Dept. of Computer Engineering

Middle East Technical Univ.

Ankara - TURKEY

Anaysis of Algorihms, Fall 2012

CEng 315

1

Recurrences (cont.)

• Inhomogeneous Recurrences: We now consider

recurrences of a slightly more general form:

a0tn + a1tn-1 + … + aktn-k = bnp(n)

(2)

• The left hand side is the same as (1), but on the

right hand side we have bnp(n),

where b is a constant and p(n) is a polynomial in n

of degree d.

Anaysis of Algorihms, Fall 2012

CEng 315

2

Recurrences (cont.)

Example: Solve the following recurrence: tn - 2tn-1 = 3n.

In this case b =3 and p(n) = 1, a polynomial of degree d=0.

A little manipulation allows us to reduce this example to the form (1). To see this, we first multiply the recurrence by 3, obtaining 3tn - 6tn-1 = 3n+1.

If we replace n by n+1 in the original recurrence, we get

tn+1 - 2tn = 3n+1.

Finally, subtracting these two equations, we have

tn+1 - 5tn + 6tn-1 = 0,

which can be solved by the above method.

The characteristic equation is

x2 - 5x + 6 = 0. That is, (x-2)(x-3) = 0.

• Intuitively we can see that the factor (x-2) corresponds to the left-hand side of the original recurrence, whereas the factor (x-3) has appeared as a result of our manipulation to get rid of the right-hand side. Anaysis of Algorihms, Fall 2012

CEng 315

3

Recurrences (cont.)

• Generalizing this approach, to solve (2):

a0tn + a1tn-1 + … + aktn-k = bnp(n) (2)

we can show that it is sufficient to take

the following characteristic equation:

(a0xk+ a1xk-1 +…+ ak )(x-b)d+1 = 0,

where d is the degree of polynomial p(n)

• Once this equation is obtained, proceed as in the

homogeneous case.

Anaysis of Algorihms, Fall 2012

CEng 315

4

Recurrences (cont.)

Example: The number of movements of a ring required in the Towers of Hanoi problem is given by the following recurrence relation: tn = 2tn-1 + 1, n 1, subject to t0 = 0.

The recurrence can be written

tn - 2tn-1 = 1, which is of the form (2) with b = 1, p(n) = 1, a polynomial of degree 0.

The characteristic equation is therefore

(x-2)(x-1) = 0.

The roots of this equation are 1 and 2, so the general solution of the recurrence is

tn = c11n +c22n.

We need two initial conditions. We know that t0 = 0; to find a second initial condition we use the recurrence itself to calculate

t1 = 2t0 +1 = 1.

We finally have c1 +c2 = 0 n = 0

c1 + 2c2 = 1 n = 1

From which we obtain the solution, tn = 2n –1, where c1= -1, c2 = 1. Then we can conclude that tn (2n).

Anaysis of Algorihms, Fall 2012

CEng 315

5

Recurrences (cont.)

• A further generalization of the same type of the

argument allows us finally to solve recurrences of the

form

a0tn + a1tn-1 + … + aktn-k = b1np1(n) + b2np2(n)+…

(3)

where the bi are distinct constants and pi(n) are

polynomials in n respectively of degree di. It suffices to

write the characteristic equation

(a0xk+ a1xk-1 +…+ ak )(x - b1)d1+1(x - b2)d2+1 … = 0,

Anaysis of Algorihms, Fall 2012

CEng 315

6

Recurrences (cont.)

Example: Solve the following recurrence:

tn = 2tn-1 + n + 2n

n 1, subject to t0 = 0.

tn - 2tn-1 = n + 2n, which is of the form (3) with

b1=1, p1(n) = n, b2 =2, p2(n) = 1.

The characteristic equation is (x-2)(x-1)2(x-2) = 0, Roots:1,1,2,2. The general solution of the recurrence is therefore of the form tn = c11n + c2n1n + c32n + c4n2n

Using the recurrence, we can calculate

t1= 3 (t1= 0+1+21 = 3), t2 = 12, t3 = 35.

c1 +c3 = 0 n = 0,

c1 + c2 + 2c3 + 2c4 = 3 n = 1

c1 + 2c2 + 4c3 + 8c4 = 12 n = 2

c1 + 3c2 + 8c3 + 24c4 = 35 n = 3 ,

arriving at tn = -2 – n + 2n+1 + n2n

Therefore, tn (n2n).

Anaysis of Algorihms, Fall 2012

CEng 315

7

Recurrences (cont.)

Change of Variable:

We write T(n) for the term of a general recurrence, and tk for the term of a new recurrence obtained by a change of variable.

Example: Solve the recurrence if n is a power of 2 and if

T(n) = 4T(n/2) + n

n > 1.

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