Cmsc 250 Intro to Algorithms
Study guide MidTerm Test: Practice Problems
You may send me the solutions of the problems for feedback
A. Efficiency
1. True or False? Circle the correct answers (may be more than one correct answer)
N2 + (logN)6 = O((N3logN) 3)
N2 + logN = O(NlogN) ,
N2+logN = O(N2)
log(N2) = O(N2 logN),
N + logN = O(N)
NlogN = O(N + logN)
(log(N2))2 = O(N2 logN),
(log(N2))2 = O(N2),
Nlog(N2) = O(N),
(log(N2))2 = O(log(N)),
log(N2) = O(log(N))
2. Fill in the blanks:
1. N +( logN)3
=(
)
2. N + NlogN
=(
)
3. N2 +N4 ( logN)3 =  (
)
4. N2+logN
=(
)
5. N3logN
=(
)
6. log(N2)
=(
)
7. (log(N2))2
=(
)
3. Determine the complexity for the following loops, using the Big-Oh notation:
a.
for(i = 0; i < N; i++)
{
for(j = i +1; j < N; j++)
{
if(a[j-1] > a[j])
{ t = a[i]; a[i] = a[j]; a[j] = t; }
}
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}
b.
private void percolateDown(int hole)
{
int child;
AnyType tmp = array[hole];
for( ; hole*2 <= currentSize; hole = child)
{
child = hole * 2;
if( child != currentSize &&
array[child+1].compareTo( array[child]) < 0)
child++;
if( array[child].compareTo(tmp) < 0)
array[hole] = array[child];
else break; // found position to insert
}
array[hole] = tmp;
}
4. Consider the following recursive algorithms
4.1.
Algorithm S(n)
//Input: A positive integer n
//Output: to be determined by you
if n = 1 return 1
else return S(n − 1) + n ∗ n ∗ n
a. What does it compute?
b. Set up a recurrence relation for the run time of the algorithm and solve it.
c. Determine the efficiency class of this algorithm
4.2.
Algorithm Q(n)
//Input: A positive integer n
if n = 1 return 1
else return Q(n − 1) + 2 ∗ n − 1
a. Set up a recurrence relation for Q(n) and solve it to determine the output of the
algorithm
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b. Set up a recurrence relation for the run time of the algorithm and solve it.
c. Determine the efficiency class of this algorithm
5. Write a pseudocode for a recursive algorithm for finding the position of the largest
element in an array of n numbers. Set up a recurrence relation for the run time of the
algorithm and solve it.
6. Write a pseudocode for a recursive algorithm that determines whether a given word is
a palindrome. A given word is a palindrome if it can be read backwards, for example
pop, rotor, noon. Set up and solve a recurrence relation for the run time of the
algorithm and solve it.
7. Write a pseudocode for a recursive algorithm that reverses the contents of an array. Set
up and solve a recurrence relation for the run time of the algorithm and solve it.
8. Write a pseudocode for a recursive algorithm that traverses a binary tree in pre-order.
Set up and solve a recurrence relation for the run time of the algorithm and solve it.
9. Write a pseudocode for a recursive algorithm that counts the leaves in a binary tree. Set
up and solve a recurrence relation for the run time of the algorithm and solve it.
10. Write a pseudocode for a recursive algorithm that computes the height of a binary tree.
Set up and solve a recurrence relation for the run time of the algorithm and solve it.
11. Solve the following recurrence relations, with initial condition T(1) = 1
11.1
T(N) = T(N/2) + LogN
11.2
T(N) = T(N/2) + N
11.3
T(N) = 2T(N/2) + N
11.4
T(N) = 2T(N/2) + NLogN
11.5
T(N) = 2T(N/2) + N2
11.6
T(N) = T(N-1) + N
11.7
T(N) = T(N-1) + 1
11.8
T(N) = 2T(N-1) + 1
B. Sorting
12. Match the descriptions on the left with the names of sorting algorithms on the right
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Assumes part of the array is sorted,
inserts the next element there.
Quicksort
Splits the array into two, sorts
recursively and then merges
MergeSort
Splits the array in two sets and a
middle element: smaller to the left,
bigger to the right. Then sorts
recursively each set
Bubble
Compares and swaps adjacent
elements. Several passes are needed
Insertion
13. Describe briefly (but adequately) the operation of each of the following sorting
algorithms: selection sort, insertion sort, bubble sort, mergesort, quick sort.
14. Selection sort is best used for (circle the correct answer):
a. Small number of records
b. Large number of records
c. Almost sorted files
d. Large records
15. Insertion sort is best used for (circle the correct answer):
a. Small number of records
b. Large number of records
c. Almost sorted files
d. Large records
16. Which algorithm is usually very fast, however in certain cases its run time may increase
to O(N2), and that is why it is usually not used for military and life-supporting
applications. Explain why the run time may increase to O(N2)
17. What is the role of the pivot in quicksort? Describe at least three ways to choose it.
Discuss their advantages and disadvantages.
18. Which algorithm is not used for sorting in main memory and why?
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19. Write down the recurrence relation for the worst case of quicksort and solve it.
20. Write down the recurrence relation for the best case of quicksort and solve it. Give the
name of another sorting algorithm with the same recurrence relation?
21. What is the best running time (in terms of Big Oh notation) of insertion sort? When
does it happen?
22. Does selection sort have best//worst-case running time? Explain your answer.
23. Does merge sort have best/ worst-case running time? Explain your answer.
24. Does bubble sort have best/ worst-case running time? Explain your answer.
25. Write down the average complexity of each algorithm using Big Oh notation.
Bubble sort
Insertion sort
Mergesort
Quick sort
Selection sort
26. Next to each code segment write the name of the corresponding sorting algorithm
26.1
void method01(int [] a){
int size = a.length;
int i, j, t;
for(i = 0; i < size; i++) {
for(j = i +1; j < size; j++)
if(a[j-1] > a[j]) {
t = a[i];
a[i] = a[j];
a[j] = t;
}
}
}
{
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26.2
void method02(int [] a){
int j, p;
int tmp;
for ( p = 1; p < N ; p++)
{
tmp = a[p];
for (j = p; j > 0 && tmp < a[j-1]; j--)
a[j] = a[j-1];
a[j] = tmp;
}
}
26.3
void method03 ( int a[], int left, int right)
{
if (right > left)
{
middle = left + (right - left)/2;
method03 (a, left, middle);
method03 (a, middle+1, right);
help_method(a, left, middle, right);
}
}
void help_method(int a[], int left, int middle, int right)
{
int b[] = new b [right – left + 1];
int i = left, j = middle+1, k = 0;
while (i < middle && j <= right)
{
if (a[i] < a[j]) b[k++] = a[i++];
else b[k++] = a[j++];
}
while(i <= middle) b[k++] = a[i++];
while(j <= right) b[k++] = a[j++];
k = 0;
for (k = 0; k < right – left + 1; k++)
a[left + k] = b[k];
}
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26.4
void method04 ( int a[], int left, int right){
if( left + 10 <= right)
{
element = help_method(array, left, right);
int i = left, j = right - 1;
for ( ; ; )
{
while (a[++i] < element) {}
while (element < a[--j] ) {}
if (i < j)
swap (a[i],a[j]);
else break;
}
swap (a[i], a[right-1]);
method04 ( a, left, i-1);
method04 (a, i+1, right);
}
else insertionsort (a, left, right);
}
int help_method (int[] array, int left, int right)
{
int middle = (left + right)/2;
if (array[left] > array[middle]) swap(array, left, middle);
if (array[middle] > array[right]) swap(array,middle,right);
if (array[left] > array[middle]) swap (array, left, middle);
swap(array, middle, right-1);
return array[right -1];
}
26.5
void method05 ( int [ ] num )
{
int i, j, first, temp;
for ( i = num.length - 1; i > 0; i-- )
{
first = 0;
for(j = 1; j <= i; j ++)
{
if( num[ j ] < num[ first ] )
first = j;
}
temp = num[ first ];
num[ first ] = num[ i ];
num[ i ] = temp;
}
}
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C. Trees
27. Construct a binary tree with 8 nodes and minimum number of levels and assign the
letters of CAMBRIDGE to the nodes so that the name can be read in a pre-order/inorder/post-order traversal.
28. Draw the binary search tree for the letters in the word CAMBRIDGE
29. Draw an AVL tree for the letters in the word CAMBRIDGE. Indicate the type of each
rotation
30. Draw 2-3-4 tree for the letters in the word CAMBRIDGE. Show the tree after insertion
of each letter.
31. What is the complexity of search in Binary Search trees, provided they are kept balanced?
32. What is the worst complexity of search in Binary Search trees, provided they are not kept
balanced? When does it happen?
33. Which traversal order would retrieve the elements in a Binary Search Tree in sorted order?
34. Give the relation (exact formulas) between the height M of a perfect binary tree and the number
of nodes N.
N=
M=
35. In a well-balanced binary tree with M levels (the root is at level 0) there are O(
Fill in the blank appropriately.
36. In a well-balanced binary tree with N nodes there are O(
) nodes.
) levels. Fill in appropriately
37. What is the minimal height of a binary tree with 38 nodes? What would be the maximal height?
38. How many links are there in a binary tree with N nodes?
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39. In a perfect binary tree with N nodes how many nodes are with height 0?
40. In a perfect binary tree, how many nodes are there at level p?
How to represent binary trees:
Drawing a tree in WORD is difficult. Here how to represent the tree (after you have drawn
it on paper):
Write down the left and the right child of each node in a table. Example: rwa the binary
search tree for “IOWA”, It should look like this:
I
/
\
A
O
\
W
Node
I
A
O
W
left
A
-
right
O
W
-
I will be able to reconstruct the tree and see if it is correct.
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