Algorithms: Greedy Algorithms

73 Pages·2012·0.6 MB·English

Algorithms: Greedy Algorithms
Amotz Bar-Noy
CUNY
Spring 2012
AmotzBar-Noy (CUNY) GreedyAlgorithms Spring2012 1/62
Greedy Algorithms
Greedy algorithms make decisions that “seem” to be the best
following some greedy criteria.
In Off-Line problems:
Thewholeinputisknowninadvance.
Possibletodosomepreprocessingoftheinput.
Decisionsareirrevocable.
In Real-Time and On-Line problems:
Thepresentcannotchangethepast.
Thepresentcannotrelyontheun-knownfuture.
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How and When to use Greedy Algorithms?
Initial solution: Establish trivial solutions for a problem of a small
size. Usually n = 0 or n = 1.
Topbottomprocedure: Foraproblemofsizen,lookforagreedy
decision that reduces the size of the problem to some k < n and
then, apply recursion.
Bottom up procedure: Construct the solution for a problem of
size n based on some greedy criteria applied on the solutions to
the problems of size k = 1,...,n−1.
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The Coin Changing Problem
Input:
Integercoindenominationsd >···>d >d =1.
n 2 1
Anintegeramounttopay: A.
Output: Number of coins n for each denomination d to get the
i i
exact amount.
A=n d +n d +n d +n d .
n n n−1 n−1 2 2 1 1
Goal: Minimize total number of coins.
N =n +···+n +n .
n 2 1
Remark: There is always a solution with N = A since d = 1.
1
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Examples
USA: d = 100, d = 50, d = 25, d = 10, d = 5, d = 1.
6 5 4 3 2 1
A=73=2·25+2·10+3·1.
N =2+2+3=7.
Old British: d = 240, d = 20, d = 1.
3 2 1
A=307=1·240+3·20+7·1.
N =1+3+7=11.
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Greedy Solution
Idea: Use the largest possible denomination and update A.
Implementation:
Coin-Changing(d > ··· > d > d = 1)
n 2 1
for i = n downto 1
n = (cid:98)A/d(cid:99)
i i
A = A mod d = A−nd
i i i
Return(N = n +···+n +n )
n 2 1
Correctness: A = n d +n d +n d +n d .
n n n−1 n−1 2 2 1 1
Complexity: Θ(n) division and mod integer operations.
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A coin system for which Greedy is not optimal:
d =4,d =3,d =1andA=6:
3 2 1
Greedy: 6=1·4+2·1 ⇒ N =3.
Optimal: 6=2·3 ⇒ N =2.
A coin system for which Greedy is very “bad”:
d =x +1,d =x,d =1andA=2x:
3 2 1
Greedy: 2x =1·(x +1)+(x −1)·1 ⇒ N =x.
Optimal: 2x =2·x ⇒ N =2.
Optimality
Greedy is optimal for the USA system.
AmotzBar-Noy (CUNY) GreedyAlgorithms Spring2012 7/62
A coin system for which Greedy is very “bad”:
d =x +1,d =x,d =1andA=2x:
3 2 1
Greedy: 2x =1·(x +1)+(x −1)·1 ⇒ N =x.
Optimal: 2x =2·x ⇒ N =2.
Optimality
Greedy is optimal for the USA system.
A coin system for which Greedy is not optimal:
d =4,d =3,d =1andA=6:
3 2 1
Greedy: 6=1·4+2·1 ⇒ N =3.
Optimal: 6=2·3 ⇒ N =2.
AmotzBar-Noy (CUNY) GreedyAlgorithms Spring2012 7/62
Optimality
Greedy is optimal for the USA system.
A coin system for which Greedy is not optimal:
d =4,d =3,d =1andA=6:
3 2 1
Greedy: 6=1·4+2·1 ⇒ N =3.
Optimal: 6=2·3 ⇒ N =2.
A coin system for which Greedy is very “bad”:
d =x +1,d =x,d =1andA=2x:
3 2 1
Greedy: 2x =1·(x +1)+(x −1)·1 ⇒ N =x.
Optimal: 2x =2·x ⇒ N =2.
AmotzBar-Noy (CUNY) GreedyAlgorithms Spring2012 7/62
Efficiency
Optimal solution: Check all possible combinations.
Notapolynomialtimealgorithm.
Another optimal solution: Polynomial in both n and A.
Notastronglypolynomialtimealgorithm.
Objective:
Findasolutionthatispolynomialonlyinn.
Probablyimpossible!?
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