ms nikita greedy agorithm

Analysis & Design of Algorithms

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Greedy Algorithms

Greedy AlgorithmsThe General MethodContinuous Knapsack Problem

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1. Greedy Algorithms

Greedy Algorithm:

A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum.

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1.Greedy Algorithms

Methodology:

Start with a solution to a small sub-problem Build up to the whole problem Make choices that look good in the short term but

not necessarily in the long term

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1.Greedy Algorithms

Advantages: When they work, they work fast Simple and easy to implement

Disadvantages: They do not always work. Short term choices may be disastrous on the long

term. Correctness is hard to prove

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1.Greedy Algorithms

Applications: Applications of greedy method are very board

Example: Sorting. Merging sorted list. Knapsack Minimum Spanning Tree (MST) Hoffman Encoding

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1.Greedy Algorithms

Characteristics and Features:To construct the solution in an optimal way. Algorithm

maintains two sets. One contains chosen items and the other contains rejected items.

The greedy algorithm consists of four (4) function.A function that checks whether chosen set of items

provide a solution.

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1.Greedy Algorithms (Cont.)

Characteristics and Features:A function that checks the feasibility of a set.The selection function tells which of the candidates is

the most promising.An objective function, which does not appear

explicitly, gives the value of a solution.

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2. The General Method

Let a[ ] be an array of elements that may contribute to a solution. Let S be a solution,

Greedy (a[ ],n) {

S = empty;for each element (i) from a[ ], i = 1:n {

x = Select (a,i);if (Feasible(S,x)) S = Union(S,x);

}return S;

}

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2. The General Method (Cont.)

Select:Selects an element from a[ ] and removes it. Selection is optimized

to satisfy an objective function.

Feasible:True if selected value can be included in the solution vector, False

otherwise.

Union:Combines value with solution and updates objective function.

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3. Knapsack Problem

There are two versions of the problem:1. 0-1 Knapsack Problem2. Fractional Knapsack Problem

i. Bounded Knapsack Problemii. Unbounded Knapsack Problem

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3. Knapsack Problem

In a knapsack problem or rucksack problem,we are given a set of 𝑛items, where each item 𝑖is specified by a size 𝑠𝑖 and a value 𝑣𝑖. We are also given a size bound 𝑆, the size of our knapsack.

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Item i Size si Value vi

1 1 8

2 3 6

3 5 5

3. Knapsack Problem

Environment:

Object (i):Total Weight wi

Total Profit pi

Fraction of object (i) is continuous (0 =< xi <= 1)A Number of Objects

1 =< i <= nA knapsack

Capacity m

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Objects

1 2

3 n

Capacity

M

3. Knapsack Problem

GreedyKnapsack ( p[ ] , w[ ] , m , n ,x[ ] ){

insert indices (i) of items in a maximum heap on value vi = pi / wi ;

Zero the vector x; Rem = m ;For k = 1..n{ remove top of heap to get index (i);

if (w[i] > Rem) then break;x[i] = 1.0 ; Rem = Rem – w[i] ;

}if (k < = n ) x[i] = Rem / w[i] ;

}// T(n) = O(n log n)

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3. Knapsack Problem

Example:

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Given:𝑛 = 4 (# of elements)𝑆 = 5 pounds (maximum size)Elements (size, value) = { (1, 200), (3, 240), (2, 140), (5, 150) }

3. Knapsack Problem

Example:

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1. Calculate Vi =vi

sifor 𝑖 = 1,2, … , 𝑛

2. Sort the items by decreasing Vi3. Find j, such that𝑠1 + 𝑠2 +⋯+ 𝑠𝑗 ≤ 𝑆

< 𝑠1 + 𝑠2 +⋯+ 𝑠𝑗+1

3. Knapsack Problem

Example:

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𝑉𝑖 =𝑣𝑎𝑙𝑢𝑒𝑖𝑠𝑖𝑧𝑒𝑖

=𝑐𝑜𝑠𝑡𝑖

𝑤𝑒𝑖𝑔ℎ𝑡𝑖

COST WEIGHT VALUES

A 200 1 200

B 240 3 80

C 140 2 70

D 150 5 30

3. Knapsack Problem

Solution is:

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1 pounds A3 pounds B2 pounds C4 pounds D

THANK YOU

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