Minimum spanning Tree
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Kruskal's algorithm finds a minimum spanning tree by growing it from a forest of trees. It works by repeatedly selecting the lowest cost edge that connects two different trees without forming a cycle and adding it to the spanning tree. This continues until only one tree remains, which is the minimum spanning tree. The algorithm considers edges in order of increasing weight, merging components with safe edges. It runs in O(ElogV) time by using a priority queue to track the lowest cost remaining edge.
![Example
8 7
b c d 0
4 9
2 Q = {a, b, c, d, e, f, g, h, i}
a 11 i 14 e
4 VA =
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8 10 Extract-MIN(Q) a
h g 2
f
1
4
8 7 key [b] = 4 [b] = a
b c d
4 9 key [h] = 8 [h] = a
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a 11 i 4 14 e
7 6
8 10 4 8
h g 2
f Q = {b, c, d, e, f, g, h, i} VA = {a}
1
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![Example
4 8 key [c] = 8 [c] = b
8 7
b c d key [h] = 8 [h] = a - unchanged
4 9
2
8 8
a 11 i 4 14 e
7 6 Q = {c, d, e, f, g, h, i} VA = {a, b}
8 10
h g f Extract-MIN(Q) c
1 2
8
key [d] = 7 [d] = c
4 8 7
8 7 key [f] = 4 [f] = c
b c d
4 2 2 9 key [i] = 2 [i] = c
a 11 i 4 14 e
7 6
8 10 7 4 8 2
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f Q = {d, e, f, g, h, i} VA = {a, b, c}
1
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![Example
4 8 7 key [h] = 7 [h] = i
8 7
b c d key [g] = 6 [g] = i
4 9
2 2 7 46 8
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Q = {d, e, f, g, h} VA = {a, b, c, i}
7 6
8 10
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1
8 6 4
7
4 8 7 key [g] = 2 [g] = f
8 7
b c d key [d] = 7 [d] = c unchanged
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2 2 10 key [e] = 10 [e] = f
a 11 i 14 e
4 7 10 2 8
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![Example
4 8 7 key [h] = 1 [h] = g
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b c d 7 10 1
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2 2 10 Q = {d, e, h} VA = {a, b, c, i, f, g}
a 11 i 4 14 e
7 6
Extract-MIN(Q) h
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![Example
4 8 7
8 7
b c d key [e] = 9 [e] = f
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2 2 10
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8 10
h g f Extract-MIN(Q) e
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24](https://image.slidesharecdn.com/finalpptforppt-111122080719-phpapp02/85/Minimum-spanning-Tree-24-320.jpg)
![Prim’s (V, E, w, r)
1. Q←
Total time: O(VlgV + ElgV) = O(ElgV)
2. for each u V
3. do key[u] ← ∞ O(V) if Q is implemented
as a min-heap
4. π[u] ← NIL
5. INSERT(Q, u)
O(lgV)
6. DECREASE-KEY(Q, r, 0) key[r] ← 0
Min-heap
7. while Q Executed |V| times
operations:
8. do u ← EXTRACT-MIN(Q) Takes O(lgV) O(VlgV)
Executed O(E) times total
9. for each v Adj[u]
Constant O(ElgV)
10. do if v Q and w(u, v) < key[v]
Takes O(lgV)
11. then π[v] ← u
12. DECREASE-KEY(Q, v, w(u, v))
25](https://image.slidesharecdn.com/finalpptforppt-111122080719-phpapp02/85/Minimum-spanning-Tree-25-320.jpg)
Minimum spanning Tree
- 1. TOPIC:- Minimum Spanning Tree 1
- 2. PROBLEM: LAYING TELEPHONE WIRE Central office 2
- 3. WIRING: NAÏVE APPROACH Central office Expensive! 3
- 4. WIRING: BETTER APPROACH Central office Minimize the total length of wire connecting the customers 4
- 6. We are interested in: Finding a tree T that contains all the vertices of a graph G spanning tree and has the least total weight over all such trees minimum-spanning tree (MST) w(T ) w((v, u)) ( v ,u ) T 6
- 7. Before discuss about MST (minimum spanning tree) lets get familiar with Graphs. 7
- 8. • A graph is a finite set of nodes with edges between nodes. • Formally, a graph G is a structure (V,E) 1 2 consisting of – a finite set V called the set of nodes, and 3 4 – a set E that is a subset of VxV. That is, E is a set of pairs of the form (x,y) where x and y are nodes in V 8
- 9. Directed vs. Undirected Graphs • If the directions of the edges matter, then we show the edge directions, and the graph is called a directed graph (or a digraph) • If the relationships represented by the edges are symmetric (such as (x,y) is edge if and only if x is a sibling of y), then we don’t show the directions of the edges, and the graph is called an undirected graph. 9
- 10. SPANNING TREES Suppose you have a connected undirected graph Connected: every node is reachable from every other node Undirected: edges do not have an associated direction ...then a spanning tree of the graph is a connected subgraph in which there are no cycles A connected, Four of the spanning trees of the graph undirected graph 10
- 11. Minimum Spanning Tree (MST) A minimum spanning tree is a subgraph of an undirected weighted graph G, such that • it is a tree (i.e., it is acyclic) • it covers all the vertices V – contains |V| - 1 edges • the total cost associated with tree edges is the minimum among all possible spanning trees • not necessarily unique 11
- 12. APPLICATIONS OF MST Cancer imaging. The BC Cancer Research Ctr. uses minimum spanning trees to describe the arrangements of nuclei in skin cells. •Cosmology at the University of Kentucky. This group works on large-scale structure formation, using methods including N-body simulations and minimum spanning trees. •Detecting actin fibers in cell images. A. E. Johnson and R. E. Valdes-Perez use minimum spanning trees for biomedical image analysis. •The Euclidean minimum spanning tree mixing model. S. Subramaniam and S. B. Pope use geometric minimum spanning trees to model locality of particle interactions in turbulent fluid flows. The tree structure of the MST permits a linear-time solution of the resulting particle-interaction matrix. 12
- 13. •Extracting features from remotely sensed images. Mark Dobie and co-workers use minimum spanning trees to find road networks in satellite and aerial imagery. •Finding quasar superstructures. M. Graham and co-authors use 2d and 3d minimum spanning trees for finding clusters of quasars and Seyfert galaxies. •Learning salient features for real-time face verification, K. Jonsson, J. Matas, and J. Kittler. Includes a minimum-spanning-tree based algorithm for registering the images in a database of faces. •Minimal spanning tree analysis of fungal spore spatial patterns, C. L. Jones, G. T. Lonergan, and D. E. Mainwaring. 13
- 14. •A minimal spanning tree analysis of the CfA redshift survey. Dan Lauer uses minimum spanning trees to understand the large- scale structure of the universe. •A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees, S. Subramaniam and S. B. Pope. •Sausages, proteins, and rho. In the talk announced here, J. MacGregor Smith discusses Euclidean Steiner tree theory and describes potential applications of Steiner trees to protein conformation and molecular modeling. •Weather data interpretation. The Insight group at Ohio State is using geometric techniques such as minimum spanning trees to extract features from large meteorological data sets 14
- 15. What is a Minimum-Cost Spanning Tree For an edge-weighted , connected, undirected graph, G, the total cost of G is the sum of the weights on all its edges. A minimum-cost spanning tree for G is a minimum spanning tree of G that has the least total cost. Example: The graph Has 16 spanning trees. Some are: The graph has two minimum-cost spanning trees, each with a cost of 6:
- 16. HOW CAN WE GENERATE A MST? 16
- 17. We have two Ways to generate a MST 17
- 18. Prim’s Algorithm Prim’s algorithm finds a minimum cost spanning tree by selecting edges from the graph one-by-one as follows: It starts with a tree, T, consisting of the starting vertex, x. Then, it adds the shortest edge emanating from x that connects T to the rest of the graph. It then moves to the added vertex and repeats the process.
- 19. Prim’s Algorithm The edges in set A always form a single tree Starts from an arbitrary “root”: VA = {a} At each step: 8 7 b c d Find a light edge crossing (VA, V - VA) 4 9 2 Add this edge to A a 11 i 14 e 4 Repeat until the tree spans all vertices 7 6 8 10 h g 2 f 1 19
- 20. Example 8 7 b c d 0 4 9 2 Q = {a, b, c, d, e, f, g, h, i} a 11 i 14 e 4 VA = 7 6 8 10 Extract-MIN(Q) a h g 2 f 1 4 8 7 key [b] = 4 [b] = a b c d 4 9 key [h] = 8 [h] = a 2 a 11 i 4 14 e 7 6 8 10 4 8 h g 2 f Q = {b, c, d, e, f, g, h, i} VA = {a} 1 8 Extract-MIN(Q) b 20
- 21. Example 4 8 key [c] = 8 [c] = b 8 7 b c d key [h] = 8 [h] = a - unchanged 4 9 2 8 8 a 11 i 4 14 e 7 6 Q = {c, d, e, f, g, h, i} VA = {a, b} 8 10 h g f Extract-MIN(Q) c 1 2 8 key [d] = 7 [d] = c 4 8 7 8 7 key [f] = 4 [f] = c b c d 4 2 2 9 key [i] = 2 [i] = c a 11 i 4 14 e 7 6 8 10 7 4 8 2 h g 2 f Q = {d, e, f, g, h, i} VA = {a, b, c} 1 8 4 21 Extract-MIN(Q) i
- 22. Example 4 8 7 key [h] = 7 [h] = i 8 7 b c d key [g] = 6 [g] = i 4 9 2 2 7 46 8 a 11 i 4 14 e Q = {d, e, f, g, h} VA = {a, b, c, i} 7 6 8 10 Extract-MIN(Q) f h g 2 f 1 8 6 4 7 4 8 7 key [g] = 2 [g] = f 8 7 b c d key [d] = 7 [d] = c unchanged 4 9 2 2 10 key [e] = 10 [e] = f a 11 i 14 e 4 7 10 2 8 7 6 8 10 Q = {d, e, g, h} VA = {a, b, c, i, f} h g 2 f 1 6 4 Extract-MIN(Q) g 22 7 2
- 23. Example 4 8 7 key [h] = 1 [h] = g 8 7 b c d 7 10 1 4 9 2 2 10 Q = {d, e, h} VA = {a, b, c, i, f, g} a 11 i 4 14 e 7 6 Extract-MIN(Q) h 8 10 h g 2 f 1 7 1 2 4 7 10 4 8 7 8 7 Q = {d, e} VA = {a, b, c, i, f, g, h} b c d 4 9 Extract-MIN(Q) d 2 2 10 a 11 i 4 14 e 7 6 8 10 h g 2 f 1 23 1 2 4
- 24. Example 4 8 7 8 7 b c d key [e] = 9 [e] = f 4 9 9 2 2 10 9 a 11 i 4 14 e 7 6 Q = {e} VA = {a, b, c, i, f, g, h, d} 8 10 h g f Extract-MIN(Q) e 1 2 1 2 4 Q= VA = {a, b, c, i, f, g, h, d, e} 24
- 25. Prim’s (V, E, w, r) 1. Q← Total time: O(VlgV + ElgV) = O(ElgV) 2. for each u V 3. do key[u] ← ∞ O(V) if Q is implemented as a min-heap 4. π[u] ← NIL 5. INSERT(Q, u) O(lgV) 6. DECREASE-KEY(Q, r, 0) key[r] ← 0 Min-heap 7. while Q Executed |V| times operations: 8. do u ← EXTRACT-MIN(Q) Takes O(lgV) O(VlgV) Executed O(E) times total 9. for each v Adj[u] Constant O(ElgV) 10. do if v Q and w(u, v) < key[v] Takes O(lgV) 11. then π[v] ← u 12. DECREASE-KEY(Q, v, w(u, v)) 25
- 26. Algorithm How is it different from Prim’s algorithm? Prim’s algorithm grows one tree all the time Kruskal’s algorithm grows tree1 multiple trees (i.e., a forest) at the same time. Trees are merged together u using safe edges Since an MST has exactly |V| - 1 v edges, after |V| - 1 merges, tree2 we would have only one component 26
- 27. Kruskal’s Algorithm 8 7 Start with each vertex being its b c d 4 9 own component 2 a 11 i 4 14 e Repeatedly merge two 6 7 8 10 components into one by h g 2 f 1 choosing the light edge that We would add connects them edge (c, f) Which components to consider at each iteration? Scan the set of edges in monotonically increasing order by weight 27
- 28. Example 1. Add (h, g) {g, h}, {a}, {b}, {c}, {d}, {e}, {f}, {i} 8 7 b c d 2. Add (c, i) {g, h}, {c, i}, {a}, {b}, {d}, {e}, {f} 4 9 3. Add (g, f) 2 {g, h, f}, {c, i}, {a}, {b}, {d}, {e} a 11 i 4 14 e 4. Add (a, b) {g, h, f}, {c, i}, {a, b}, {d}, {e} 8 7 6 5. Add (c, f) {g, h, f, c, i}, {a, b}, {d}, {e} 10 h g f 6. Ignore (i, g) {g, h, f, c, i}, {a, b}, {d}, {e} 1 2 7. Add (c, d) {g, h, f, c, i, d}, {a, b}, {e} 1: (h, g) 8: (a, h), (b, c) 8. Ignore (i, h) {g, h, f, c, i, d}, {a, b}, {e} 2: (c, i), (g, f) 9: (d, e) 9. Add (a, h) {g, h, f, c, i, d, a, b}, {e} 4: (a, b), (c, f) 10: (e, f) 10. Ignore (b, c) {g, h, f, c, i, d, a, b}, {e} 11. Add (d, e) 6: (i, g) 11: (b, h) {g, h, f, c, i, d, a, b, e} 12. Ignore (e, f) 7: (c, d), (i, h) 14: (d, f) {g, h, f, c, i, d, a, b, e} 13. Ignore (b, h) {g, h, f, c, i, d, a, b, e} {a}, {b}, {c}, {d}, {e}, {f}, {g}, {h}, {i} 14. Ignore (d, f) {g, h, f, c, i, d, a, b, e} 28
- 29. Thank You Bibliography- http://www.cs.brown.edu/ en.wikipedia.org/wiki/ 29