Particularly, you can find the shortest path from a node called the "source node" to all other nodes in the graph , producing a shortest-path tree. This algorithm is used in GPS devices to find the shortest path between the current location and the destination.
It has broad applications in industry, specially in domains that require modeling networks. This algorithm was created and published by Dr. Edsger W. Dijkstra , a brilliant Dutch computer scientist and software engineer. In , he published a 3-page article titled "A note on two problems in connexion with graphs" where he explained his new algorithm. In just 20 minutes, Dr. Dijkstra designed one of the most famous algorithms in the history of Computer Science. Dijkstra's Algorithm can only work with graphs that have positive weights.
This is because, during the process, the weights of the edges have to be added to find the shortest path. If there is a negative weight in the graph, then the algorithm will not work properly.
Once a node has been marked as "visited", the current path to that node is marked as the shortest path to reach that node. And negative weights can alter this if the total weight can be decremented after this step has occurred. Now that you know more about this algorithm, let's see how it works behind the scenes with a a step-by-step example.
The algorithm will generate the shortest path from node 0 to all the other nodes in the graph. We will have the shortest path from node 0 to node 1 , from node 0 to node 2 , from node 0 to node 3 , and so on for every node in the graph.
We also have this list see below to keep track of the nodes that have not been visited yet nodes that have not been included in the path :. Since we are choosing to start at node 0 , we can mark this node as visited. Equivalently, we cross it off from the list of unvisited nodes and add a red border to the corresponding node in diagram:. Now we need to start checking the distance from node 0 to its adjacent nodes. As you can see, these are nodes 1 and 2 see the red edges :.
Before adding a node to this path, we need to check if we have found the shortest path to reach it. We are simply making an initial examination process to see the options available.
We need to update the distances from node 0 to node 1 and node 2 with the weights of the edges that connect them to node 0 the source node. These weights are 2 and 6, respectively:. If we check the list of distances, we can see that node 1 has the shortest distance to the source node a distance of 2 , so we add it to the path.
We mark it with a red square in the list to represent that it has been "visited" and that we have found the shortest path to this node:. Now we need to analyze the new adjacent nodes to find the shortest path to reach them.
We will only analyze the nodes that are adjacent to the nodes that are already part of the shortest path the path marked with red edges. Node 3 and node 2 are both adjacent to nodes that are already in the path because they are directly connected to node 0 and node 1 , respectively, as you can see below. These are the nodes that we will analyze in the next step. Since we already have the distance from the source node to node 2 written down in our list, we don't need to update the distance this time.
We only need to update the distance from the source node to the new adjacent node node 3 :. To find the distance from the source node to another node in this case, node 3 , we add the weights of all the edges that form the shortest path to reach that node:.
Now that we have the distance to the adjacent nodes, we have to choose which node will be added to the path. We must select the unvisited node with the shortest currently known distance to the source node. From the list of distances, we can immediately detect that this is node 2 with distance 6 :. We also mark it as visited by adding a small red square in the list of distances and crossing it off from the list of unvisited nodes:. Now we need to repeat the process to find the shortest path from the source node to the new adjacent node, which is node 3.
Let's see how we can decide which one is the shortest path. Node 3 already has a distance in the list that was recorded previously 7, see the list below. But now we have another alternative. Clearly, the first existing distance is shorter 7 vs. Related Articles. Table of Contents. Save Article. Improve Article. Like Article. Recommended Articles. Shortest path from source to destination such that edge weights along path are alternatively increasing and decreasing.
Construct an N-ary Tree having no pair of adjacent nodes with same weight from given weights. It is different from the minimum spanning tree as the shortest distance among two vertices might not involve all the vertices of the graph. And therefore if any of the weights are introduced to be negative on the edges of the graph, the algorithm would never work properly. However, some algorithms like the Bellman-Ford Algorithm can be used in such cases. It is also a known fact that breadth-first search BFS could be used for calculating the shortest path for an unweighted graph, or for a weighted graph that has the same cost at all its edges.
But if the weighted graph has unequal costs at all its edges, then BFS infers uniform-cost search. Now what? Instead of extending nodes in order of their depth from the root, uniform-cost search develops the nodes in order of their costs from the root. Also, the estimated distance to every node is always an overvalue of the true distance and is generally substituted by the least of its previous value with the distance of a recently determined path.
It uses a priority queue to greedily choose the nearest node that has not been visited yet and executes the relaxation process on all of its edges. For example, an individual wants to calculate the shortest distance between the source, A, and the destination, D, while calculating a subpath which is also the shortest path between its source and destination.
It works on the fact that any subpath, let say a subpath B to D of the shortest path between vertices A and D is also the shortest path between vertices B and D, i. This way the algorithm deploys a greedy approach by searching for the next plausible solution and expects that the end result would be the appropriate solution for the entire problem.
The algorithm maintains the track of the currently recognized shortest distance from each node to the source code and updates these values if it identifies another shortest path. This process is being continued till all the nodes in the graph have been added to the path, as this way, a path gets created that connects the source node to all the other nodes following the plausible shortest path to reach each node.
Visit also: Types of Statistical Analysis. Mark the picked starting node with a current distance of 0 and the rest nodes with infinity,. 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,.
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. 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. Now the neighbours of node C will be checked, i. 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. Assign Node B a minimum distance value.
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. Assign Node A a minimum distance value. The same is repeated with node D, and marked 2 as lowest value at D.
Assign Node D a minimum distance value. Since, all the neighbours of node C have checked, so node C is marked as visited with a green check mark.
Marked Node C as visited. 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 neighbour 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.
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