Dynamic Programming: Interview-Level Concepts Quiz Quiz

Explore dynamic programming concepts often encountered in coding interviews, including optimal substructure, overlapping subproblems, and common application scenarios. This quiz is designed to help candidates identify key characteristics and pitfalls in dynamic programming problems.

1. Identifying Optimal Substructure

   Which property best describes a problem suitable for dynamic programming, such as finding the shortest path in a weighted graph?

   1. The problem has exponential time complexity only
   2. The problem has optimal substructure
   3. The problem relies on brute-force enumeration
   4. The problem involves randomization

   Explanation: Optimal substructure means an optimal solution can be formed from optimal solutions to its subproblems, which is essential for dynamic programming. Brute-force enumeration is inefficient and does not exploit subproblem reuse. Randomization is unrelated to dynamic programming design. While dynamic programming can sometimes reduce exponential time complexity, having exponential complexity alone does not mean a problem is suitable for this technique.

2. Overlapping Subproblems Recognition

   Why is memoization effective for the calculation of Fibonacci numbers using recursion?

   1. Because the problem cannot be divided into subproblems
   2. Because Fibonacci numbers are always sorted
   3. Because recursion always guarantees efficiency
   4. Because subproblems are repeatedly solved

   Explanation: Memoization is effective when the same subproblems are solved multiple times, as in the case of the recursive Fibonacci calculation. The division into subproblems is possible here, so saying it's impossible is incorrect. Fibonacci numbers being sorted is unrelated to the efficiency provided by memoization. Recursion does not always guarantee efficiency, especially without memoization.

3. Space Optimization in Dynamic Programming

   When computing the nth value in the Fibonacci sequence using dynamic programming, how can you reduce space usage from O(n) to O(1)?

   1. By only storing the previous two values
   2. By increasing the number of recursive calls
   3. By storing all computed values in an array
   4. By sorting the intermediate results

   Explanation: Only the previous two Fibonacci values are needed at each step, so storing just them reduces space to O(1). Storing all values uses O(n) space, not O(1). Increasing recursive calls can increase stack usage and is inefficient. Sorting does not impact the memory usage required for computation.

4. Tabulation vs. Memoization

   Which statement correctly describes a main difference between tabulation (bottom-up) and memoization (top-down) in dynamic programming?

   1. Tabulation cannot be used for optimization problems
   2. Tabulation fills a data structure iteratively, while memoization relies on recursion
   3. Memoization is slower than brute-force solutions
   4. Tabulation always uses more memory than memoization

   Explanation: Tabulation builds solutions iteratively, typically filling out a table, while memoization uses a recursive approach and caches results. Memory usage is not always higher with tabulation; it depends on the specific problem and implementation. Memoization, when implemented well, is generally faster than brute-force, not slower. Tabulation is commonly used for optimization problems in dynamic programming.

5. Recognizing a Dynamic Programming Applicability

   Given a problem where you must count the number of ways to climb a staircase by taking 1 or 2 steps at a time, why is dynamic programming an appropriate solution strategy?

   1. Because the problem requires a priority queue
   2. Because random choices must be made at every step
   3. Because sorting the steps improves efficiency
   4. Because the total number of ways depends on the solutions to smaller step counts

   Explanation: The count for n steps is based on the sum of ways to climb n-1 and n-2 steps, which is ideal for dynamic programming. Using a priority queue is unnecessary here since the problem does not involve ordering by priority. Random choices do not feature in this problem, and sorting the steps doesn't affect the core recurrence or efficiency.