Cs 1102 Tutorial 5

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CS1102 Tut 4 Recursion and

Big Oh

Max Tan

[email protected]

S15-03-07 Tel:65164364http://www.comp.nus.edu.sg/~tanhuiyi
mailto:[email protected]:[email protected]

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Groups Assignment

Question 1Group 1

Question 2Group 2

Question 3Group 3

Question 4Group 4

Question 5

Group 1 Question 6Group 2

Question 7Group 3

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First 15 minutes

Recap on MergeSort and QuickSort

Insertion sort, selection sort, bubble

sort are very simple O(n2) algorithms.

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First 15 minutes

How does mergeSort work?

MergeSort(A[i..j])

Precondition : Nil

Postcondition : A[i..j] is sorted

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First 15 minutes

MergeSort(A[i..j])

Precondition : Nil


MergeSort on leftHalf

MergeSort on rightHalf

Merge 2 sorted halves.

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First 15 minutes

Unsorted array

mergeSort : Sorts an imput array

Unsorted array

dividemergeSort mergeSort

After mergesort is done, array is sorted!

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First 15 minutes

Unsorted array


Sorted arrays


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First 15 minutes

Unsorted array


Sorted arrays


Merge

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First 15 minutes

Unsorted array


Sorted arrays


Merge

Output

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First 15 minutes

Time complexity analysis

h

Assume the height of this

tree is h

Number of elements in each

array in first level = n

Number of elements in each

array in 2ndlevel = n/2

in kthlevel = n/2k-1

In the hthlevel, n/2h-1= 1

n = 2h-1

lg n = h-1

h = lg n + 1

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First 15 minutes

Time complexity analysis

h

h = lg n + 1

Each level has O(n)computations

Hence h levels has O(hn)

computations

h = log n, therefore

Time complexity is O(nlog n)

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First 15 minutes

QuickSort(A[i..j])Precondition : Nil


Find a pivot and position the elementsaccordingly

QuickSort on left

QuickSort on right

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First 15 minutes

Unsorted array

quickSort : Sorts an imput array

Find pivot

pivot

quickSort quickSort

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First 15 minutes

Unsorted array

quickSort : Sorts an imput array

Find pivot

pivot

quickSort quickSort

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First 15 minutes

QuickSort

QuickSort(A[i..j])

Precondition : Nil


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Question 1

Given the following recursive function:

int f(int n, int m) {

if (n == 0) return m;

else return f(n-1, 1-m);}

What is the result of f(3,7)?

(a) 0

(b) 3(c) 7

(d) -7

(e) None of the above

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Question 1Solution

(e)

f(3,7)= f(2,-6)

= f(1, 7)

= f(0, -6)

= -6

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Question 2

Given the following recursive function:

int g(int n) {

if (n == 0) return 1

else return g(n-1) + g(n-2);

}

Which of the following statement is true?

(a) It does not have a base case

(b) It is not recursive

(c) The function does not terminate for all n

(d) None of the above

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Question 2

Solution

(d)

g(n) where n is 0 terminates

As g(0) = 1All other values do not terminate.

g(-1) does not terminate, so g(1) also does not

terminate

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Question 3

Consider the following function where the input n is

assumed to be positive integer.

f(n) = n2+ (n-1)2+ + 22+ 12

Write a recursive function to implement it and a main()

function to display the first 10 function values. What if n

is not positive?

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Question 3

Solution

int f(int n) {

if (n

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Question 4

int h(int n) {

if (n

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Question 4

Solution (1)

h(10) = 1 + h(9) + h(7)

h(9) = 1 + h(8) + h(6)

h(8) = 1 + h(7) + h(5)

h(7) = 1 + h(6) + h(4)

h(6) = 1 + h(5) + h(3)h(5) = 1 + h(4) + h(2)

h(4) = 1 + h(3) + h(1)

h(3) = 1

h(2) = 1

h(1) = 1

The same computations are repeatedly executed when itis called with the same value and hence the recursivesolution is inefficient

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Question 4

Solution (1)

To improve its performance, use iterative solution.

int h(int n) {

if (n

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Question 4

Solution (3)

compute h(0), h(1), , until h(10) will be easier to get theanswer

h(0) = 1

h(1) = 1

h(2) = 1h(3) = 1

h(4) = 3

h(5) = 5

h(6) = 7

h(7) = 11h(8) = 17

h(9) = 25

h(10) = 37

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Question 5

int method( int n) {

if (n < = 1) return 1;

return method(n/2) + method(1);

}

The tightest time complexity of this method is:

(a) O(log n)

(b) O(n)

(c) O(n2)

(d) O(2n)

(e) O(n2n)

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Question 5

Solution

(a)

Domain of interest is reduced by half for each

recursive call.

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Question 6

Solution

a. Computing the sum of the first n even integersby using a for loop

b. Displaying all n integers in an array

c. Displaying all n integers in a sorted linked list

d. Displaying all n names in a circular linked liste. Displaying one array element

f. Displaying the last integer in a linked list

g. Searching an array of n integers for aparticular value by using a binary search

h. Adding an item to a stack of n itemsi. Adding an item to a queue of n items

a. O(n)

b. O(n)

c. O(n)

d. O(n)

e. O(1)

f. O(n)

g. O( log n)

h. O(1)

i. O(1)

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Question 7

Consider the following method f, which calls the method swap.Assume that swap exists and simply swaps the contents of thetwo arguments. Do not be concerned with fs purpose.

void f(int [] theArray, int n) {

for (int j = 0; j < n; ++j) {

int i = 0;

while (i < 1) {

if (theArray[i] < theArray[j])

swap(theArray[i], theArray[j]);++i;

} // end while

} // end for

} // end f

How many comparisons does f perform?

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Question 7

Solution

O(n)

as the statements in the while loop only execute once

for each j because of the condition i

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Cs 1102 Tutorial 5