Solve the following recurrence relations. a. x(n) = x(n − 1) + 3
for n > 1, x(1) = 0 b. x(n) = 5x(n − 1) for n > 1, x(1) = 6
c. x(n) = x(n/5) + 1 for n > 1, x(1) = 1 (solve for n = 5k )
1. Using domain and range transformations, solve the following
recurrence relations:
a) T(1) = 1, T(n) = 2T(n/2) + 6n - 1
b) T(1) = 1, T(n) = 3T(n/2) + n^2 - n
Using the backward substitution method, solve the following
recurrence relations: a.T(n)= T(n−1)+3forn>1 ,T(1)=0
b.T(n)=3T(n−1) forn>1 ,T(1)=7 c.T(n)= T(n−1)+n for n>0
,T(0)=0 d.T(n)= T(n/2)+n for n>1 ,T(1)=1(solve for n=2k) e.T(n)=
T(n/3)+1forn>1 ,T(1)=1(solve for n=3k)
Solve the following recurrence relations: (find an asymptotic
upper bound O(?) for each one)
a. T(n) = T(2n/3)+T(n/3) + n^2
b. T(n) = √nT(√n) + n
c. T(n) = T(n-1)+T(n/2) + n
The base case is that constant size problems can be solved in
constant time (O(1)). You can use the induction, substitution or
recursion tree method
Solve the recurrence equations by Substitution
a) T(n) = 4T (n/2) + n, T (1) = 1
b) T(n) = 4T (n/2) + n2 , T (1) = 1
c) T(n) = 4T (n/2) + n3 , T (1) = 1
Solve the given non-homogeneous recurrence relations:
an = an-1 + 6an-2 + f(n)
a)
an = an-1 + 6an-2 -
2n+1 with a0 = -4, a1= 5
b)
an = an-1 + 6an-2 + 5 x
3n with a0 = 2, a1 = 5
c)
an = an-1 + 6an-2 - 36n with
a0 = 10, a1= 40