Backtracking Algorithms MCQs

Backtracking is mainly used for solving:

A. Greedy problems B. Optimization problems C. Constraint satisfaction problems D. Numerical computation problems

Correct Answer: C. Constraint satisfaction problems

Explanation:
Backtracking is particularly useful for solving constraint satisfaction problems where we need to find a solution that satisfies all given constraints. It systematically searches for a solution by trying different possibilities and backtracking when constraints are violated.

Which of the following is a characteristic of backtracking algorithms?

A. Solve only linear problems B. Explore all possibilities and backtrack when a solution path fails C. Selects the best option at each step D. Always uses recursion

Correct Answer: B. Explore all possibilities and backtrack when a solution path fails

Explanation:
Backtracking algorithms work by incrementally building candidates to solutions and abandoning a candidate ("backtracking") as soon as it determines that the candidate cannot possibly lead to a valid solution. This is different from greedy algorithms which make locally optimal choices at each step.

Backtracking is often implemented using:

A. Stack B. Queue C. Tree D. Heap

Correct Answer: A. Stack

Explanation:
Backtracking is typically implemented using recursion, which inherently uses the call stack to keep track of the state at each decision point. When a path doesn't lead to a solution, the algorithm "backtracks" by popping the stack to return to a previous state.

The n-Queens problem requires placing queens on an n×n chessboard such that:

A. No two queens are on the same column B. No two queens are on the same row or diagonal C. Queens are placed in alternate rows only D. All of the above

Correct Answer: B. No two queens are on the same row or diagonal

Explanation:
The n-Queens problem requires that no two queens threaten each other, which means they cannot share the same row, column, or diagonal. The standard solution already ensures queens are in different columns by placing one queen per column, so the main constraints are rows and diagonals.

What is the base case in the recursive solution of the n-Queens problem?

A. No queen placed B. One queen placed C. All queens placed successfully D. The first queen is on (0,0)

Correct Answer: C. All queens placed successfully

Explanation:
The base case occurs when all n queens have been placed on the chessboard without threatening each other. This means we've found a valid solution to the problem and can record or print this configuration.

The number of solutions for 8-Queens problem is:

A. 8 B. 92 C. 64 D. 24

Correct Answer: B. 92

Explanation:
The 8-Queens problem has 92 distinct solutions. These include fundamental solutions and their rotations and reflections. There are 12 unique fundamental solutions, with the remaining 80 being symmetrical variants of these.

Which data structure is commonly used to represent a solution in n-Queens problem?

A. Queue B. Array C. HashMap D. Linked List

Correct Answer: B. Array

Explanation:
An array is typically used to represent the positions of queens in the n-Queens problem. A common approach is to use a 1D array where the index represents the column and the value at that index represents the row where the queen is placed.

The Hamiltonian Circuit problem involves:

A. Visiting all vertices without repeating any edge B. Visiting all vertices exactly once and returning to the start C. Finding shortest path between two nodes D. Covering all edges in minimum time

Correct Answer: B. Visiting all vertices exactly once and returning to the start

Explanation:
A Hamiltonian Circuit is a cycle in a graph that visits each vertex exactly once and returns to the starting vertex. This is different from an Eulerian circuit which visits every edge exactly once.

The Hamiltonian Circuit problem is:

A. NP B. NP-Complete C. NP-Hard D. P

Correct Answer: B. NP-Complete

Explanation:
The Hamiltonian Circuit problem is a classic NP-Complete problem. This means that: 1) It's in NP (solutions can be verified quickly) 2) It's as hard as the hardest problems in NP (any NP problem can be reduced to it)

In backtracking for the Hamiltonian Circuit problem, pruning occurs when:

A. A vertex is repeated B. A dead end is reached C. The edge does not exist D. All of the above

Correct Answer: D. All of the above

Explanation:
In backtracking for Hamiltonian Circuit, we prune the search tree when: 1) A vertex is repeated (violates the "exactly once" condition) 2) We reach a dead end (no unvisited vertices left but haven't completed the cycle) 3) The required edge doesn't exist in the graph This pruning makes the algorithm more efficient than brute-force search.

Which of the following problems is similar to Hamiltonian Circuit?

A. Shortest Path Problem B. Minimum Spanning Tree C. Traveling Salesman Problem D. Assignment Problem

Correct Answer: C. Traveling Salesman Problem

Explanation:
The Traveling Salesman Problem (TSP) is very similar to the Hamiltonian Circuit problem. In fact, TSP can be reduced to the Hamiltonian Circuit problem - if we can solve Hamiltonian Circuit, we can solve TSP by checking for a Hamiltonian cycle with total weight less than some value.

Subset-Sum Problem is an example of:

A. Greedy Algorithm B. Divide and Conquer C. NP-Complete Problem D. Dynamic Programming only

Correct Answer: C. NP-Complete Problem

Explanation:
The Subset-Sum problem is a classic NP-Complete problem. It has two key properties: 1) It's in NP (a proposed solution can be verified quickly) 2) It's NP-Hard (other NP problems can be reduced to it) While it can be solved with dynamic programming for small inputs, it remains intractable for large inputs.

In Subset-Sum, the goal is to:

A. Maximize the sum of subsets B. Find a subset whose sum is equal to a given number C. Sort elements in a subset D. Find all subsets

Correct Answer: B. Find a subset whose sum is equal to a given number

Explanation:
The Subset-Sum problem specifically looks for a subset of numbers that adds up exactly to a target sum. For example, given the set {3, 34, 4, 12, 5, 2} and target sum 9, the solution would be {4, 5}. The problem doesn't require finding all subsets or maximizing the sum.

Which of the following strategies can solve the Subset-Sum problem?

A. Backtracking B. Dynamic Programming C. Brute Force D. All of the above

Correct Answer: D. All of the above

Explanation:
The Subset-Sum problem can be approached with multiple strategies: 1) Backtracking - systematically explores possibilities and backtracks when sum exceeds target 2) Dynamic Programming - builds a solution table for smaller subproblems (pseudo-polynomial time) 3) Brute Force - checks all possible subsets (exponential time) Each has different time/space tradeoffs.

In the backtracking approach to the Subset-Sum problem, the problem tree:

A. Contains only sorted elements B. Represents inclusion and exclusion of each element C. Only considers positive numbers D. Uses the greedy method

Correct Answer: B. Represents inclusion and exclusion of each element

Explanation:
The backtracking approach builds a binary tree where: - Each level represents a decision point for one element - Left branches represent including the element in the subset - Right branches represent excluding the element The tree is pruned when the current sum exceeds the target, making it more efficient than brute force.

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