Write the Dijkstra’s shortest path algorithm. Also, find the time complexity of this shortest path algorithm. Find the shortest paths from the vertex ‘A’ using Dijkstra’s shortest path algorithm for the graph given in Figure 1. Show all the steps of computation.
### **Algorithm Steps** 1. Assign tentative distance for all vertices: 0 for source, ∞ for others. 2. Set source as current. For all neighbors not yet visited, update their distance if path through current is smaller. 3. Mark current as visited. Set next unvisited vertex with smallest tentative distance as current. 4. Repeat until all vertices visited. **Time Complexity:** - With Min Heap: $$O((V+E)\log V)$$ - With array: $$O(V^2)$$ *** ### **Apply Dijkstra’s Algorithm ________ ___ ___ __________ ______.
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Explain the divide and conquer approach of multiplying two large integers. Compute the time complexity of this approach. Also, explain the binary search algorithm and find its time complexity
Explain the algorithm to find the optimal Binary Search Tree. Demonstrate this algorithm to find the Optimal Binary Search Tree for the following probability data (where pi represents the probability that the search will be for the key node ki, whereas qi represents that the search is for dummy node di. Make suitable assumptions, if any)
| i | 0 | 1 | 2 | 3 | 4 |
| pi | 0.10 | 0.15 | 0.20 | 0.10 | |
| qi | 0.05 | 0.10 | 0.10 | 0.10 | 0.10 |
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Write the Dijkstra’s shortest path algorithm. Also, find the time complexity of this shortest path algorithm. Find the shortest paths from the vertex ‘A’ using Dijkstra’s shortest path algorithm for the graph given in Figure 1. Show all the steps of computation.
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