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Lecture 04

the divide and conquer is used for reoccurences else known as recursive functions

3 different methods

Substitution

gues the form of the solution first then use mathematical induction to prove the solution can be used to find constants

Recursion-tree

Master

Exercise 1

Consider the following recurrence T(n) = T(2n/3) + Θ(1). Prove that T(n) = O(lg n).

a = 1

b = 3/2

d = 0

log3/2(1) == 0

O(nd lg n) == O(n0 lg n) == O(lg n)

Exercise 2

Consider the following recurrence

T(n) =  {
        {    Θ(1)                if n = 1
        {    T(n − 1) + Θ(n)     if n > 1   
        {
Prove that T (n) = O(n2) using the substitution method.

guess == O(n^2)

c =

n0 > 1

T(n − 1) + Θ(n)

<= c(n-1)2 * b*n

we assume b <= c

<= c(n-1)2 * cn

== cn2 − 2cn + c + cn

we assume c <= cn

<= cn2 − 2cn + cn + cn

== cn2

exercise 3

Consider the following recurrence:

T(n) =  {
        { Θ(1)                             if n ≤ 1
        { T(⌊n/2⌋) + T(⌈n/2⌉ − 1) + Θ(n)     if n > 1
        {
Use the substitution method to prove that T(n) = O(n lg n).

n0 > 1

T(⌊n/2⌋) + T(⌈n/2⌉ − 1) + Θ(n)

==T(⌊n/2⌋) + T(⌈n/2⌉ − 1) + bn