3.mathematical analysis of nonrecursive algorithms

7/30/2019 3.Mathematical Analysis of Nonrecursive Algorithms

1/22

CS3024-FAZ 1

Mathematical Analysis ofNonrecursive Algorithms

Design and Analysis of Algorithms(CS3024)

23/02/2006


2/22

CS3024-FAZ 2

Learning by Examples

We systematically apply the general

framework outlined before to analyzing the

efficiency of nonrecursive algorithms.


3/22

CS3024-FAZ 3

Important Sum Manipulations (1)

)(2

1

2

)1(...21

limitsintegerupper&lower:;11

)(

22

10

111

11

nnnn

nii

ullu

baba

acca

n

i

n

i

u

li

u

i

i

u

i

i

u

i

ii

u

i

i

u

i

i


4/22

CS3024-FAZ 4

Important Sum Manipulations (2)

n

i

n

n

i

i

nn

n

i

i

kkkkn

i

k

n

i

nni

n

aa

aaaaa

n

k

ni

nnnnni

1

12

1

1

1

12

0

1

1

3222

1

2

lglg

...5772.0;ln...1

)1(1

1...1

1

1...21

31

6)12)(1(...21


5/22

CS3024-FAZ 5

Example 1

Algorithm MaxElement(A[0..n-1])

// determine the value of the largest element in a

// given array

// input: An array A[0..n-1] of real number// output: the value of the largest element in A

maxval A[0]

fori 1 to n-1 do

ifA[i] > maxval

maxval A[i]

return maxval


6/22

CS3024-FAZ 6

Exp1: Analysis (1)

Input size = number of elements = n

Most often executed inside the forloop

The comparison A[i] > maxvalThe assignment maxval A[i]

Basic operation = the comparison

Executed on each repetition of the loopThe assignment are not

The number of comparison will be the

same for all array of size n


7/22CS3024-FAZ 7

Exp1: Analysis (2)

C(n) = the number of times this comparison

is executed

The algorithm makes one comparison on

each execution of the loop within the

bounds between 1 and n-1 (inclusively)

Thus, we have:

)(11)(1

1

nnnCn

i


8/22CS3024-FAZ 8

Analyzing Efficiency of

Nonrecursive Algorithms (1)

1. Decide on a parameter(s) indicating an inputs

size

2. Identify the algorithms basic operation

(typically, it is located in its inner most loop)3. Check whether the number of times the basic

operation is executed depends only on the size

of an input. If it also depend on some

additional property, the worst-case, average-

case, and, if necessary, the best-case

efficiencies have to be investigated separately


9/22CS3024-FAZ 9

Analyzing Efficiency of

Nonrecursive Algorithms (2)

4. Set up a sum expressing the number of

times the algorithms basic operation is

executed

5. Using standard formulas and rules of

sum manipulation, either find a closed-

form formula for the count or, at the very

least, establish its order of growth


10/22CS3024-FAZ 10

Example 2

Algorithm UniqueElements(A[0..n-1])

//checks whether all the elements in a given

// array are distinct

//input: an array A[0..n-1]//output: returns true if all elements in A are

// distinct, and false otherwise

fori 0 to n 2 doforji+1 to n 1 do

ifA[i] = A[j] return false

return true


11/22CS3024-FAZ 11

Exp2: Analysis (1)

Inputs size = n

Innermost loop contains a single operation

Basic operation = comparison

The number of comparison will depend not

only on n but also on whether there are

equal elements in the array and, if there

are, which array positions they occupy

We will limit our investigation on the worst-

case only


12/22CS3024-FAZ 12

Exp2: Analysis (2)

The worst-case input is an array for which

Cworst(n) is the largest among all arrays of

size n

There are two kinds of worst-case inputs:

Array with no equal elements

Array in which the last two elements are the

only pair of equal elements


13/22CS3024-FAZ 13

Exp2: Analysis (3)

)(21

2)1(1...)2()1(

)1(]1)1()1[(1)(

22

2

0

2

0

2

0

1

1

nnnnnn

ininnCn

i

n

i

n

i

n

ij

worst


14/22CS3024-FAZ 14

Example 3

AlgorithmMatrixMultiplication(A[0..n-1, 0..n-1],B[0..n-1, 0..n-1])//multiplies two square matrices of order n by the

// definition-based algorithm

//input: two n-by-n matrices A and B

//output: matrix C = AB

fori 0 to n-1 do

forj 0 to n-1 do

C[i,j] 0,0

fork 0 to n-1 do

C[i,j] C[i,j] + A[i,k]* B[k,j]

return C


15/22CS3024-FAZ 15

Exp3: Analysis (1)

Inputs size = matrix order n

In the innermost loop: multiplication &

addition basic operation candidates

MUL & ADD executed exactly once on each

repetition on innermost loop we dont have

to choose between these two operations

Sum of total number of multiplication

1

0

321

0

1

0

1

0

1

0

1

0

1)(n

i

n

i

n

j

n

i

n

j

n

k

nnnnM


16/22CS3024-FAZ 16

Exp3: Analysis (2)

Estimate the running time of the algorithmon a particular machine

More accurate estimation (include addition)

cm: the time of one multiplication

ca: the time of one addition

3)()( ncnMcnTmm

333 )()()()( nccncncnAcnMcnT amamam


17/22CS3024-FAZ 17

Example 4

Algorithm Binary(n)

//input: a positive decimal integern

//output: the number of binary digits in nsbinary representation

count 1

while n > 1 do

countcount+ 1n n/2

return count


18/22CS3024-FAZ 18

Exp4: Analysis (1)

The most frequent executed operation is

the comparison n > 1

The number of times the comparison will

be executed is larger than the number of

repetition of the loops body by exactly 1


19/22CS3024-FAZ 19

Exp4: Analysis (2)

The value of n is about halved on each

repetition of the loop about log2 n

The exact formula: log2

n + 1

Another approach analysis techniques

based on recurrence relation


20/22CS3024-FAZ 20

Exercises (1)

1. Compute the following sums:

1 + 3 + 5 + 7 ++ 999

2 + 4 + 6 + 8 ++ 1024

2. Compute order of growth of the following

sums

n

i

n

j

n

i

n

j

jn

i

n

i

ijiii1 1

1

0 1

11

3

1

3

;3);1(;;1

;2)1(;lg;)1(1

11

2

21

0

22

n

i

in

i

n

i

iii


21/22CS3024-FAZ 21

Exercises (2)

3. Algorithm Mystery(n)

//input: a nonnegative integer n

S 0

fori 1 to n do

S S + i * i

return S

a. What does this algorithm compute?

b. What is its basic operation?

c. How many times is the basic op executed?

d. What is the efficiency class of this algorithm?

e. Can you make any improvement?


22/22

CS3024 FAZ 22

Exercises (3)

4. Algorithm Secret(A[0..n-1])

//input: an array A[0..n-1] of n real number

mi A[0]; ma A[0]fori 1 to n-1 do

ifA[i] < mi then mi A[i]ifA[i] > ma then ma A[i]

return ma - mi

a. What does this algorithm compute?b. What is its basic operation?

c. How many times is the basic op executed?

d. What is the efficiency class of this algorithm?

e. Can you make any improvement?

3.mathematical analysis of nonrecursive algorithms

Documents