ms nikita greedy agorithm
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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
