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Assume that a CPU can process 108 operations per second. Suppose you have to sort an array with
106 elements. Which of the following is true?
Insertion sort will always take more than 2.5 hours while merge sort will always take less than 1
second.
Insertion sort will always take more than 2.5 hours while quicksort will always take less than 1 second
Insertion sort could take more than 2.5 hours while merge sort will always take less than 1 second.
Insertion sort could take more than 2.5 hours while quicksort will always take less than 1 second.
2
points
7) Suppose our aim is to sort an array in descending order. Which of the following statements is true?
Input in descending order is worst case for both selection sort and insertion sort.
Input in ascending order is worst case for both selection sort and insertion sort.
Input in descending order is worst case for insertion sort but not for selection sort.
Input in ascending order is worst case for selection sort but not for insertion sort.
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points
8) Suppose we want to sort an array in descending order and we implement quicksort so that we always
choose the last element in the array as the pivot element. Assume that the input is a permutation of {1,
2, …, n}. Which of the following would be a worst case permutation of this input for this implementation
of quicksort?
{1, 2,…, n} with all even numbers in random order followed by all odd numbers in descending order.
{1, 2, . . . , n} in descending order.
{1,2,...,n} in some random order.
{1, 2, . . . , n} with all odd numbers in ascending order followed by all even numbers in random order.
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points
9) Which of the following statements is not true?
For every fixed strategy to choose a pivot for quicksort, we can construct a worst case input that
requires time O(n2).
If we randomly choose a pivot element each time, quicksort will always terminate in time O(n log n).
If we could find the median in time O(n), quicksort would have worst case complexity O(n log n).
Quicksort and merge sort are both examples of divide and conquer algorithms.
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points
10)We have a list of points [(7,1),(3,5),(6,1),(6,5),(0,2),(9,0)]. We sort these in ascending order by the
second coordinate. Which of the folllowing corresponds to a stable sort of this input?
2
points
Week 2 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
file:///Users/madhavan/mirror/projects/NPTEL/algorithms-2015/DAA_allquizzes.html Page 3 of 9
[(9,0),(7,1),(6,1),(0,2),(3,5),(6,5)]
[(0,2),(3,5),(6,1),(6,5),(7,1),(9,0)]
[(9,0),(6,1),(7,1),(0,2),(3,5),(6,5)]
[(9,0),(7,1),(6,1),(0,2),(6,5),(3,5)]
11)An undirected graph G has 100 nodes and there is a path from every vertex to every other vertex in G.
Suppose v1 and v2 are two vertices in G. Then, what can we say about the shortest path from v1 to v2?
The path has at most 2 edges.
The path has at most 50 edges.
The path has at most 99 edges.
None of the above.
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points
12)An undirected graph G has 300 edges. The degree of a vertex v, deg(v), is the number of edges
connected to v. Suppose V = {v1,v2,...,v30}. What is the value of deg(v1)+ deg(v2) + · · · + deg(v30)?
300
600
900
None of these
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points
13)Which of the following is not true of depth-first search (DFS) starting at a vertex v?
DFS identifies all vertices reachable from v.
Using an adjacency list instead of an adjacency matrix can improve the worst case complexity.
DFS will identify the shortest paths from v in any graph without edge weights.
DFS numbering can be used to identify cycles in the graph.
2
points
14)How many topological orderings does the following directed acyclic graph have?
6
12
18
24
2
points
15)Anjali has enrolled for a part-time Masters programme where she has to complete 8 courses, numbered
1, 2, . . . , 8. Each course takes a full semester to complete. She can take as many or as few courses as
she wants in each semester.
Some courses are prerequisites for other courses. If course A is a prerequisite for course B, she can
take course B the semester after she finishes course A, or any time after that, but not before.
Given the following information about prerequisites, compute the minimum number of semesters she
needs to complete these courses
Prerequisites for course 1: course 2,4,5,7
Prerequisites for course 2: course 3
Prerequisites for course 3: course 5,6
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points
Week 3 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
file:///Users/madhavan/mirror/projects/NPTEL/algorithms-2015/DAA_allquizzes.html Page 4 of 9
Prerequisites for course 4: course 8
Prerequisites for course 5: course 8
Prerequisites for course 6: course 7,8
3
4
5
6
16)Consider the following fragment of code for a graph algorithm on an undirected graph.
for each vertex i in V
mark i as visited
for each edge (j,i) pointing into i
update weight(j,i) to weight(j,i) + k
Which of the following is the most accurate description of the complexity of this fragment. (Recall that n
is the number of vertices, m is the number of edges.)
O(n)
O(nm)
O(n+m)
O(m)
2
points
17)Consider the following strategy to solve the single source shortest path problem with edge weights from
source s.
1. Replace each edge with weight w by w edges of weight 1 connected by new intermediate nodes
2. Run BFS(s) on the modified graph to find the shortest path to each of the original vertices in the graph.
Which of the following statements is correct?
This strategy will solve the problem correctly but is not as efficient as Dijkstra's algorithm.
This strategy will solve the problem correctly and is as efficient as Dijkstra's algorithm.s st
This strategy will not solve the problem correctly.
This strategy will only work if the graph is acyclic.
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points
18)Consider the following strategy to convert a graph with negative edge weights to one that does not have
negative edge weights. Let the maximum magnitude negative edge weight in the graph be -k. Then, for
each edge in the graph with weight w, update the weight to w+k+1. Consider the following claim:
To solve the shortest path problem in the original graph, we can run Dijkstra's algorithm on the modified
graph and subtract the added weights to get the original distances.
Which of the following is not correct.
The claim is not true in general.
The claim is true for all graphs.
The claim is true for connected acyclic graphs.
The claim is not true in general for connected graphs with cycles.
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points
19)Consider the following algorithm on a graph with edge weights. 2
Week 4 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
file:///Users/madhavan/mirror/projects/NPTEL/algorithms-2015/DAA_allquizzes.html Page 5 of 9
Sort the edges as [e1,e2,...,em] in decreasing order of cost.
Start with the original graph. Consider each edge ej. If this edge is part of a cycle delete it.
Which of the following is not true.
After processing all m edges, the resulting graph is connected.
What remains at the end is a minimum cost spanning tree.
Exactly m-n+1 edges will be deleted.
At most n-1 edges will be deleted.
points
20)Suppose we have a graph with negative edge weights. We take the largest magnitude negative edge
weight -k and reset each edge weight w to w+k+1. Which of the following is true?
Kruskal's algorithm will identify the same spanning tree on the new graph as on the old graph.
Minimum cost spanning trees in the original graph will not correspond to minimum cost spanning
trees in the new graph.
Prim's algorithm will not work on the original graph but will work on the modified graph.
There are more minimum cost spanning trees in the modified graph than in the original graph.
2
points
21)Suppose we want to extend the union-find data structure to support the operation Reset(c), which takes
as input the name of a component c and then breaks up c into singleton components, like
MakeUnionFind(). For instance if c = 3 and c currently consists of {1,3,7}, then Reset(c) will produce
three components called 1, 3 and 7 consisting of {1}, {3} and {7}, respectively.
Which of the following is correct about the cost of adding Reset(c) to the array and pointer
implementations of union-find discussed in the lecture?
Array representation: O(n), Pointer representation: O(n)
Array representation: O(size(c)), Pointer representation: O(n)
Array representation: O(n), Pointer representation: O(size(c))
Array representation: O(size(c)), Pointer representation: O(size(c))
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points
22)Suppose we want to delete an arbitrary element from a max heap. (Assume we have auxiliary arrays
NodeToHeap[] and HeapToNode[] required for the update() operation.)
Consider the following strategies.
Strategy 1: Remove the element from the array, compress the array and reheapify.
Strategy 2: Update the value of this node to the current maximum value in the heap + 1, then
delete_max.
Strategy 1 takes time O(n log n) and Strategy 2 takes time O(n)
Strategy 1 takes time O(n) and Strategy 2 takes time O(n log n)
Strategy 1 takes time O(log n) and Strategy 2 takes time O(n)
Strategy 1 takes time O(n) and Strategy 2 takes time O(log n)
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points
23)Suppose we want to support the operations predecessor and successor in a heap. Given a value v in the
heap, pred(v) tells us the next smaller value currently in the heap and succ(v) tells us the next larger
value currently in the heap.
In both min heaps and max heaps, both operations take time O(n).
In both min heaps and max heaps, both operations take time O(log n).
In a min heap, pred(v) takes time O(log n) and succ(v) takes O(n) whereas in a max heap pred(v)
takes time O(n) and succ(v) takes O(log n).
In a min heap, pred(v) takes time O(n) and succ(v) takes O(log n) whereas in a max heap pred(v)
2
points
Week 5 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
file:///Users/madhavan/mirror/projects/NPTEL/algorithms-2015/DAA_allquizzes.html Page 6 of 9
takes time O(log n) and succ(v) takes O(n).
24)Suppose we do merge sort with a three-way split: divide the array into 3 equal parts, sort each part and
do a 3 way merge. What would the worst-case complexity of this version be?
O(n2)
O(n2 log3n)
O(n log2n)
O(n (log2n)2)
2
points
25)In the closest pair of points problem, we have assumed that no two points have the same x or y
coordinate. Which of the following steps would become more complicated to justify without this
assumption.
Constructing QY and RY from PY in time O(n) in the divide step.
Constructing QX and RX from PX in time O(n) in the divide step.
Constructing SY from QY and RY in time O(n) the combine step.
Arguing that every d/2 side square in the d-band around the separator can have at most one point.
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points
26)What is the complexity of inorder traversal for a binary search tree with n nodes?
O(log n) whether the tree is balanced or unbalanced.
O(log n) if the tree is balanced, O(n) otherwise.
O(n) whether the tree is balanced or unbalanced.
O(n) if the tree is balanced, O(n log n) otherwise.
2
points
27)Suppose we do not have a parent pointer in the nodes of a search tree, only left-child and right-child.
Which of the following operations can be computed in time O(log n) for a balanced search tree?
find, insert, delete, but not min, max, pred, succ
find, insert, delete, min, max, but not pred, succ
find, insert, delete, pred, succ but not min, max
All of find, insert, delete, min, max, pred, succ
2
points
28)Consider the following algorithm to build a balanced search tree from a sorted sequence.
Make the mid-point of the sequence the root of the tree
Recursively construct balanced search trees from elements to the left and right of the midpoint and make
these the left and right subtrees of the root.
What is the complexity of this procedure where the sorted sequence is a sorted array?
O(n)
O(n log n)
O(n2)
Depends on the contents of the original sequence
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points
29)Consider the following algorithm to build a balanced search tree from a sorted sequence.
Make the mid-point of the sequence the root of the tree
Recursively construct balanced search trees from elements to the left and right of the midpoint and make
these the left and right subtrees of the root.
2
points
Week 6 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
file:///Users/madhavan/mirror/projects/NPTEL/algorithms-2015/DAA_allquizzes.html Page 7 of 9
What is the complexity of this procedure where the sorted sequence is a sorted list?
O(n)
O(n log n)
O(n2)
Depends on the contents of the original sequence.
30)Which of the following is not a greedy algorithm?
Dijkstra's algorithm for single source shortest paths
Bellman-Ford algorithm for single source shortest paths
Prim's algorithm for minimum cost spanning tree
Kruskal's algorithm for minimum cost spanning tree
2
points
31)We want to write down a recursive expression for Profit[i].
Profit[N] = 0
For 1 = i < N, Profit[i] = ??
max(Profit[i+1], Profit[i+2] + Price[i+1] - Price[i] - 2)
max(Profit[i+2] + Price[i+1] - Price[i] - 2, Profit[i+3] + Price[i+2] - Price[i] - 2, ... Profit[N] + Price[n-1] -
Price[i] - 2, Price[n] - Price[i] - 2)
max(Profit[i+1], Profit[i+2] + Price[i+1] - Price[i] - 2, Profit[i+3] + Price[i+2] - Price[i] - 2, ... Profit[N] +
Price[n-1] - Price[i] - 2)
max(Profit[i+1], Profit[i+2] + Price[i+1] - Price[i] - 2, Profit[i+3] + Price[i+2] - Price[i] - 2, ... Profit[N] +
Price[n-1] - Price[i] - 2, Price[n] - Price[i] - 2)
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points
32)What is the size of the memo table for this problem?
N
N-1
N+1
N2
2
points
33)What is a good order to compute the values of Profit[i] using dynamic programming?
From Profit[1] to Profit[N]
From Profit[N] to Profit[1]
Either from Profit[1] to Profit[N] or from Profit[N] to Profit[1]
None of these
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points
34)How much time will dynamic programming take to compute Profit[1]?
O(N)
O(N log N)
O(N2)
O(N2 log N)
2
points
35)What is the value of Profit[1] in the example given in the problem statement?
6
7
8
2
points
Week 7 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
file:///Users/madhavan/mirror/projects/NPTEL/algorithms-2015/DAA_allquizzes.html Page 8 of 9
9
36)Which of the following is not a linear constraint?
8x = 3y + 25
5x + 6xy = 5
9x = 7y + 8z
5y + 3x = 33
2
points
37)When we model a graph problem using LP, mapping each path to a variable is not a good strategy
because:
Paths can be hard to compute.
We have to be careful to avoid cycles.
A graph has exponentially many paths.
Edges may be directed.
2
points
38)Suppose we compute the maximum s-t flow F in a network. Then, which of the following is true of s-t
cuts?
From F, we can identify a minimum s-t cut in polynomial time.
From F, we can identify all minimum s-t cuts in polynomial time.
From F, we know the capacity of the minimum s-t cut, but identifying such a cut can take exponential
time.
F gives a lower bound for the capacity of the minimum s-t cut but not the exact capacity.
2
points
39)We have an exponential time algorithm for problem A, and problem A reduces in polynomial time to
problem B. From this we can conclude that:
B has an exponential time algorithm.
B cannot have a polynomial time algorithm.
A cannot have a polynomial time algorithm.
None of the above.
2
points
40)Suppose SAT reduces to a problem C. To claim that C is NP-complete, we additionally need to show
that:
There is a checking algorithm for C.
Every instance of C maps to an instance of SAT.
Every instance of SAT maps to an instance of C.
C does not have an efficient algorithm.
2
points
Week 8 Quiz
Design and Analysis of Algorithms - - Assessment 16/04/15 7:01 am
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