Design and Analysis of AlgorithmsRecurrences

Haidong XueSummer 2012, at GSU

What is a recurrence?

• “an equation or inequality that describes a function in terms of its value on smaller inputs”

• Time complexity of divide and conquer algorithms can be represented as recurrence equations.

• What is the recurrence of fact2

fact2(n)if(n<=1) return 1;return n*fact2(n-1);

How to solve recurrences?

• What can be omitted?– Floor, ceiling and boundary conditions– Since the solution typically doesn’t change by

more than a constant factor• Example: merge sort

How to solve recurrence?

• Three methods to solve recurrence– Substitution method– Recursion-tree method– Master method

Substitution method-examples

1. Guess the solution2. Use mathematical induction to prove it

Substitution method-examples

Guess: 1. Basis– When n=2,

2. Inductive steps– Inductive hypothesis: when n<=k, k>=2, – The statement needs to be proved: when n=k+1,

Substitution method-examples

Proof:• When n=k+1, • Since , • • So, =O((k+1)lg(k+1)),

Substitution method-examples

• It could be very hard prove even if one made the right guess

• 4.3-2

• How to make a guess?– Recursion-tree

Recursion-tree

• Recursion tree: each node represents the known cost of a sub problem

• Help us to make a guess of the total cost

Recursion-tree𝑇 (𝑛 )=2𝑇 (𝑛2 )+Θ (𝑛)

Θ (𝑛)

Θ (𝑛/2 ) Θ (𝑛/2 )

Θ (𝑛/4 ) Θ (𝑛/4 ) Θ (𝑛/4 ) Θ (𝑛/4 )

……

……

lg (𝑛)

Θ (𝑛)

=

=

= Θ (𝑛/2lg (𝑛)) Θ (𝑛/2lg (𝑛)) Θ (𝑛/2lg (𝑛)) Θ (𝑛/2lg (𝑛)) Θ (𝑛/2lg (𝑛))

Totally?

The master theorem

Let a>=1 and b>1 be constants, let f(n) be a function, and let T(n) be defined on the nonnegative integers by the recurrence:

• If for some constant >0, then • If , then • If for some constant >0, and if for some

constant c<1 and all sufficiently large n, then

The master theorem

𝑇 (𝑛 )=𝑎𝑇 (𝑛𝑏 )+ 𝑓 (𝑛)Recurrence f(n) T(n)

𝑇 (𝑛)=9𝑇 (𝑛 /3)+𝑛𝑇 (𝑛)=𝑇 (2𝑛/3)+1

𝑇 (𝑛)=2𝑇 (𝑛/2)+𝑛𝑙𝑔𝑛𝑇 (𝑛)=2𝑇 (𝑛/2)+Θ(𝑛)

𝑛 𝑛𝑙𝑜𝑔 39=𝑛2 Θ()

1 𝑛𝑙𝑜𝑔 3/21=1 Θ(lg(n))

𝑛𝑙𝑔(𝑛) 𝑛𝑙𝑜𝑔 22 (cannot use master method)

Θ(𝑛) 𝑛𝑙𝑜𝑔 22 Θ()