Recurrence Relation 1

 

Recurrence Relation

A recurrence is an equation or inequality that describes a function in terms of its values on smaller inputs. To solve a Recurrence Relation means to obtain a function defined on the natural numbers that satisfy the recurrence.

For Example, the Worst Case Running Time T(n) of the MERGE SORT Procedures is described by the recurrence.

T (n) = θ (1) if n=1
 2T + θ (n) if n>1

There are four methods for solving Recurrence:

1. Substitution Method
2. Iteration Method
3. Recursion Tree Method
4. Master Method

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1. Substitution Method:

The Substitution Method Consists of two main steps:

1. Guess the Solution.
2. Use the mathematical induction to find the boundary condition and shows that the guess is correct.

For Example1 Solve the equation by Substitution Method.

T (n) = T + n

We have to show that it is asymptotically bound by O (log n).

Solution:

For T (n) = O (log n)

We have to show that for some constant c

T (n) ≤c logn. 

Put this in given Recurrence Equation.

	T (n) ≤c log+ 1
  		≤c log+ 1 = c logn-clog2 2+1
  		≤c logn for c≥1
Thus T (n) =O logn.

Example2 Consider the Recurrence

T (n) = 2T+ n n>1

Find an Asymptotic bound on T.

Solution:

We guess the solution is O (n (logn)).Thus for constant 'c'.
 T (n) ≤c n logn
Put this in given Recurrence Equation.
Now,
  T (n) ≤2clog +n
      ≤cnlogn-cnlog2+n
      =cn logn-n (clog2-1)
      ≤cn logn for (c≥1)
Thus T (n) = O (n logn).

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2. Iteration Methods

It means to expand the recurrence and express it as a summation of terms of n and initial condition.

Example1: Consider the Recurrence

1. T (n) = 1  if n=1  
2.       = 2T (n-1) if n>1  

Solution:

  
T (n) = 2T (n-1)
      = 2[2T (n-2)] = 22T (n-2)
      = 4[2T (n-3)] = 23T (n-3)
      = 8[2T (n-4)] = 24T (n-4)   (Eq.1)

Repeat the procedure for i times

T (n) = 2i T (n-i)
Put n-i=1 or i= n-1 in    (Eq.1)
T (n) = 2n-1 T (1)
      = 2n-1 .1    {T (1) =1 .....given}
      = 2n-1 

Example2: Consider the Recurrence

1. T (n) = T (n-1) +1 and T (1) =  θ (1).  

Solution:

 T (n) = T (n-1) +1
       = (T (n-2) +1) +1 = (T (n-3) +1) +1+1
       = T (n-4) +4 = T (n-5) +1+4
       = T (n-5) +5= T (n-k) + k
Where k = n-1
   T (n-k) = T (1) = θ (1)
   T (n) = θ (1) + (n-1) = 1+n-1=n= θ (n).