Complexity Theory: Master the Fundamentals of Computational Complexity

Question 1

Which of the following measures the number of steps an algorithm takes to complete based on its input size?

Accepted Answer

Time complexity

Answer

Space complexity

Answer

NP-completeness

Answer

Reducibility

Question 2

Which complexity class encompasses problems solvable in polynomial time by non-deterministic Turing machines?

Accepted Answer

NP

Answer

P

Answer

NP-complete

Answer

NP-hard

Question 3

Which technique simplifies the analysis of complex algorithms by reducing them to simpler, well-understood problems?

Accepted Answer

Reducibility

Answer

Asymptotic analysis

Answer

Divide-and-conquer

Answer

Dynamic programming

Question 4

Which of the following statements accurately describes an NP-complete problem?

Accepted Answer

All other NP problems can be reduced to it.

Answer

It possesses an exponential-time solution.

Answer

It remains challenging to solve even for small inputs.

Answer

It represents the most difficult problem within NP.

Question 5

Which algorithm efficiently solves the knapsack problem?

Accepted Answer

Dynamic programming

Answer

Greedy algorithm

Answer

Divide-and-conquer

Answer

Backtracking

Question 6

What is the time complexity of the bubble sort algorithm?

Accepted Answer

O(n^2)

Answer

O(n)

Answer

O(n log n)

Answer

O(n^3)

Question 7

Which data structure balances space efficiency with the ability to insert and delete elements efficiently?

Accepted Answer

Linked list

Answer

Array

Answer

Stack

Answer

Queue

Question 8

Which technique optimizes recursive algorithms by avoiding redundant computations?

Accepted Answer

Memoization

Answer

Branch and bound

Answer

Divide-and-conquer

Answer

Iteration

Question 9

Which problem is considered NP-hard?

Accepted Answer

Traveling Salesman Problem

Answer

Minimum Spanning Tree

Answer

Dijkstra's Shortest Path

Answer

Topological Sort

Question 10

Define time complexity and explain its significance in evaluating algorithm efficiency.

Accepted Answer

Time complexity measures the number of basic operations an algorithm performs as a function of the input size.

Answer

Time complexity is a measure of the memory space occupied by an algorithm during execution.

Answer

Time complexity is independent of the input size and always constant for any algorithm.

Question 11

Which of the following is NOT a measure of time complexity?

Accepted Answer

Zeta notation

Answer

Omega notation

Answer

Little-o notation

Answer

Theta notation

Question 12

Consider an algorithm with the following time complexity: f(n) = n^3 + 2n^2 + 5n + 100. Using the big-O notation, what is the time complexity of this algorithm?

Accepted Answer

O(n^3)

Answer

O(n)

Answer

O(n^2)

Answer

O(n^4)

Question 13

An algorithm has a time complexity of O(n log n). What does this indicate?

Accepted Answer

The worst-case time requirement for the algorithm grows proportionally to n log n.

Answer

The best-case time requirement for the algorithm grows proportionally to n log n.

Answer

The average-case time requirement for the algorithm grows proportionally to n log n.

Answer

None of the above statements are true.

Question 14

Which characteristic is NOT associated with NP-complete problems?

Accepted Answer

They can be solved in polynomial time on a deterministic Turing machine.

Answer

They are among the most difficult problems in NP.

Answer

Any other NP problem can be reduced to them in polynomial time.

Answer

They play a significant role in approximation algorithm theory.

Question 15

Which technique is commonly used to demonstrate that a problem is NP-complete?

Accepted Answer

Reduction from a known NP-complete problem

Answer

Solving the problem using a polynomial-time algorithm

Answer

Applying brute-force search to find a solution

Answer

Analyzing the problem's structure and complexity

Question 16

An algorithm's space complexity is O(n^2). What does this imply?

Accepted Answer

The algorithm requires up to n^2 units of memory space in the worst case.

Answer

The algorithm requires exactly n^2 units of memory space in the best case.

Answer

The algorithm requires an average of n^2 units of memory space.

Answer

The algorithm requires n^2 units of memory space only if n is a prime number.

Question 17

Which complexity class is widely used for problems solvable in linear time?

Accepted Answer

P

Answer

NP

Answer

EXP

Answer

NEXP

Question 18

Is the halting problem decidable?

Accepted Answer

No

Answer

Yes

Question 19

How does time complexity generally relate to algorithm efficiency?

Accepted Answer

Lower time complexity usually implies improved algorithm efficiency.

Answer

Higher time complexity usually implies improved algorithm efficiency.

Answer

Time complexity is unrelated to algorithm efficiency.

Answer

The relationship between time complexity and algorithm efficiency varies depending on the problem being solved.

Question 20

How does Complexity Theory influence software development decisions?

Accepted Answer

Complexity Theory helps developers choose and optimize algorithms, considering computational limits.

Answer

Complexity Theory is primarily theoretical and has no practical value in software development.

Answer

Complexity Theory ensures efficient algorithms exist for all problems.

Answer

Understanding Complexity Theory is crucial for all areas of computer science.

Question 21

Which space complexity measure accounts for the total memory utilized by an algorithm during its execution?

Accepted Answer

Total Space

Answer

Auxiliary Space

Answer

Input Space

Answer

Output Space

Question 22

How do non-deterministic Turing machines contribute to defining the NP complexity class?

Accepted Answer

NP is the set of decision problems for which a non-deterministic Turing machine can quickly verify a solution.

Answer

Non-deterministic Turing machines are used to define the complexity class of problems solvable in polynomial time.

Answer

The halting problem for non-deterministic Turing machines is undecidable.

Answer

Non-deterministic Turing machines are only used to define the complexity class P.

Question 23

Which of the following is NOT a measure of time complexity used to describe the efficiency of an algorithm?

Accepted Answer

Number of iterations

Answer

Big O notation

Answer

Big Omega notation

Answer

Big Theta notation

Question 24

Which of the following problems is known to be polynomial-time reducible to all other NP problems?

Accepted Answer

Boolean Satisfiability Problem

Answer

Traveling Salesman Problem

Answer

Integer Factorization Problem

Answer

Knapsack Problem

Question 25

Which of the following is a defining characteristic of NP-complete problems?

Accepted Answer

They can be reduced to any other problem in NP in polynomial time.

Answer

They have efficient (polynomial-time) solutions.

Answer

They require exponential time to solve.

Answer

They are relatively easy to approximate.

Question 26

Which of the following algorithms has an optimal worst-case time complexity for sorting an array of n elements?

Accepted Answer

Merge Sort

Answer

Bubble Sort

Answer

Insertion Sort

Answer

Selection Sort

Question 27

Which of the following problems is known to be PSPACE-complete?

Accepted Answer

Solving a Sudoku puzzle.

Answer

Determining if a graph is connected.

Answer

Finding the shortest path in a graph.

Answer

Determining if a string contains a palindrome.

Question 28

Which of the following is a lower bound on the time complexity of any algorithm solving a given problem?

Accepted Answer

Big Omega notation

Answer

Big O notation

Answer

Big Theta notation

Answer

The number of instructions executed by the algorithm

Question 29

Which of the following complexity classes represents decision problems that can be solved in polynomial time by a deterministic Turing machine?

Accepted Answer

P

Answer

NP

Answer

NP-hard

Answer

NP-complete

Question 30

The time complexity of a nondeterministic Turing machine is typically expressed using the notation:

Accepted Answer

O(2^n)

Answer

O(n^k)

Answer

O(log n)

Answer

O(n log n)

Question 31

Which of these options provides an example of a space complexity measure?

Accepted Answer

Number of memory cells used by a Turing machine

Answer

Time taken (in milliseconds) to solve a problem

Answer

Total instructions executed by a Turing machine

Answer

Size of the input to a problem

Question 32

To demonstrate that a problem is NP-hard, which technique is commonly employed?

Accepted Answer

Polynomial-time reduction

Answer

Direct mathematical proof

Answer

Exhaustive search

Answer

Approximation algorithms

Question 33

The halting problem for Turing machines falls under which category of problems?

Accepted Answer

Undecidable

Answer

NP-complete

Answer

P-complete

Answer

Recursively enumerable

Question 34

For an algorithm performing a binary search on a sorted array of size n, its time complexity can be expressed as:

Accepted Answer

O(log n)

Answer

O(n)

Answer

O(n^2)

Answer

O(n log n)

Question 35

Which of the following measures describes the lower bound of the time required by an algorithm?

Accepted Answer

Omega notation

Answer

Big Omega notation

Answer

Big O notation

Answer

Theta notation

Question 36

Which property distinguishes NP-complete problems from other NP problems?

Accepted Answer

They can be reduced to every other NP problem in polynomial time

Answer

They are all in the class P

Answer

They can be solved in polynomial time

Answer

They are all decision problems

Question 37

Which method is most commonly used to prove that a problem is NP-complete?

Accepted Answer

Polynomial-time reduction

Answer

Brute force search

Answer

Random guessing

Answer

Divide and conquer

Question 38

Which of the following problems is not considered NP-complete?

Accepted Answer

Sorting

Answer

Hamiltonian cycle

Answer

Traveling salesman

Answer

Satisfiability

Question 39

Which type of reduction is essential in Complexity Theory?

Accepted Answer

Polynomial-time reduction

Answer

Linear-time reduction

Answer

Constant-time reduction

Answer

Logarithmic-time reduction

Question 40

Which property describes a key characteristic of space complexity?

Accepted Answer

It can be measured in terms of bits or cells

Answer

It is not affected by the input size

Answer

It is always independent of time complexity

Answer

It is always greater than time complexity

Question 41

Which technique is commonly employed to reduce space complexity in algorithms?

Accepted Answer

Dynamic programming

Answer

Divide and conquer

Answer

Backtracking

Answer

Recursion

Question 42

Which of the following is a limitation of Complexity Theory?

Accepted Answer

It does not provide precise bounds on the computational resources required by algorithms.

Answer

It is not applicable to analyzing real-world problems

Answer

It only pertains to deterministic algorithms

Question 43

Which area exemplifies a significant application of Complexity Theory?

Accepted Answer

The design and analysis of efficient algorithms and data structures

Answer

Predicting the execution time of programs with certainty

Answer

Analyzing the complexity of natural phenomena

Answer

Developing artificial intelligence systems

Question 44

Which complexity class contains all problems that can be solved in polynomial time?

Accepted Answer

P

Answer

NP

Answer

NP-complete

Question 45

Which of the following is known to be an NP-complete problem?

Accepted Answer

Traveling Salesman Problem

Answer

Sorting an array

Answer

Checking if a number is prime

Answer

Finding the shortest path in a graph

Question 46

What is the primary purpose of using the concept of reducibility in Complexity Theory?

Accepted Answer

To group and classify problems based on their inherent computational difficulty

Answer

To design algorithms that can solve NP-complete problems efficiently

Answer

To optimize the performance of algorithms

Answer

To guarantee the correctness of algorithms

Question 47

Which measure is used to quantify the amount of memory occupied by an algorithm?

Accepted Answer

Space complexity

Answer

Output size

Answer

Time complexity

Answer

Input size

Question 48

Is it true that all NP-complete problems can be solved in polynomial time?

Accepted Answer

False

Answer

True

Answer

Possibly

Answer

Cannot be determined

Question 49

Which is an example of a heuristic algorithm?

Accepted Answer

A* search

Answer

Merge sort

Answer

Dijkstra's algorithm

Answer

Binary search

Question 50

What is the fundamental objective of Complexity Theory?

Accepted Answer

To understand the inherent difficulty of computational problems

Answer

To create new computer architectures

Answer

To develop efficient algorithms

Question 51

Can Complexity Theory provide guarantees that all algorithms can be designed to be efficient for all possible inputs?

Accepted Answer

False

Answer

True

Answer

Possibly