Dijkstra Algorithm Computer science engineering

Dijkstra Algorithm Computer science engineering

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  Dijkstra’s Algorithm : FindingSingle – Source Shortest Paths
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  The Single Source-ShortestPath Algorithm:Overview The Single-Source Shortest Path problem focuses on finding the shortest path from one designated source vertex to every other vertex in a weighted graph. Unlike the All-Pairs Shortest Path (APSP) problem which finds routes between all node pairs, SSSP is ideal for "point A to everywhere else" scenarios. Why Do We Need It? • Many real-world problems don't require knowing the shortest path between every possible pair of locations. Instead, they need the most efficient routes from a single starting point. Applications like GPS navigation, network broadcasting, and finding the closest available resources are all based on the SSSP model. Limitations of Simpler Algorithms While algorithms like Breadth-First Search (BFS) can find the shortest path in unweighted graphs, they fail when edges have different "costs" or weights. Dijkstra's algorithm was developed to solve this by systematically handling weighted edges to find the truly optimal path, not just the one with the fewest steps.
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  Introducing the DijkstraAlgorithm Finds the Shortest Path Its main purpose is to find the most efficient path from a single starting point (the "source") to every other node in a weighted network or graph. Uses a Greedy Strategy The algorithm works by always choosing the nearest unvisited node. It then explores paths from that node, continuously updating its list of shortest distances until all nodes have been visited. Requires Non- Negative Weights A critical rule for Dijkstra's is that it only functions correctly when the "cost" of traveling between any two nodes (the edge weight) is zero or positive. It cannot handle negative weights.
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  Key Advantages ofDijkstra Algorithm Efficient and Reliable Dijkstra's algorithm is a highly efficient and reliable method for finding the absolute shortest path from a single source in graphs with non-negative weights. Graph Versatility The algorithm is compatible with both directed and undirected graphs, providing broad applicability. Optimized for Sparse Graphs In real-world scenarios like road or computer networks, graphs are often "sparse" (meaning they have far fewer connections than the maximum possible). Predictable Performance The time complexity is consistently O(V^2), where V represents the number of vertices, making its performance predictable even with increasing graph size.
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  Core Idea: GreedyApproach with Iterative Refinement The fundamental principle of Dijkstra's algorithm is to iteratively build a set of nodes with known shortest paths. It always "greedily" selects the unvisited node with the smallest known distance from the source and uses it to update, or "relax," the distances of its neighbors.
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  Step-by-Step Implementation ofDijkstra Algorithm Initialize First, set up your data structures. Create a distances table or array to store the shortest known distance from the source node to every other node. Set the distance to the source node itself as 0 and all other distances to infinity ( ). All ∞ nodes start in the "unvisited" set. Select the Closest Node From the set of unvisited nodes, identify the node with the smallest known distance from the source. This becomes your "current" node for this iteration. In the very first step, this will always be the source node itself. Update Neighbor Distances For the current node, examine all of its unvisited neighbors. For each neighbor, calculate the distance to it by adding the current node's distance to the weight of the edge connecting them. If this new calculated distance is shorter than the neighbor's currently recorded distance, update it in your distances table. This process is often called "relaxing the Mark as Visited and Repeat After checking all of the current node's neighbors, move it from the "unvisited" set to the "visited" set. This means its shortest path has been finalized. Repeat steps 2 and 3 until all reachable nodes have been visited or the unvisited set is empty. The final distances table will then contain the shortest path from the source to all other nodes.
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  Example: Finding theShortest Path from Node 1 Let's apply Dijkstra's algorithm to the following graph, starting from Node 1. Our goal is to find ‍ ♂️ ‍ ️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ ‍ ♂️ the shortest distance from Node 1 to all other nodes. We'll use a table to track the shortest distance found so far to each node and which nodes have been visited.
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  1 2 34 1 0 3 ∞ 7 2 8 0 2 ∞ 3 5 ∞ 0 1 = 4 2 ∞ ∞ 0
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  1 2 34 Iteration 1 1 0 3 ∞ 7 2 8 0 2 15 3 5 8 0 1 = STEPS : 1. Keep the first row and first column the same as in . 2. Update the remaining entries by checking if going through vertex 1 provides a shorter path. 3. For each pair , compare the current distance with the distance . 4. Record the smaller of the two values. 4 2 5 ∞ 0
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  1 2 34 Iteration 2 1 0 3 5 7 2 8 0 2 15 STEPS : 1.Keep the second row and second column the same as in . 2.Update the remaining entries by checking if going through vertex 2 provides a shorter path. 3.For each pair , compare the current distance with . 4.Record the smaller value. 5.You may also use previously updated paths through vertex 1 if that yields an even shorter distance. 3 5 8 0 1 = 4 2 5 7 0
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  1 2 34 Iteration 3 1 0 3 5 6 2 7 0 2 3 STEPS: 1. Keep the third row and third column the same as in . 2. Update the remaining entries by checking if going through vertex 3 provides a shorter path. 3. Compare with . 4. Record the smaller value. 5. Paths that already use vertices 1 or 2 can also contribute to a shorter distance. 3 5 8 0 1 = 4 2 5 7 0
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  1 2 34 Iteration 4 1 0 3 5 6 2 5 0 2 3 3 3 6 0 1 = STEPS: 1. Keep the fourth row and column the same as in A3. 2. Update the remaining entries by checking if going through vertex 4 provides a shorter path. 3. Compare with . 4. Record the smaller value. 5. Paths that include vertices 1, 2, or 3 may also combine to yield the shortest final distances. 4 2 5 7 0
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  Final Shortest Distances Matrix Afterthe Floyd-Warshall algorithm completes its iterations, the matrix reflects the shortest distances between all pairs: 1 2 3 4 1 0 3 5 6 2 5 0 2 3 3 3 6 0 1 4 2 5 7 0
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  Detecting Negative Cycles The Floyd-Warshallalgorithm is not only useful for finding shortest paths but also for detecting the presence of negative cycles within a graph. How to Detect If, at any point during or after the iterations, any of the diagonal elements in the distance matrix (i.e., dist[i][i]) becomes negative, it indicates the existence of a negative cycle reachable from vertex i. Implication of Negative Cycles When a negative cycle is present, the concept of a "shortest path" becomes undefined. This is because traversing a negative cycle repeatedly can indefinitely decrease the total path distance, leading to infinite loops and no true shortest path. Algorithm's Role While Floyd-Warshall can detect negative cycles, it cannot provide a meaningful shortest path solution in their presence. Its output will simply reflect the infinitely decreasing nature of paths involving such cycles.
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  Real-World Applications ofDijkstra Algorithm GPS and Mapping Services The most common application is in GPS navigation. When you ask for the fastest route from your location to a destination, services like Google Maps or Waze use Dijkstra's algorithm (or a variation like A*) to model the road network as a graph. Network Routing n computer networking, Dijkstra's is used in routing protocols like OSPF (Open Shortest Path First). Routers use the algorithm to find the shortest path to send data packets between different points in a network. Social Network Analysis Helps in understanding connectivity and relationships within social graphs, determining the shortest "degrees of separation" between individuals. Flight Itineraries Airlines and travel websites use Dijkstra's to find the cheapest or fastest flight routes. Airports are treated as nodes and flights as edges, with weights representing cost or travel time.