Example: Suppose you have to choose among three algorithms to solve a problem:
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•
•
Algorithm A solves an instance of size 𝑛 by recursively solving eight instances of
!
size ! , and then combining their solutions in time Ο(𝑛! ).
Algorithm B solves an instance of size 𝑛 by recursively solving twenty instances
!
of size ! , and then combining their solutions in time Ο(𝑛! ).
Algorithm C solves an instance of size 𝑛 by recursively solving two instances of
size 2𝑛, and then combining their solutions in time Ο(𝑛).
Which one is preferable and why?
Solution:
•
Algorithm A:
𝑇 𝑛 = 8𝑇
!
!
+ 𝑛!
!
!
𝑎 = 8; 𝑏 = 2; 𝑛!"#! = 𝑛!"#! = 𝑛!
!
𝑓 𝑛 = 𝑛! → 𝑓 𝑛 = Θ 𝑛!"#! = Θ 𝑛!
𝑇 𝑛 = Θ(𝑛! 𝑙𝑔𝑛)
•
Algorithm B:
𝑇 𝑛 = 20𝑇
!
+ 𝑛!
!
!
!"
𝑎 = 20; 𝑏 = 3; 𝑛!"#! = 𝑛!"#! = 𝑛!.!"
!
𝑓 𝑛 = 𝑛! → 𝑓 𝑛 = Ω 𝑛!"#! !! = Ω 𝑛!.!"!!
!"
𝑇 𝑛 = Θ 𝑛!"#!
•
𝐶𝑎𝑠𝑒 2 𝑜𝑓 𝑀𝑎𝑠𝑡𝑒𝑟 𝑀𝑒𝑡ℎ𝑜𝑑
= Θ(𝑛!.!" )
Algorithm C:
𝑇 𝑛 = 2𝑇 2𝑛 + 𝑛
The running time of this algorithm is “infinite”
Algorithm B is the preferable.
𝐶𝑎𝑠𝑒 3 𝑜𝑓 𝑀𝑎𝑠𝑡𝑒𝑟 𝑀𝑒𝑡ℎ𝑜𝑑
𝑓𝑜𝑟 0 ≺ 𝜀 ≼ 0.73