Greedy Algorithm-Dijkstra's algo
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This document discusses Dijkstra's algorithm for finding the shortest paths between nodes in a graph. It begins with pseudocode that initializes all distances as infinite except the source node, and iterates through selecting the unvisited node with the lowest distance. It updates distances and predecessors until all nodes are visited. The document provides an example applying Dijkstra's algorithm to a graph, showing the distances and predecessors updated at each step until all are permanent. It concludes that Dijkstra's algorithm builds a shortest path tree and the optimal path can be found by tracing back predecessors.
![Dijkstra's Algorithm:
dist[s] ←0 (distance to source vertex is zero)
for all v ∈ V–{s}
do dist[v] ←∞ (set all other distances to infinity)
S←∅ (S, the set of visited vertices is initially empty)
Q←V (Q, the queue initially contains all vertices)
while Q ≠∅ (while the queue is not empty)
do u ← mindistance(Q,dist) (select the element of Q with the min. distance)
S←S∪{u} (add u to list of visited vertices)
for all v ∈ neighbors[u]
do if dist[v] > dist[u] + w(u, v) (if new shortest path found)
then d[v] ←d[u] + w(u, v) (set new value of shortest path)
(if desired, add traceback code)
return dist](https://image.slidesharecdn.com/dijkstrasalgo-151009141022-lva1-app6892/85/Greedy-Algorithm-Dijkstra-s-algo-2-320.jpg)
Greedy Algorithm-Dijkstra's algo
- 1. Analysis and Design of Algorithms (2150703) Presented by : Jay Patel (130110107036) Gujarat Technological University G.H Patel College of Engineering and Technology Department of Computer Engineering Greedy Algorithms Guided by: Namrta Dave
- 2. Dijkstra's Algorithm: dist[s] ←0 (distance to source vertex is zero) for all v ∈ V–{s} do dist[v] ←∞ (set all other distances to infinity) S←∅ (S, the set of visited vertices is initially empty) Q←V (Q, the queue initially contains all vertices) while Q ≠∅ (while the queue is not empty) do u ← mindistance(Q,dist) (select the element of Q with the min. distance) S←S∪{u} (add u to list of visited vertices) for all v ∈ neighbors[u] do if dist[v] > dist[u] + w(u, v) (if new shortest path found) then d[v] ←d[u] + w(u, v) (set new value of shortest path) (if desired, add traceback code) return dist
- 3. Dijkstra's Algorithm: Ex: 1 2 3 4 5 6 2 4 21 3 4 2 3 2 Initialize 1 0 Select the node with the minimum temporary distance label.
- 4. 1 43 2 2 3 4 5 6 2 4 21 3 4 2 3 2 2 4 0 1 1 3 4 5 6 2 4 21 3 4 2 3 2 2 4 0 2 1 2 3 4 5 6 2 4 21 3 4 2 3 2 2 6 43 0 The predecessor of node 3 is now node 2 1 2 4 5 6 2 4 21 3 4 2 3 2 2 3 6 4 0 3
- 5. 5 87 6 1 2 4 5 6 2 4 21 3 4 2 3 2 0 d(5) is not changed. 3 2 3 6 4 1 2 4 6 2 4 21 3 4 2 3 2 0 3 2 3 6 4 5 1 2 4 6 2 4 21 3 4 2 3 2 0 3 2 3 6 4 5 d(4) is not changed 6 1 2 6 2 4 21 3 4 2 3 2 0 3 2 3 6 4 5 6 4
- 6. 9 11 10 1 2 6 2 4 21 3 4 2 3 2 0 3 2 3 6 4 5 6 4 d(6) is not updated 1 2 2 4 21 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 There is nothing to update 1 2 2 4 21 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors
- 7. There are some methods left: • Huffman’s Algorithm • Task scheduling • Travelling salesman Problem etc. • Dynamic Greedy Problems Greedy Algorithms: We can find the optimized solution with Greedy method which may be optimal sometime.
- 8. THANK YOU