Minimum spanning tree

MINIMUM SPANNING TREES
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Presenting by:-
Muhammed Ahmed
MCA IIIrd sem.

PROBLEM: LAYING TELEPHONE WIRE
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Central office

WIRING: NAIVE APPROACH
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Central office
Expensive!

WIRING: BETTER APPROACH
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Central office
Minimize the total length of wire connecting the customers

MINIMUM SPANNING TREE (MST)
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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

APPLICATIONS OF MST
 Any time you want to visit all vertices in a graph at
minimum cost (e.g., wire routing on printed circuit
boards, sewer pipe layout, bridges routing table, road
planning…)
 Internet content distribution
 it is a hot research topic
 Idea: publisher produces web pages, content distribution
network replicates web pages to many locations so consumers
can access at higher speed
 MST may not be good enough!
 content distribution on minimum cost tree may take a long time!
 Provides a heuristic for traveling salesman problems.
The optimum traveling salesman tour is at most twice
the length of the minimum spanning tree.
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HOW CAN WE GENERATE A MST?
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a
c
e
d
b
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45
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a
c
e
d
b
2
45
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PRIM’S ALGORITHM
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Initialization
a. Pick a vertex r to be the root
b. Set D(r) = 0, parent(r) = null
c. For all vertices v  V, v  r, set D(v) = 
d. Insert all vertices into priority queue P,
using distances as the keys
a
c
e
d
b
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e a b c d
0    
Vertex Parent
e -

PRIM’S ALGORITHM
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While P is not empty:
1. Select the next vertex u to add to the tree
u = P.deleteMin()
2. Update the weight of each vertex w adjacent to
u which is not in the tree (i.e., w  P)
If weight(u,w) < D(w),
a. parent(w) = u
b. D(w) = weight(u,w)
c. Update the priority queue to reflect
new distance for w

PRIM’S ALGORITHM
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a
c
e
d
b
2
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d b c a
4 5 5 
Vertex Parent
e -
b e
c e
d e
The MST initially consists of the vertex e, and we update
the distances and parent for its adjacent vertices
Vertex Parent
e -
b -
c -
d -
d b c a
   
e
0

PRIM’S ALGORITHM
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a
c
e
d
b
2
45
9
6
4
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a c b
2 4 5
Vertex Parent
e -
b e
c d
d e
a d
d b c a
4 5 5 
Vertex Parent
e -
b e
c e
d e

PRIM’S ALGORITHM
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a
c
e
d
b
2
45
9
6
4
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c b
4 5
Vertex Parent
e -
b e
c d
d e
a d
a c b
2 4 5
Vertex Parent
e -
b e
c d
d e
a d

PRIM’S ALGORITHM
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a
c
e
d
b
2
45
9
6
4
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b
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Vertex Parent
e -
b e
c d
d e
a d
c b
4 5
Vertex Parent
e -
b e
c d
d e
a d

PRIM’S ALGORITHM
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Vertex Parent
e -
b e
c d
d e
a d
a
c
e
d
b
2
45
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The final minimum spanning tree
b
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Vertex Parent
e -
b e
c d
d e
a d

KRUSKAL’S ALGORITHM
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Initialization
a. Create a set for each vertex v  V
b. Initialize the set of “safe edges” A
comprising the MST to the empty set
c. Sort edges by increasing weight
a
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F = {a}, {b}, {c}, {d}, {e}
A = 
E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}

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For each edge (u,v)  E in increasing order
while more than one set remains:
If u and v, belong to different sets U and V
a. add edge (u,v) to the safe edge set
A = A  {(u,v)}
b. merge the sets U and V
F = F - U - V + (U  V)
Return A

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E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}
Forest
{a}, {b}, {c}, {d}, {e}
{a,d}, {b}, {c}, {e}
{a,d,c}, {b}, {e}
{a,d,c,e}, {b}
{a,d,c,e,b}
A

{(a,d)}
{(a,d), (c,d)}
{(a,d), (c,d), (d,e)}
{(a,d), (c,d), (d,e), (b,e)}
a
c
e
d
b
2
45
9
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COMPLEXITY
 Kruskal's algorithm can be shown to run
in O(E log E) time, or equivalently, O(E log V)
time. Where E is the number of edges in the
graph and V is the number of vertices.
 The time complexity of prim’s Algorithm is
given by:
T(n)=2(n-1)(n-1) є Ө(n2)
where n=no. of vertices.
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DIFFERENCE
 The basic difference is in which edge you choose to add next to
the spanning tree in each step.
In Prim's, we always keep a connected component, starting with a
single vertex. we look at all edges from the current component to
other vertices and find the smallest among them. we then add the
neighboring vertex to the component, increasing its size by 1. In
N-1 steps, every vertex would be merged to the current one if we
have a connected graph.
In Kruskal's, we do not keep one connected component but a
forest. At each stage, we look at the globally smallest edge that
does not create a cycle in the current forest. Such an edge has to
necessarily merge two trees in the current forest into one. Since
we start with N single-vertex trees, in N-1 steps, they would all
have merged into one if the graph was connected.
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GREEDY APPROACH
 Both Prim’s and Kruskal’s algorithms are greedy
algorithms.
 The greedy approach works for the MST problem;
however, it does not work for many other
problems!
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Minimum spanning tree

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