minimum spanning tree
DESCRIPTION
Algorithm used in data structure of minimum Spanning treeTRANSCRIPT
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UIT – RGPV BHOPAL
TOPIC:- Minimum Spanning Tree
PRESENTED BY- Narendra Singh Patel
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PROBLEM: LAYING TELEPHONE WIRE
Central office
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WIRING: NAÏVE APPROACH
Central office
Expensive!
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WIRING: BETTER APPROACH
Central office
Minimize the total length of wire connecting the customers
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Minimum Spanning Trees
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We are interested in:
Finding a tree T that contains all the vertices of a graph G spanning treeand has the least total weight over allsuch trees minimum-spanning tree (MST)
Tuv
uvwTw),(
)),(()(
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Before discuss about MST (minimum spanning tree) lets get familiar with Graphs.
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Graph-• A graph is a finite set of nodes with
edges between nodes.• Formally, a graph G is a structure
(V,E) consisting of – a finite set V called the set of nodes, and– 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
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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.
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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 cyclesA connected,
undirected graph
Four of the spanning trees of the graph
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• 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
A minimum spanning tree is a subgraph of an undirected weighted graph G, such that
Minimum Spanning Tree (MST)
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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.
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• 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.
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•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
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:
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HOW CAN WE GENERATE A MST?
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We have two Ways to generate a MST
1. Prims algorithm2. Kruskals
algorithm
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.
Prim’s Algorithm
The edges in set A always form a single tree
Starts from an arbitrary “root”: VA = {a}
At each step:
Find a light edge crossing (VA, V - VA)
Add this edge to A
Repeat until the tree spans all vertices
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Example
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0 Q = {a, b, c, d, e, f, g, h,
i} VA =
Extract-MIN(Q) a
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h g f
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8 7
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a
b c d
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h g f
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8 7
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key [b] = 4 [b] = a
key [h] = 8 [h] = a
4 8 Q = {b, c, d, e, f, g, h, i} VA = {a}
Extract-MIN(Q) b
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Example
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key [c] = 8 [c] = b
key [h] = 8 [h] = a - unchanged
8 8 Q = {c, d, e, f, g, h, i} VA = {a, b}
Extract-MIN(Q) c
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h g f
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8 7
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key [d] = 7 [d] = c
key [f] = 4 [f] = c
key [i] = 2 [i] = c
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Q = {d, e, f, g, h, i} VA = {a, b, c}
Extract-MIN(Q) i
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Example
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a
b c d
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h g f
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key [h] = 7 [h] = i
key [g] = 6 [g] = i
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Q = {d, e, f, g, h} VA = {a, b, c, i}
Extract-MIN(Q) f
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h g f
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key [g] = 2 [g] = f
key [d] = 7 [d] = c unchanged
key [e] = 10 [e] = f
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Q = {d, e, g, h} VA = {a, b, c, i, f}
Extract-MIN(Q) g
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7 6 4
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Example
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a
b c d
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h g f
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8 7
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key [h] = 1 [h] = g
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Q = {d, e, h} VA = {a, b, c, i, f, g}
Extract-MIN(Q) h
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h g f
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Q = {d, e} VA = {a, b, c, i, f, g, h}
Extract-MIN(Q) d
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Example
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a
b c d
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h g f
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8 7
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key [e] = 9 [e] = f
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Q = {e} VA = {a, b, c, i, f, g, h, d}
Extract-MIN(Q) eQ = VA = {a, b, c, i, f, g, h, d, e}
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Prim’s (V, E, w, r)
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1. Q ←
2. for each u V
3. do key[u] ← ∞
4. π[u] ← NIL
5. INSERT(Q, u)
6. DECREASE-KEY(Q, r, 0) ► key[r] ← 0
7. while Q
8. do u ← EXTRACT-MIN(Q)
9. for each v Adj[u]
10. do if v Q and w(u, v) < key[v]
11. then π[v] ← u
12. DECREASE-KEY(Q, v, w(u, v))
O(V) if Q is implemented as a min-heap
Executed |V| times
Takes O(lgV)
Min-heap operations:O(VlgV)
Executed O(E) times total
Constant
Takes O(lgV)
O(ElgV)
Total time: O(VlgV + ElgV) = O(ElgV)
O(lgV)
Algorithm
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How is it different from Prim’s algorithm?Prim’s algorithm grows one tree all the timeKruskal’s algorithm grows multiple trees (i.e., a forest) at the same time.Trees are merged together using safe edgesSince an MST has exactly |V| - 1 edges, after |V| - 1 merges, we would have only one component
u
v
tree1
tree2
Kruskal’s Algorithm
Kruskal’s Algorithm
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Start with each vertex being its own component
Repeatedly merge two components into one by choosing the light edge that connects them
Which components to consider at each iteration?Scan the set of edges in
monotonically increasing order by weight
We would addedge (c, f)
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Example
1. Add (h, g)
2. Add (c, i)
3. Add (g, f)
4. Add (a, b)
5. Add (c, f)
6. Ignore (i, g)
7. Add (c, d)
8. Ignore (i, h)
9. Add (a, h)
10. Ignore (b, c)
11. Add (d, e)
12. Ignore (e, f)
13. Ignore (b, h)
14. Ignore (d, f)
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a
b c d
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h g f
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8 7
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1: (h, g)
2: (c, i), (g, f)
4: (a, b), (c, f)
6: (i, g)
7: (c, d), (i, h)
8: (a, h), (b, c)
9: (d, e)
10: (e, f)
11: (b, h)
14: (d, f)
{g, h}, {a}, {b}, {c}, {d}, {e}, {f}, {i}
{g, h}, {c, i}, {a}, {b}, {d}, {e}, {f}
{g, h, f}, {c, i}, {a}, {b}, {d}, {e}
{g, h, f}, {c, i}, {a, b}, {d}, {e}
{g, h, f, c, i}, {a, b}, {d}, {e}
{g, h, f, c, i}, {a, b}, {d}, {e}
{g, h, f, c, i, d}, {a, b}, {e}
{g, h, f, c, i, d}, {a, b}, {e}
{g, h, f, c, i, d, a, b}, {e}
{g, h, f, c, i, d, a, b}, {e}
{g, h, f, c, i, d, a, b, e}
{g, h, f, c, i, d, a, b, e}
{g, h, f, c, i, d, a, b, e}
{g, h, f, c, i, d, a, b, e}{a}, {b}, {c}, {d}, {e}, {f}, {g}, {h}, {i}
Thank You
Bibliography- http://www.cs.brown.edu/en.wikipedia.org/wiki/
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