Week -7-8 Topic - Graph Algorithms CSE – 5311. Prepared by:- Sushruth Puttaswamy Lekhendro Lisham . Contents. Different Parts of Vertices used in Graph Algorithms Analysis of DFS Analysis of BFS Minimum Spanning Tree Kruskal’s Algorithm Shortest Path’s Dijkshitra’s Algorithm.
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Different Parts of Vertices (used in Graph Algorithms)
Vertices of the Graph are kept in 3 parts as below:
QUEUE is used.
a b e
Graph with 5 Vertices
(U/V/F) 1-D Array
For a Graph G=(V, E) and n = |V| & m=|E|
Hence DFS is the champion algorithm for all connectivity problems
Proof by contradiction:
Let this be a MST. Also let us assume that the shortest edge is not part of this tree. Consider node g & h.
All edges on the path between these nodes are longer.(w1,w2,w3)
Let us take any edge (ex-w1) & delete it. We will then connect g & h directly.
But this edge has a lighter weight then already existing edges which is a contradiction.
Step 1: Initially all the edges of the graph are sorted based on their weights.
Step2: Select the edge with minimum weight from the sorted list in step 1.Selected edge shouldn’t form a cycle. Selected edge is added into the tree or forest.
Step 3: Repeat step 2 till the tree contains all nodes of the graph.
We have n vertices & m edges the following steps are followed.
O(mlogm)= O(mlogn )= O(mlog n).
If we use path compression then find & union become m times inverse Ackerman’s function.
But sorting always takes mlogm time.
DIJKSTRA (G, w, s)
The algorithm repeatedly selects the vertex u with the minimum shortest-path estimate, adds u to S, and relaxes all edges leaving u.