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Transcript of GRAPHS Prof. Muhammad Saeed Analysis of Algorithms Analysis Of Algorithms1.
![Page 1: GRAPHS Prof. Muhammad Saeed Analysis of Algorithms Analysis Of Algorithms1.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5514d863550346b0338b5452/html5/thumbnails/1.jpg)
GRAPHS
Prof. Muhammad Saeed
Analysis of Algorithms
Analysis Of Algorithms 1
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Analysis Of Algorithms 2
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Analysis Of Algorithms 3
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Analysis Of Algorithms 4
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Analysis Of Algorithms 5
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Topological Sort
Vertex 1 2 3 4 5 6 7V1 0 0 0 0 0 0 0
V2 1 0 0 0 0 0 0
V3 2 1 1 1 0 0 0
V4 3 2 1 0 0 0 0
V5 1 1 0 0 0 0 0
V6 3 3 3 3 2 1 0
V7 2 2 2 1 0 0 0
enqueue V1 V2 V5 V4 V3, V7 V6
dequeue V1 V2 V5 V4 V3 V7 V6
Indegree Before Dequeue
Analysis Of Algorithms 6
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological SortStep by Step
Step 1
Analysis Of Algorithms 7
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
Step by Step
Step 2
Analysis Of Algorithms 8
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
0011032
V5
V5
Step by Step
Step 3
Analysis Of Algorithms 9
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
0011032
V5
V5
0010031
V4
V4
Step by Step
Step 4
Analysis Of Algorithms 10
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
0011032
V5
V5
0010031
V4
V4
0000020
V3, V7
V3
Step by Step
Step 5
Analysis Of Algorithms 11
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
0011032
V5
V5
0010031
V4
V4
0000020
V3, V7
V3
0000010
V7
Step by Step
Step 6
Analysis Of Algorithms 12
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
0011032
V5
V5
0010031
V4
V4
0000020
V3, V7
V3
0000010
V7
0000000
V6
V6
Step by Step
Step 7
Analysis Of Algorithms 13
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Indegree Before Dequeue
Vertex 1 2 3 4 5 6 7V1 0V2 1V3 2V4 3V5 1V6 3V7 2
enqueue V1
dequeue V1
Topological Sort
0012132
V2
V2
0011032
V5
V5
0010031
V4
V4
0000020
V3, V7
V3
0000010
V7
0000000
V6
V6
Analysis Of Algorithms 14
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Analysis Of Algorithms 15
Complexity Analysis
Topological Sort
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Shortest Path AlgorithmUnweighted Graphs
Analysis Of Algorithms 16
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Shortest Path AlgorithmUnweighted Graphs
Initial State
v Known dv pv
v1 0 0
v2 0 0
v3 0 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v3
Step by Step
Step 1
Analysis Of Algorithms 17
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Shortest Path AlgorithmUnweighted Graphs
Initial State
v Known dv pv
v1 0 0
v2 0 0
v3 0 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v3
v3 dequeued
Known dv pv
0 1 v3
0 0
1 0 0
0 0
0 0
0 1 v3
0 0
v1, v6
Step by Step
Step 2
Analysis Of Algorithms 18
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Shortest Path AlgorithmUnweighted Graphs
Initial State
v Known dv pv
v1 0 0
v2 0 0
v3 0 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v3
v3 dequeued
Known dv pv
0 1 v3
0 0
1 0 0
0 0
0 0
0 1 v3
0 0
v1, v6
v1 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
0 1 v3
0 0
v6, v2, v4
Step by Step
Step 3
Analysis Of Algorithms 19
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Shortest Path AlgorithmUnweighted Graphs
Initial State
v Known dv pv
v1 0 0
v2 0 0
v3 0 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v3
v3 dequeued
Known dv pv
0 1 v3
0 0
1 0 0
0 0
0 0
0 1 v3
0 0
v1, v6
v1 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
0 1 v3
0 0
v6, v2, v4
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
1 1 v3
0 0
v2, v4
Step by Step
Step 4
Analysis Of Algorithms 20
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Shortest Path AlgorithmUnweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0 0
Q v4, v5
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
1 1 v3
0 0
v2, v4
Step by Step
Step 5
Analysis Of Algorithms 21
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Shortest Path AlgorithmUnweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0 0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
1 1 v3
0 0
v2, v4
Step by Step
Step 6
Analysis Of Algorithms 22
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Shortest Path AlgorithmUnweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0 0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v5 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
0 3 v4
v7
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
1 1 v3
0 0
v2, v4
Step by Step
Step 7
Analysis Of Algorithms 23
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Shortest Path AlgorithmUnweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0 0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v5 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
0 3 v4
v7
v7 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
1 3 v4
empty
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
1 1 v3
0 0
v2, v4
Step by Step
Step 8
Analysis Of Algorithms 24
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Shortest Path AlgorithmUnweighted Graphs
v2 dequeued
v Known dv pv
v1 1 1 v3
v2 1 2 v1
v3 1 0 0
v4 0 2 v1
v5 0 3 v2
v6 1 1 v3
v7 0 0
Q v4, v5
v4 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
0 3 v2
1 1 v3
0 3 v4
v5, v7
v5 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
0 3 v4
v7
v7 dequeued
Known dv pv
1 1 v3
1 2 v1
1 0 0
1 2 v1
1 3 v2
1 1 v3
1 3 v4
empty
v6 dequeued
Known dv pv
1 1 v3
0 2 v1
1 0 0
0 2 v1
0 0
1 1 v3
0 0
v2, v4
n = |V|T(n) = O(n2) for arraysT(n) = O(|V| + |E|) for adjacency List
Analysis Of Algorithms 25
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Shortest Path AlgorithmUnweighted Graphs
Analysis Of Algorithms 26
Complexity Analysis
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Analysis Of Algorithms 27
END OF (Shortest Path Unweighted Graphs)Algorithm
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
Analysis Of Algorithms 28
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v1
Step by Step
Step 1
Analysis Of Algorithms 29
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v1
v1 dequeued
Known dv pv
1 0 0
0 2 v1
0 0 0
0 1 v1
0 0
0 0
0 0
v2, v4
Step by Step
Step 2
Analysis Of Algorithms 30
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
v2 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 v4
0 1 v1
0 3 v4
0 9 v4
0 5 v4
v2, v5,v3, v6, v7,
Step by Step
Step 3v1dequeued
Known dv pv
1 0 0
0 2 v1
0 0 0
0 1 v1
0 0
0 0
0 0
v4, v2
Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v1
Analysis Of Algorithms 31
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
v2 dequeued
Known dv pv
1 0 0
1 2 v1
0 0 0
0 1 v1
0 12 v2
0 0
0 0
v4, v5
Step by Step
Step 4v1dequeued
Known dv pv
1 0 0
0 2 v1
0 0 0
0 1 v1
0 0
0 0
0 0
v2, v4
Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
Q v1
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 v4
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7Analysis Of Algorithms 32
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s AlgorithmStep by Step
Step 5v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 v4
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7 Analysis Of Algorithms 33
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s AlgorithmStep by Step
Step 6v3 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 v4
1 1 v1
1 3 v4
0 8 v3
0 5 v4
v6,v7
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 0
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7 Analysis Of Algorithms 34
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
v6 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 v4
1 1 v1
1 3 v4
1 8 v3
0 5 v4
v7
Step by Step
Step 7v3 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 0
1 1 v1
1 3 v4
0 8 v3
0 5 v4
v6,v7
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 0
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7 Analysis Of Algorithms 35
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
v6 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 0
1 1 v1
1 3 v4
1 8 v3
0 5 v4
v7
Step by Step
Step 8v3 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 0
1 1 v1
1 3 v4
0 8 v3
0 5 v4
v6,v7
v5 dequeued
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 0 3 0
v4 1 1 v1
v5 1 3 v4
v6 0 9 v4
v7 0 5 v4
Q v3, v6,v7
v4 dequeued
Known dv pv
1 0 0
1 2 v1
0 3 0
1 1 v1
0 3 v4
0 9 v4
0 5 v4
v5, v3, v6,v7
v7 dequeued
Known dv pv
1 0 0
1 2 v1
1 3 v4
1 1 v1
1 3 v4
1 6 v7
1 5 v4
EmptyAnalysis Of Algorithms 36
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Shortest Path AlgorithmWeighted Graphs
Dijkstra’s Algorithm
Analysis Of Algorithms 37
Complexity Analysis
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Analysis Of Algorithms 38
END OF Dijkstra’s Algorithm
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Analysis Of Algorithms 39
Shortest Path Algorithm Negative Weighted Graphs
Bellman-Ford’s Algorithm
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Analysis Of Algorithms 40
BELLMAN-FORD (G, w, s)
INITIALIZE-SINGLE-SOURCE (G, s)for each vertex i = 1 to V[G] - 1 do for each edge (u, v) in E[G] do RELAX (u, v, w)For each edge (u, v) in E[G] do if d[u] + w(u, v) < d[v] then return FALSEreturn TRUE
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Analysis Of Algorithms 41
s
zy
6
7
8-3
72
9
-2xt
-4
5
………….Bellman-Ford’s Algorithm
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Analysis Of Algorithms 42
s
zy
6
7
8-3
72
9
-2xt
-4
5
s
zy
6
7
8-3
72
9
-2xt
-4
5
………….Bellman-Ford’s Algorithm
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Analysis Of Algorithms 43
s
zy
6
7
8-3
72
9
-2xt
-4
5
s
zy
6
7
8-3
72
9
-2xt
-4
5
………….Bellman-Ford’s Algorithm
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Analysis Of Algorithms 44
s
zy
6
7
8-3
72
9
-2xt
-4
s
zy
6
7
8-3
72
9
-2xt
-4
5
………….Bellman-Ford’s Algorithm
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Analysis Of Algorithms 45
Complexity Analysis
Bellman-Ford Algorithm
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Analysis Of Algorithms 46
END OF Bellman-Ford Algorithm
A
C
B
ED
2
-1
4-3
2 3
5 3
-1
4
π: nild: 0
π: nil Ad: ∞ -1
π: nil Cd: ∞ 2
π: nil Ad: ∞ 4
π: nild: ∞
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Minimum Spanning TreeWeighted and Undirected Graphs
Prim’s Algorithm
Analysis Of Algorithms 47
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Prim’sAlgorithm
Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
v1
Step by Step
Step 1
Minimum Spanning TreeWeighted and Undirected Graphs
Analysis Of Algorithms 48
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Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
v1
Minimum Spanning TreeWeighted and Undirected Graphs
v1 known
Known dv pv
1 0 0
0 2 v1
0 4 v1
0 1 v1
0 0
0 0
0 0
v2 , v3 ,v4
Prim’sAlgorithmStep by Step
Step 2
Analysis Of Algorithms 49
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Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
v1
Minimum Spanning TreeWeighted and Undirected Graphs
v1 known
Known dv pv
1 0 0
0 2 v1
0 4 v1
0 1 v1
0 0
0 0
0 0
v2 , v3 ,v4
Prim’sAlgorithmStep by Step
Step 3
v4 known
Known dv pv
1 0 0
0 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v2 ,v3 , v5 , v6 , v7 Analysis Of Algorithms 50
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Initial State
v Known dv pv
v1 0 0 0
v2 0 0
v3 0 0
v4 0 0
v5 0 0
v6 0 0
v7 0 0
v1
Minimum Spanning TreeWeighted and Undirected Graphs
v1 known
Known dv pv
1 0 0
0 2 v1
0 4 v1
0 1 v1
0 0
0 0
0 0
v2 , v3 ,v4
Prim’sAlgorithmStep by Step
Step 4
v4 known
Known dv pv
1 0 0
0 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v2 ,v3 , v5 , v6 , v7
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7 Analysis Of Algorithms 51
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v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
Minimum Spanning TreeWeighted and Undirected GraphsPrim’sAlgorithmStep by Step
Step 5
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7 Analysis Of Algorithms 52
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v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
Minimum Spanning TreeWeighted and Undirected Graphs
v7 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
0 1 v7
1 4 v4
v5 , v6
Prim’sAlgorithmStep by Step
Step 6
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7 Analysis Of Algorithms 53
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v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
Minimum Spanning TreeWeighted and Undirected Graphs
v7 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
0 1 v7
1 4 v4
v5 , v6
Prim’sAlgorithmStep by Step
Step 7
v6 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
1 1 v7
1 4 v4
v5
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7 Analysis Of Algorithms 54
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Minimum Spanning TreeWeighted and Undirected GraphsPrim’sAlgorithmStep by Step
Step 8
v5 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
1 6 v7
1 1 v7
1 4 v4
v2known
Known dv pv
1 0 0
1 2 v1
0 2 v4
1 1 v1
0 7 v4
0 8 v4
0 4 v4
v3 , v5 , v6 , v7
v3 known
v Known dv pv
v1 1 0 0
v2 1 2 v1
v3 1 2 v4
v4 1 1 v1
v5 0 7 v4
v6 0 5 v3
v7 0 4 v4
v5 , v6 , v7
v7 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
0 1 v7
1 4 v4
v5 , v6
v6 known
Known dv pv
1 0 0
1 2 v1
1 2 v4
1 1 v1
0 6 v7
1 1 v7
1 4 v4
v5 Total Cost = 16Analysis Of Algorithms 55
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Prim’s Algorithm
Analysis Of Algorithms 56
Complexity Analysis
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Analysis Of Algorithms 57
END OF Prim’s Algorithm
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Minimum Spanning TreeWeighted and Undirected Graphs
Kruskal’s Algorithm
Analysis Of Algorithms 58
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Kruskal’s AlgorithmStep by Step
Step 1
Minimum Spanning Tree Weighted Undirected Graphs
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Analysis Of Algorithms 59
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Minimum Spanning TreeWeighted and Undirected Graphs
Kruskal’s Algorithm
Analysis Of Algorithms 60
Complexity Analysis
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Analysis Of Algorithms 61
End of Kruskal’s Algorithm
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EndEnd GRAPHS
Analysis Of Algorithms 62