited and has a smallest recent distance. We repeat the algorithm and check for node B and E.
For node B, we add 2 to 5, get 7 and compare it with the minimum distance value of B, since 7>4, so leave the smallest distance value at node B as 4.
For node E, we obtain 2+ 7= 9, and compare it with the minimum distance of E which is infinity, and mark the smallest value as node E as 9. The node D is marked as visited with a green check mark.
The current node is set as node B, here we need to check only node E as it is unvisited and the node D is visited. We obtain 4+ 1=5, compare it with the minimum distance of the node.
As 9 > 5, leave the smallest value at node node E as 5.
We mark D as visited node with a green check mark, and node E is set as the current node.
Since it doesn’t have any unvisited neighbors, so there is not any requirement to check anything. Node E is marked as a visited node with a green mark.
So, we are done as no unvisited node is left. The minimum distance of each node is now representing the minimum distance of that node from node C.
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