C13
1. Solve the following recurrence equation using the iteration method. Show all the
steps in your derivation.
c n = 1
n T
n
) ( = aT √+cn n > 1
b↵
Substitute the value of T(n) from the recurrence equation:
aT(n/b) + cn
⇒ a(aT((n/b)/b) + c(n/b)) + cn
⇒ a2T(n/b2) + cn(a/b) + cn
⇒ a2T(n/b2) + cn((a/b) + 1)
⇒ a2(aT((n/b2)/b) + cn/b2) + cn((a/b) + 1)
⇒ a3T(n/b3) + cn(a2/b2) + cn((a/b) + 1)
⇒ a3T(n/b3) + cn((a2/b2)+ (a/b )+ 1)
…
⇒ akT(n/bk) + cn((ak-1/bk-1)+ (ak-2/bk-2)+ … + (a2/b2)+ (a/b) + 1)
When k = logb n,
⇒ n = bk
T(n) = akT(1) + cn(ak-1/bk-1 + ... + a2/b2 + a/b + 1)
= akc + cn(ak-1/bk-1 + ... + a2/b2 + a/b + 1)
= cak + cn(ak-1/bk-1 + ... + a2/b2 + a/b + 1)
= cnak/bk + cn(ak-1/bk-1 + ... + a2/b2 + a/b + 1)
= cn(ak/bk + ... + a2/b2 + a/b + 1)
docsity.com