Computational Complexity: A Comprehensive Guide for Beginners

Question 1

What is a key characteristic that distinguishes NP-complete problems from other problems in the NP complexity class?

Accepted Answer

NP-complete problems can be reduced to any other NP-complete problem in polynomial time.

Answer

NP-complete problems are those that can be solved in polynomial time on a deterministic Turing machine.

Answer

NP-complete problems are equivalent to NP-hard problems.

Answer

NP-complete problems are the easiest problems in NP.

Question 2

Which of the following is not a formally defined complexity class in computational complexity theory?

Accepted Answer

undecidable

Answer

P

Answer

NP

Answer

EXPTIME

Question 3

Which of the following accurately describes the relationship between the complexity classes P and NP?

Accepted Answer

P is a subset of NP.

Answer

NP is a subset of P.

Answer

P and NP are disjoint sets.

Answer

P and NP are equivalent.

Question 4

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

Accepted Answer

They can be reduced to any other NP-complete problem.

Accepted Answer

They have no known polynomial-time algorithms.

Answer

They can be solved in polynomial time.

Answer

They are the hardest problems in NP.

Question 5

Which characteristic is a defining feature of NP-complete problems?

Accepted Answer

NP-complete problems represent the most challenging problems within the NP complexity class.

Answer

NP-complete problems can be solved efficiently using brute-force approaches.

Answer

NP-complete problems have a polynomial-time relationship between problem size and solution time.

Answer

NP-complete problems can be solved efficiently on quantum computers.

Question 6

Which of the following is true about NP-hard problems?

Accepted Answer

All NP-complete problems are also NP-hard.

Answer

They can always be solved in polynomial time.

Answer

They always have polynomial-time approximation algorithms.

Answer

They measure the time complexity of a problem.

Question 7

What is the time complexity of an algorithm that sequentially scans an array of size n to find a specific element?

Accepted Answer

O(n)

Answer

O(1)

Answer

O(n^2)

Answer

O(log n)

Question 8

Which of these is NOT a recognized complexity class in computational theory?

Accepted Answer

Z

Answer

P

Answer

NP

Answer

NP-complete

Question 9

Determine the time complexity of the Python code below, which calculates the sum of numbers from 1 to n:

```
def sum_to_n(n):
  sum = 0
  for i in range(1, n + 1):
    sum += i
  return sum
```

Accepted Answer

O(n)

Answer

O(1)

Answer

O(log n)

Answer

O(n^2)

Question 10

Which complexity class includes problems solvable using algorithms that require polynomial space?

Accepted Answer

PSPACE

Answer

P

Answer

NP

Answer

NP-complete

Question 11

What is the defining characteristic that sets NP-complete problems apart from other problem classes?

Accepted Answer

NP-complete problems can be verified efficiently if a solution is provided.

Answer

NP-complete problems can be solved efficiently for any input size.

Answer

NP-complete problems always have algorithms that run in linear time.

Answer

NP-complete problems are known to have efficient algorithms for large inputs.

Question 12

Which characteristic defines NP-complete problems in computational complexity?

Accepted Answer

They serve as a basis for solving other NP problems through polynomial-time reductions.

Answer

They can be solved efficiently in polynomial time.

Answer

They encompass all NP problems.

Answer

They have no practical applications in real-world scenarios.

Question 13

In computational complexity analysis, how are NP-complete and NP-hard problems related?

Accepted Answer

All NP-complete problems are NP-hard, but not all NP-hard problems are NP-complete.

Answer

NP-hard problems are a subset of NP-complete problems.

Answer

NP-complete problems are unrelated to NP-hard problems.

Answer

Every NP-hard problem can be solved in polynomial time if and only if every NP-complete problem can be solved in polynomial time.

Question 14

Which of the following is not a complexity class?

Accepted Answer

O(n^2)

Answer

P

Answer

NP

Answer

NP-complete

Question 15

An algorithm's execution time is O(n²) for an input of size n. Which computational complexity class does this algorithm belong to?

Accepted Answer

Quadratic-time complexity

Answer

Constant-time complexity

Answer

Linear-time complexity

Answer

Exponential-time complexity

Question 16

Explain the relationship between NP-completeness and NP-hardness.

Accepted Answer

NP-hard problems are always NP-complete, but the reverse is not always true.

Answer

NP-complete problems are always solvable in polynomial time.

Answer

NP-hard problems are never NP-complete, and NP-complete problems are never solvable in polynomial time.

Answer

NP-completeness and NP-hardness are unrelated concepts.

Question 17

In computational complexity, which complexity class represents problems solvable by a deterministic Turing machine in polynomial time?

Accepted Answer

P

Answer

NP

Answer

EXP

Answer

NEXP

Question 18

Determine the time complexity of the insertion sort algorithm. Which statement accurately describes its complexity?

Accepted Answer

O(n) in the best case and O(n^2) in the worst case

Answer

O(1) for all inputs

Answer

O(n log n) regardless of the input

Answer

O(n^2) for all inputs

Question 19

What is a defining characteristic of an NP-hard problem?

Accepted Answer

It can be reduced from any NP-complete problem in polynomial time.

Answer

It can be solved efficiently in polynomial time.

Answer

It has an exponential time complexity.

Answer

It can only be solved by non-deterministic algorithms.

Question 20

An algorithm's running time can grow exponentially with input size, reaching up to 2ⁿ steps for an input of size n. Which complexity class describes this algorithm's behavior?

Accepted Answer

O(2<sup>n</sup>)

Answer

O(n)

Answer

O(n<sup>2</sup>)

Answer

O(n!)

Question 21

Which type of Turing machine can solve decision problems in the complexity class NP in polynomial time?

Accepted Answer

Non-deterministic

Answer

Deterministic

Answer

Quantum

Answer

Parallel

Question 22

Which of the following statements accurately describes the Theta notation, Θ(f(n))?

Accepted Answer

Θ(f(n)) represents functions that grow at the same asymptotic rate as f(n), both in the upper and lower bounds.

Answer

Θ(f(n)) represents functions that grow asymptotically faster than f(n), but not exponentially faster.

Answer

Θ(f(n)) represents functions that grow asymptotically slower than f(n), but not exponentially slower.

Answer

Θ(f(n)) represents functions that grow asymptotically at a rate significantly different from f(n).

Question 23

Two algorithms, A and B, have time complexities of O(2ⁿ) and O(3ⁿ), respectively. Which statement accurately describes the relationship between their running times?

Accepted Answer

The running times of Algorithms A and B are asymptotically equivalent.

Answer

Algorithm A runs faster than Algorithm B for all input sizes.

Answer

Algorithm B runs faster than Algorithm A for all input sizes.

Answer

The relationship between the running times of Algorithms A and B depends on the specific input size.

Question 24

An algorithm with exponential time complexity exhibits which of the following runtime behaviors?

Accepted Answer

Its running time increases exponentially with the input size.

Answer

Its running time increases steadily with the input size.

Answer

Its running time remains constant regardless of the input size.

Question 25

Asymptotic analysis estimates a function's behavior as its input size approaches infinity. Which of the following is NOT a commonly used asymptotic notation type to describe a function's growth rate?

Accepted Answer

Delta Notation

Answer

Big O Notation

Answer

Theta Notation

Answer

Omega Notation

Question 26

Which of the following best describes an NP-hard problem?

Accepted Answer

Can be reduced from a known NP-complete problem

Answer

Can be solved in polynomial time

Answer

Has a fixed time complexity of O(n^2)

Question 27

Which of the following problems is NP-complete?

Accepted Answer

Finding the shortest path in a graph

Accepted Answer

Finding the minimum spanning tree of a graph

Answer

Sorting a list of numbers

Answer

Finding the maximum value in a list of numbers

Question 28

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

Accepted Answer

NP-hard problems can be reduced to any other NP-hard problem in polynomial time.

Answer

NP-hard problems can be solved efficiently in polynomial time on a deterministic Turing machine.

Answer

All NP-hard problems are also NP-complete problems.

Answer

NP-hard problems can be solved efficiently using quantum computing.

Question 29

Regarding the P versus NP problem, which statement is accurate?

Accepted Answer

If P equals NP, all NP-complete problems can be solved in polynomial time.

Answer

P and NP represent the same complexity class.

Answer

NP-complete problems can be solved in polynomial time.

Answer

The P versus NP problem has been solved.

Question 30

Which characteristic is unique to NP-complete problems?

Accepted Answer

They can be transformed into each other in polynomial time.

Answer

They can be solved efficiently using known algorithms.

Question 31

Which complexity class includes decision problems solvable in polynomial time by a deterministic Turing machine?

Accepted Answer

P

Answer

NP

Answer

NP-complete

Answer

NP-hard

Question 32

In parameterized complexity, a problem is considered fixed-parameter tractable if its running time is:

Accepted Answer

Bounded by a polynomial function of the input size and a constant function of the parameter

Answer

Bounded by a polynomial function of the input size only

Answer

Bounded by an exponential function of the input size

Question 33

Which historical event marked a pivotal moment in the development of computational complexity theory?

Accepted Answer

Alan Turing's formulation of the Turing machine

Answer

The invention of the first mechanical computer

Answer

The introduction of microprocessors

Answer

The advent of quantum computing

Question 34

What is a primary benefit of using approximation algorithms?

Accepted Answer

Approximation algorithms efficiently provide a close approximation of the optimal solution for complex problems.

Answer

Approximation algorithms guarantee the optimal solution for any problem.

Answer

Approximation algorithms can solve NP-hard problems exactly.

Answer

Approximation algorithms make time complexity irrelevant.

Question 35

Which complexity class specifically analyzes the memory usage of an algorithm during its execution?

Accepted Answer

SPACE

Answer

P

Answer

NP

Question 36

When demonstrating the NP-hardness of a decision problem X, which proof technique is commonly employed?

Accepted Answer

Indirect proof (proof by contradiction)

Answer

Direct proof

Answer

Proof by cases

Answer

Proof by exhaustion

Question 37

How do an algorithm's time complexity and space complexity relate?

Accepted Answer

They can be independent of each other.

Answer

They are always directly proportional.

Answer

They are always inversely proportional.

Question 38

Identify the option that does not represent a valid complexity class.

Accepted Answer

EXPONENTIAL

Answer

P

Answer

NP

Answer

EXPTIME

Question 39

Which method is most appropriate for analyzing the time complexity of an algorithm?

Accepted Answer

Establishing the asymptotic upper bound for the number of operations

Answer

Counting the number of code lines

Answer

Measuring execution time on a particular machine

Answer

Using a profiler to monitor the time consumption of each algorithm component

Question 40

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

Accepted Answer

Traveling salesman problem

Answer

Sorting a list of numbers

Answer

Finding the shortest path in a graph

Answer

Checking if a graph is connected

Question 41

Which of the following is not classified as a complexity class?

Accepted Answer

UNDECIDABLE

Answer

P

Answer

NP

Answer

EXPTIME

Question 42

Determine the time complexity of an algorithm that employs binary search on a sorted array of size n.

Accepted Answer

O(log n)

Answer

O(n)

Answer

O(n^2)

Answer

O(1)

Question 43

Which problem is known to be NP-complete?

Accepted Answer

Traveling Salesman Problem

Answer

Sorting a list of numbers

Answer

Finding the shortest path in a graph

Answer

Checking if a graph is connected

Question 44

Define the Halting Problem.

Accepted Answer

Determining whether a given program will halt or run indefinitely.

Answer

Finding the most efficient algorithm for a task.

Answer

Determining the complexity of a given algorithm.

Answer

Proving the correctness of a given program.

Question 45

What is the purpose of Big-O notation?

Accepted Answer

Express the upper bound on the worst-case time complexity of an algorithm.

Answer

Express the average-case time complexity of an algorithm.

Answer

Express the space complexity of an algorithm.

Answer

Express the efficiency of an algorithm in terms of input size.

Question 46

Define a non-deterministic Turing machine.

Accepted Answer

A Turing machine capable of making random choices during computation.

Answer

A Turing machine restricted to moving its tape head in one direction.

Answer

A Turing machine that can only write one symbol to its tape.

Answer

A Turing machine that can only read one symbol from its tape.

Question 47

Differentiate between P and NP problems. Which can be solved in polynomial time?

Accepted Answer

P problems

Answer

NP problems

Question 48

Python: Which of the following languages possesses Turing-completeness?

Accepted Answer

Python

Answer

Java

Answer

C++

Answer

Assembly

Question 49

Church-Turing Thesis claims:

Accepted Answer

Any computation executable by a physical computer is also executable by a Turing machine.

Answer

Any problem solvable by a Turing machine is also solvable by a physical computer.

Answer

Any problem solvable by a physical computer is also solvable by a quantum computer.

Answer

Any problem solvable by a quantum computer is also solvable by a Turing machine.

Question 50

In practice, how do heuristics contribute to solving NP-complete problems? Discuss the benefits and drawbacks of using heuristic algorithms.

Accepted Answer

Heuristics offer approximate solutions when exact solutions are impractical due to time or resource limitations.

Answer

Heuristics guarantee optimal solutions for NP-complete problems.

Answer

Heuristic algorithms outperform exact algorithms for NP-complete problems in all cases.

Answer

Heuristics should always be used over exact algorithms for NP-complete problems.

Question 51

What is the significance of NP-completeness in algorithm design? Discuss its implications for developing and using algorithms for NP-complete problems.

Accepted Answer

NP-completeness suggests finding an efficient algorithm for an NP-complete problem is unlikely.

Answer

NP-completeness guarantees that brute-force search can solve the problem efficiently.

Answer

Developing algorithms for NP-complete problems is always impossible.

Answer

NP-completeness has no significant implications for algorithm design.

Question 52

Which of the given options is not a complexity class?

Accepted Answer

NP-complete

Answer

P

Answer

NP

Answer

EXPTIME

Question 53

Analyze the time complexity of the following algorithm: for i = 1 to n: for j = 1 to n: print(i, j)

Accepted Answer

O(n^2)

Answer

O(n)

Answer

O(n^3)

Answer

O(2^n)

Question 54

Which problem exemplifies NP-completeness?

Accepted Answer

Solving the knapsack problem

Answer

Finding the shortest path in a graph

Answer

Determining if a graph is bipartite

Answer

Sorting a list of numbers

Question 55

Indicate the fundamental limitation of Turing machines:

Accepted Answer

They can only address problems representable as algorithms.

Answer

They require an unbounded amount of memory to function.

Answer

They lack the ability to perform random access memory operations.

Answer

They are excessively slow for practical use.

Question 56

Identify an example of a problem that is computationally intractable:

Accepted Answer

Solving the 3-SAT problem

Answer

Finding the minimum spanning tree of a graph

Answer

Determining if a given number is prime

Answer

Finding the roots of a polynomial equation

Question 57

Define the relationship between P and NP:

Accepted Answer

The relationship between P and NP is undetermined.

Answer

P is a subset of NP.

Answer

NP is a subset of P.

Answer

P and NP are disjoint sets.

Question 58

What is the fundamental significance of the halting problem?

Accepted Answer

It showcases the inherent limitations of computation.

Answer

It allows us to definitively determine whether a given algorithm will terminate.

Answer

It provides a comprehensive solution to all computational problems.

Answer

It has no practical relevance or application.

Question 59

Which characteristic is not desirable in an algorithm?

Accepted Answer

Excessive complexity

Answer

Efficiency

Answer

Accuracy

Answer

Generality

Question 60

Describe the breakthrough achieved by the Cook-Levin theorem in the area of computational complexity theory.

Accepted Answer

It introduces the concept of NP-complete problems and demonstrates that SAT is the first known NP-complete problem.

Answer

It proves the equivalence of P and NP complexity classes.

Answer

It presents an algorithm for solving all NP-complete problems in polynomial time.

Answer

It provides a comprehensive inventory of all NP-hard problems.

Question 61

Which complexity class includes all languages accepted by deterministic Turing machines within exponential time?

Accepted Answer

EXPTIME

Answer

P

Answer

NP

Answer

RE

Question 62

Determine the time complexity of the following algorithm: for i from 1 to n, for j from 1 to n, print(i+j).

Accepted Answer

O(n^2)

Answer

O(n)

Answer

O(n^3)

Answer

O(1)

Question 63

What is the significance of the halting problem in the context of computational complexity?

Accepted Answer

It illustrates the limits of computation

Answer

It provides a framework for solving any problem

Answer

It is used to analyze the performance of algorithms

Answer

It is unrelated to the study of complexity

Question 64

Which technique is commonly employed to analyze the worst-case time complexity of an algorithm?

Accepted Answer

Asymptotic analysis

Answer

Amortized analysis

Answer

Input size analysis

Answer

Probabilistic analysis