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Engineering
Data Structures and Algorithms
### Graph Analysis and Algorithms #### Graph Description The graph contains nodes labeled from \(a\) to \(l\) and edges connecting these nodes, with values representing the weights of each edge. - Nodes: \(a, b, c, d, e, f, g, h, i, j, k, l\). - Edges and their weights: - \(a-d\): 5 - \(a-e\): 3 - \(a-h\): 7 - \(b-c\): 3 - \(b-e\): 2 - \(b-f\): 6 - \(c-g\): 3 - \(c-f\): 1 - \(d-e\): 4 - \(e-f\): 7 - \(e-i\): 6 - \(h-i\): 2 - \(h-k\): 3 - \(i-l\): 2 - \(j-l\): 3 - \(k-l\): 4 - \(f-g\): 5 - \(g-j\): 5 #### Tasks **(i) Minimum Spanning Tree (MST)** Using either Prim’s or Kruskal’s algorithm, determine the Minimum Spanning Tree of the graph. State which algorithm you used. **(ii) Shortest Path from A to L** Using Dijkstra’s algorithm, find the shortest path from \(A\) to \(L\) and show all workings. #### Graph Explanation The graph forms a complex network with multiple connections between nodes. The objective is to simplify this network using the minimum spanning tree and to identify the shortest path between two specific nodes. The MST minimizes the total edge weight, ensuring all nodes remain connected. Dijkstra's algorithm identifies the shortest path by continuously selecting the nearest node that hasn't been visited yet, thereby efficiently finding the shortest distance from the starting node to all other nodes. For educational purposes, understanding the illustration and implementation of these algorithms is crucial in solving such graph-related problems.

### Graph Analysis and Algorithms #### Graph Description The graph contains nodes labeled from \(a\) to \(l\) and edges connecting these nodes, with values representing the weights of each edge. - Nodes: \(a, b, c, d, e, f, g, h, i, j, k, l\). - Edges and their weights: - \(a-d\): 5 - \(a-e\): 3 - \(a-h\): 7 - \(b-c\): 3 - \(b-e\): 2 - \(b-f\): 6 - \(c-g\): 3 - \(c-f\): 1 - \(d-e\): 4 - \(e-f\): 7 - \(e-i\): 6 - \(h-i\): 2 - \(h-k\): 3 - \(i-l\): 2 - \(j-l\): 3 - \(k-l\): 4 - \(f-g\): 5 - \(g-j\): 5 #### Tasks **(i) Minimum Spanning Tree (MST)** Using either Prim’s or Kruskal’s algorithm, determine the Minimum Spanning Tree of the graph. State which algorithm you used. **(ii) Shortest Path from A to L** Using Dijkstra’s algorithm, find the shortest path from \(A\) to \(L\) and show all workings. #### Graph Explanation The graph forms a complex network with multiple connections between nodes. The objective is to simplify this network using the minimum spanning tree and to identify the shortest path between two specific nodes. The MST minimizes the total edge weight, ensuring all nodes remain connected. Dijkstra's algorithm identifies the shortest path by continuously selecting the nearest node that hasn't been visited yet, thereby efficiently finding the shortest distance from the starting node to all other nodes. For educational purposes, understanding the illustration and implementation of these algorithms is crucial in solving such graph-related problems.

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### Graph Analysis and Algorithms #### Graph Description The graph contains nodes labeled from \(a\) to \(l\) and edges connecting these nodes, with values representing the weights of each edge. - Nodes: \(a, b, c, d, e, f, g, h, i, j, k, l\). - Edges and their weights: - \(a-d\): 5 - \(a-e\): 3 - \(a-h\): 7 - \(b-c\): 3 - \(b-e\): 2 - \(b-f\): 6 - \(c-g\): 3 - \(c-f\): 1 - \(d-e\): 4 - \(e-f\): 7 - \(e-i\): 6 - \(h-i\): 2 - \(h-k\): 3 - \(i-l\): 2 - \(j-l\): 3 - \(k-l\): 4 - \(f-g\): 5 - \(g-j\): 5 #### Tasks **(i) Minimum Spanning Tree (MST)** Using either Prim’s or Kruskal’s algorithm, determine the Minimum Spanning Tree of the graph. State which algorithm you used. **(ii) Shortest Path from A to L** Using Dijkstra’s algorithm, find the shortest path from \(A\) to \(L\) and show all workings. #### Graph Explanation The graph forms a complex network with multiple connections between nodes. The objective is to simplify this network using the minimum spanning tree and to identify the shortest path between two specific nodes. The MST minimizes the total edge weight, ensuring all nodes remain connected. Dijkstra's algorithm identifies the shortest path by continuously selecting the nearest node that hasn't been visited yet, thereby efficiently finding the shortest distance from the starting node to all other nodes. For educational purposes, understanding the illustration and implementation of these algorithms is crucial in solving such graph-related problems.
Transcribed Image Text:### Graph Analysis and Algorithms #### Graph Description The graph contains nodes labeled from \(a\) to \(l\) and edges connecting these nodes, with values representing the weights of each edge. - Nodes: \(a, b, c, d, e, f, g, h, i, j, k, l\). - Edges and their weights: - \(a-d\): 5 - \(a-e\): 3 - \(a-h\): 7 - \(b-c\): 3 - \(b-e\): 2 - \(b-f\): 6 - \(c-g\): 3 - \(c-f\): 1 - \(d-e\): 4 - \(e-f\): 7 - \(e-i\): 6 - \(h-i\): 2 - \(h-k\): 3 - \(i-l\): 2 - \(j-l\): 3 - \(k-l\): 4 - \(f-g\): 5 - \(g-j\): 5 #### Tasks **(i) Minimum Spanning Tree (MST)** Using either Prim’s or Kruskal’s algorithm, determine the Minimum Spanning Tree of the graph. State which algorithm you used. **(ii) Shortest Path from A to L** Using Dijkstra’s algorithm, find the shortest path from \(A\) to \(L\) and show all workings. #### Graph Explanation The graph forms a complex network with multiple connections between nodes. The objective is to simplify this network using the minimum spanning tree and to identify the shortest path between two specific nodes. The MST minimizes the total edge weight, ensuring all nodes remain connected. Dijkstra's algorithm identifies the shortest path by continuously selecting the nearest node that hasn't been visited yet, thereby efficiently finding the shortest distance from the starting node to all other nodes. For educational purposes, understanding the illustration and implementation of these algorithms is crucial in solving such graph-related problems.
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