Commonly Asked Interview Questions on Dynamic Programming

Dynamic Programming (DP) is a method used to solve problems by breaking them down into simpler subproblems and solving each subproblem just once, storing the solutions for future reference. It is particularly useful in problems with overlapping subproblems and optimal substructure. A solid grasp of DP and its applications is crucial for solving many algorithmic problems efficiently.

Top Coding Problems on Dynamic Programming

Easy Problems

• Nth Catalan Number
• Minimum Operations
• Minimum steps to delete using palindrome substrings
• Minimum number of Coins
• Maximum Product Cutting
• Ways to Cover a Distance
• Minimum number of deletions and insertions to transform
• Minimum sum subsequence with one of every four

Medium Problems

• Subset Sum Problem
• Longest Common Subsequence
• Longest Increasing Subsequence
• Edit Distance
• Longest Path In Matrix
• Optimal Strategy for a Game
• 0-1 Knapsack Problem
• Shortest Common Supersequence
• Partition problem
• Rod Cutting
• Coin change problem
• Word Break Problem
• Dice Throw Problem
• Box Stacking
• Egg Dropping Puzzle
• Max length chain
• Longest Common Substring
• Interleaved strings
• Maximum sum increasing subsequence
• Minimum number of jumps
• Count subsequences of type a^i, b^j, c^k
• Get Minimum Squares
• Nth Fibonacci Number
• Longest Palindromic Substring
• Total Decoding Messages
• Unique BST’s
• Player with max score
• Form a palindrome
• Word Wrap Problem
• Count Palindromic Subsequences
• Minimum time to finish tasks without skipping two consecutive

Hard Problems

• Minimum Partition
• Boolean Parenthesization Problem
• Matrix Chain Multiplication
• Longest Zig-Zag Sub Sequence
• Maximum Profit
• Maximum Path Sum in Matrix
• The painter’s partition problem
• Palindrome Partitioning
• Array Partition
• Maximum difference of zeros and ones in binary string
• Count digit groupings of a number

Top Theoretical Interiew Question on Dynamic Programming

1. What is Dynamic Programming?

Dynamic Programming is a technique used to solve problems by dividing them into smaller subproblems and storing the results of these subproblems to avoid redundant computations. It is commonly used for optimization and counting problems.

2. When should Dynamic Programming be used?

Dynamic Programming is used when:

• A problem has overlapping subproblems, meaning the same subproblem is solved multiple times.
• The problem has an optimal substructure, meaning the optimal solution can be constructed from optimal solutions of subproblems.

3. What do you understand by overlapping subproblems in DP?

Overlapping subproblems means that the same subproblems are solved multiple times in the process of solving the main problem. DP optimizes this by solving each subproblem only once and storing its result for future reference.

4. What are Memoization and Tabulation?

• Memoization is a top-down approach where recursive calls are made, and results are cached to avoid redundant calculations.
• Tabulation is a bottom-up approach where all subproblems are solved iteratively and stored in a table.

5. What is the importance of memoization in Dynamic Programming?

Memoization improves the efficiency of recursive solutions by storing the results of subproblems so that they don’t need to be recalculated. It reduces the time complexity from exponential to polynomial by avoiding redundant calculations.

6. What are the time and space complexities of DP?

• Time Complexity: The time complexity depends on the number of subproblems and how often they are solved. In general, it’s proportional to the size of the problem, for example, suppose there are n subproblems and each takes O(1) to solve, then total complexity is O(n).
• Space Complexity: Space complexity depends on how many subproblems need to be stored. In some cases, DP solutions require O(n) or O(n^2) space.

7. What is the difference between bottom-up and top-down DP approaches?

• Top-down (Memoization): Starts from the main problem and recursively solves subproblems, caching results as they are computed.
• Bottom-up (Tabulation): Starts from the smallest subproblems and builds up to the main problem iteratively, storing results in a table.

8. What is a DP table, and how do you decide its dimensions for a problem?

A DP table is an array (1D, 2D, or multi-dimensional) used to store the results of subproblems. The dimensions depend on the problem's variables and the states of the subproblems. For example, for a problem involving two sequences, a 2D table is often used.

9. What is the difference between 1d DP and 2d DP

• In 1D DP, the state of the problem is represented using a single array (1-dimensional table). It is typically used when the problem can be broken down into a single variable (e.g., the number of steps, index of a sequence).
• In 2D DP, the state of the problem is represented using a two-dimensional table (array). It is used when the problem involves two parameters or variables that influence the subproblem, such as sequences, matrices, or grids.

10. What are the main advantages of using DP over brute force methods?

• Efficiency: DP avoids redundant calculations by storing intermediate results, making it more efficient than brute force.
• Optimized Time Complexity: DP typically reduces the time complexity from exponential to polynomial.