application of greedy method
TRANSCRIPT
APPLICATION OF GREEDY METHOD
KRUSKAL’S ALGORITHM TO FIND MINIMUM
COST SPANNING TREES
INTRODUCTION
An algorithm is a mathematical procedure often used to solve optimization problems in a finite number of steps.
Most solution algorithms make choices based on a global overview of all current and future possibilities aiming at reaching the single global optimum solution.
Greedy algorithms make choices that look best at that every moment.
When compared to algorithms that guarantee to yield a global optimum solution, greedy algorithms have several advantages:
They are easier to implement, they require much less computing resources, they are much faster to execute.
Their only disadvantage being that they do not always reach the global optimum solution
when the global optimum solution is not reached, most of the times the reached sub- optimal solution is a very good solution.
Several classical optimization problems like minimum spanning tree and optimal prefix codes for data compression yield global optimum solutions using greedy algorithm.
SPANNING TREES
A tree T is said to be a spanning tree of a connected graph G if T is a subgraph of G and T contains all vertices of G.
Since the vertices of G are barely hanging together in a spanning tree, it is a sort of skeleton of the original graph g. so, a spanning tree is called a skeleton or scaffolding of G.
Among the spanning trees of G, one with the smallest weight is called minimal spanning tree or shortest spanning tree or shortest-distance spanning tree.
KRUSKAL’S ALGORITHM
A second greedy algorithm was developed by Joseph Kruskal.
Steps in Kruskal’s algorithm: list all edges of G in order of increasing weight. Select a smallest edge of G. For each successive step, select (from all
remaining edges of G) another smallest edge that makes no circuit
with the previously selected edges. Continue until n-1 edges have been selected.
At each stage, the algorithm chooses an edge
to add to its current partial solution. To do so, it needs to test each candidate edge (u ,v ) to
see whether the endpoints u and v lie in different components; otherwise the edge produces a
cycle.
Kruskal's Algorithm
EXAMPLE 1
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ALGORITHM.
enqueue edges of G in a queue in increasing order of cost.T = ;while(queue is not empty){
dequeue an edge e; if(e does not create a cycle with edges in T)
add e to T;}return T;
Kruskal's Algorithm
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Kruskal's Algorithm
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cycle!!
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minimum- spanning tree
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EXAMPLE 2
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MINIMAL SPANNING TREEE
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Kruskal Running Time
Initialization O(V) time Sorting the edges Q(E lg E) = Q(E lg V) (why?) O(E) calls to FindSet Union costs
Let t(v) – the number of times v is moved to a new cluster
Each time a vertex is moved to a new cluster the size of the cluster containing the vertex at least doubles: t(v) £ log V
Total time spent doing Union Total time: O(E lg V)
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( ) logv V
t v V V
Applications
Where ever the minimum distance is needed to
achieve there this algorithm gets place.
For example.
In telephone lining under the ground, they have to
connect all the cities with the minimum amount of wire.
If the new connection is in position to add between cities,
there we have to consider the minimum cost distance.
Assume, if the minimum distance is not calculated, the
wastage of wires will be in peak.
Railways track construction.
National highway construction.
Flyover bridge construction.
Network between two intranets.
The another non-technical example will be as,
The by-pass rider or point to point buses will move on with
the minimum distance path to reach the destination much
faster. They will never circulate their bus through
unwanted towns and villages.