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Vellore institute of Technology University
**Specialization:
**Computer Science Engineering**Downloads:
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GREEDY TECHNIQUE
Definition:
Greedy technique is a general algorithm design strategy, built on following elements:
• configurations: different choices, values to find
• objective function: some configurations to be either maximized or minimized
The method:
•
•
•
•
•
•
Applicable to optimization problems ONLY
Constructs a solution through a sequence of steps
Each step expands a partially constructed solution so far, until a complete solution
to the problem is reached.
On each step, the choice made must be
Feasible: it has to satisfy the problem’s constraints
Locally optimal: it has to be the best local choice among all feasible choices
available on that step
Irrevocable: Once made, it cannot be changed on subsequent steps of the
algorithm
NOTE:
• Greedy method works best when applied to problems with the greedy-choice
property
• A globally-optimal solution can always be found by a series of local
improvements from a starting configuration.
Greedy method vs. Dynamic programming method:
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LIKE dynamic programming, greedy method solves optimization problems.
LIKE dynamic programming, greedy method problems exhibit optimal
substructure
UNLIKE dynamic programming, greedy method problems exhibit the greedy
choice property -avoids back-tracing.
Applications of the Greedy Strategy:
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Optimal solutions:
Change making
Minimum Spanning Tree (MST)
Single-source shortest paths
Huffman codes
Approximations:
Traveling Salesman Problem (TSP)
Fractional Knapsack problem
1

Spanning Tree
Definition:
Spanning tree is a connected acyclic sub-graph (tree) of the given graph (G) that includes
all of G’s vertices
Example: Consider the following graph
1
b
a
2
5
c
d
3
The spanning trees for the above graph are as follows:
1
1
b
b
a
a
2
5
d
5
3
c
d
c
Weight (T2) = 8
Weight (T1) = 9
1
a
b
2
d
c
3
Weight (T3) = 6
Minimum Spanning Tree (MST)
Definition:
MST of a weighted, connected graph G is defined as: A spanning tree of G with
minimum total weight.
Example: Consider the example of spanning tree:
For the given graph there are three possible spanning trees. Among them the spanning
tree with the minimum weight 6 is the MST for the given graph
Question: Why can’t we use BRUTE FORCE method in constructing MST ?
Answer: If we use Brute force method• Exhaustive search approach has to be applied.
• Two serious obstacles faced:
1. The number of spanning trees grows exponentially with graph size.
2. Generating all spanning trees for the given graph is not easy.
2

MST Applications:
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Network design.
Telephone, electrical, hydraulic, TV cable, computer, road
Approximation algorithms for NP-hard problems.
Traveling salesperson problem, Steiner tree
Cluster analysis.
Reducing data storage in sequencing amino acids in a protein
Learning salient features for real-time face verification
Auto config protocol for Ethernet bridging to avoid cycles in a network, etc
Prim’s Algorithm
-to find minimum spanning tree
Some useful definitions:
Fringe edge: An edge which has one vertex is in partially constructed tree Ti and
the other is not.
• Unseen edge: An edge with both vertices not in Ti
•
Algorithm:
ALGORITHM Prim (G)
//Prim’s algorithm for constructing a MST
//Input: A weighted connected graph G = { V, E }
//Output: ET the set of edges composing a MST of G
// the set of tree vertices can be initialized with any vertex
VT { v0}
ET Ø
for i 1 to |V| - 1 do
Find a minimum-weight edge e* = (v*, u*) among all the edges (v, u) such
that v is in VT and u is in V - VT
VT VT U { u*}
ET ET U { e*}
return ET
3

The method:
STEP 1: Start with a tree, T0, consisting of one vertex
STEP 2: “Grow” tree one vertex/edge at a time
• Construct a series of expanding sub-trees T1, T2, … Tn-1.
• At each stage construct Ti + 1 from Ti by adding the minimum weight edge
connecting a vertex in tree (Ti) to one vertex not yet in tree, choose from
“fringe” edges (this is the “greedy” step!)
Algorithm stops when all vertices are included
Example:
Apply Prim’s algorithm for the following graph to find MST.
1
b
3
c
4
5
a
4
6
5
f
d
2
8
6
e
Solution:
Tree
vertices
Remaining
vertices
a ( -, - )
b(a,3)
c(-,∞)
d(-,∞)
e(a,6)
f(a,5)
b ( a, 3 )
c(b,1)
d(-,∞)
e(a,6)
f(b,4)
Graph
b
3
a
1
3
b
c
a
1
c ( b, 1 )
d(c,6)
e(a,6)
f(b,4)
b
c
3
4
a
f
4

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