Meljun Cortes Algorithm Greedy Algorithm Design Technique II

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Greedy Algorithm Design Technique

Design and Analysis of Algorithms

* Property of STI

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Definition of Greedy algorithm

Importance of Greedy algorithm design technique

Types of Greedy algorithm

Prim’s algorithm

Kruskal’s algorithm

Dijktra’s algorithm

Huffman trees

Sample problem using the application of Prim’s

algorithm, Kruskal’s algorithm, Dijktra’s algorithm,

and Huffman trees


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Greedy algorithms are simple and straight forward

[www.personal.kent.edu].

Most greedy algorithms are easy to develop,

implement, and time efficient.

Suggests constructing a solution through a

sequence of steps, each expanding a partially

constructed solution obtained so far, until a

complete solution to the problem is reached.


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On each step, which is the central point of this

technique, the choices made must be:

Feasible

Locally optimal

Irrevocable

Each step, it suggest a “greedy” grab of the best

alternative available.


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Example: Making Change Problem

Consisting of available dollar coins (100 cents),

quarters (25 cents), dimes (10 cents), nickels (5

cents) and pennies (1 cent), make a change of a

given amount using the smallest possible number

of coins.

To solve this problem, we use simple and formal

algorithms as follows:

Informal Algorithm Formal Algorithm


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Greedy algorithm:

Make a change for 2.89 (289 cents) where n =

2.89 and the solution contains 2 dollars, 3

quarters, 1 dime and 4 pennies.

The algorithm is greedy because at every stage it

chooses the largest coin without worrying about

the consequences.


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The greedy algorithm consists of four functions:

1. A function that validates whether the selected

set of items provides a solution.

2. A function that validates the feasibility of a

set.

3. The selection function that tells which of the

choices is the most capable.

4. The objective function which does not appear

explicitly, but gives the value of a solution.


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Prim s algorithm: is an algorithm in graph theory

that finds a minimum spanning tree for a

connected weighted graph.

Spanning tree of a connected graph is its

connected acyclic subgraph like a tree that

contains all the vertices of the graph [LEV07].

Minimum spanning tree (MST) of a weighted

connected graph is its spanning tree of thesmallest weight, where the weight of a tree is

defined as the sum of the weights on all its edges

[LEV07].

Minimum spanning tree problem is the problem of

finding a minimum spanning tree for a givenweighted connected graph [LEV07].


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The illustration below is an example of minimum

spanning tree:

Minimum spanning tree from the given graph


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Prim’s algorithm efficiency:

O(n2) for weight matrix representation of graph

and array implementation of priority queue

O(m log n) for adjacency lists representation

of graph with n vertices and m edges and min-

heap implementation of the priority queue


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an algorithm in graph theory that finds a minimum

spanning tree for a connected weighted graph

[www.wordiq.com]

introduced by Joseph Kruskal

an algorithm which finds a subset of the edges

that forms a tree that includes every vertex, where

the total weight of all the edges in the tree is

minimized

If in case the graph is not connected, it finds a

minimum spanning forest connected component.


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Step 1. Observe that in the given graph, the edge

(g, h) is the shortest. In this case, either vertex g or

vertex h could be represented.

Step 2. The edge (c, i) creates the second tree.

Choose vertex c as representative for second tree.


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Step 3. Edge (g, g) is the next shortest edge. Add

this edge and choose vertex g as representative.

Step 4. Edge (a, b) creates a third tree.


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Step 5. Add edge (c, f) and merge two trees. Vertex

c is chosen as the representative.

Step 6. Edge (g, i) is the next cheapest, but if we

add this edge a cycle would be created. Vertex c is

the representative of both.


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Step 7. Instead, add edge (c, d).

Step 8. If we add edge (h, i), this would make a

cycle.


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Step 9. Instead of adding edge (h, i) add edge (a,

h).

Step 10. Again, if we add edge (b, c), it would

create a cycle. Add edge (d, e) instead to complete

the spanning tree.


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introduced in 1959 by Edsger Dijktra, who is a

Dutch computer scientist

a graph search algorithm that solves the single-

source shortest path problem for a graph with

nonnegative edge path costs, producing a shortest

path tree [www.wikipedia.org]

finds the shortest paths to a graph’s vertices in

order of their distance from a given source it finds the shortest path form the source to a vertex

nearest to it, then to the second and so on


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The algorithm repeats the following steps until all

vertices are processed:


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Huffman tree: is a binary tree which are based on

the probability distribution of a symbol set and the

principle that symbols occurring more frequently

will be represented by shorter codes than other,

less probable ones.

Assume that the frequency of the character E in a

text is 15% and the frequency of the character T is

12%, as illustrated in the table below:


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Three basic steps in constructing a tree:

Step 1: Arrange the entire character set into a row

and should be ordered according to its frequency

from lowest to highest or vice versa.

Step 2: Subsequently, we will look for the two

nodes with the smallest combined frequencyweights and combine them to form a third node,

ensuing a simple two-level tree.

Step 3: Repeat Step 2 until all of the nodes on

each level are combined in a single tree.


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The process for building a tree starts with the leaf-

level nodes representing the original characters

arranged in descending order of value.


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Huffman tree – part 2


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Huffman tree – part 3


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Meljun Cortes Algorithm Greedy Algorithm Design Technique II