Dynamic programming (DP) is a powerful algorithmic technique that involves breaking down complex problems into smaller, overlapping subproblems and storing the solutions to these subproblems to avoid recomputation. This approach often leads to efficient solutions for problems that would be computationally expensive to solve using other methods.
Because of this, DP is a very sought-after skill in software development jobs, especially for positions that involve designing and optimizing algorithms. We’ve put together a list of the 20 most common dynamic programming interview questions, along with short answers and links to full answers, to help you get ready for your next interview.
Level 1 Warming Up with the Fundamentals
- Nth Catalan Number: Explain the concept of Catalan numbers and write a recursive function to calculate the Nth Catalan number.
- Minimum Operations: Given a string consisting of only ‘X’ and ‘Y’, find the minimum number of operations required to make the string alternating (XYXY…).
- Minimum steps to delete a string after repeated deletion of palindrome substrings: Given a string, find the minimum number of steps required to delete the entire string by repeatedly deleting palindrome substrings.
- Minimum number of Coins: Given a set of coin denominations and a target amount, find the minimum number of coins required to make up the target amount.
- Maximum Product Cutting: Given a rope of length n, find the maximum product you can obtain by cutting the rope into smaller pieces and multiplying their lengths.
- Ways to Cover a Distance: Given a distance and a set of steps you can take, find the number of ways you can cover the distance using those steps.
- Minimum number of deletions and insertions to transform one string into another: Given two strings, find the minimum number of deletions and insertions required to transform one string into the other.
- Minimum sum subsequence such that at least one of every four consecutive elements is picked: Given an array of integers, find the minimum sum of a subsequence such that at least one element from every four consecutive elements is picked.
Level 2: Stepping Up the Challenge
- Subset Sum Problem: Given a set of integers and a target sum, determine whether there exists a subset of the integers that adds up to the target sum.
- Longest Common Subsequence (LCS): Given two strings, find the longest common subsequence (LCS) that appears in both strings.
- Longest Increasing Subsequence (LIS): Given an array of integers, find the longest increasing subsequence (LIS) that can be formed from the array.
- Edit Distance: Given two strings, find the minimum number of edits (insertions, deletions, or substitutions) required to transform one string into the other.
- Longest Path In Matrix: Given a matrix of integers, find the longest path from the top-left corner to the bottom-right corner such that you can only move down or right.
- Optimal Strategy for a Game: Given a game where two players take turns picking coins from a pile, determine the optimal strategy for the first player to maximize their winnings.
- 0/1 Knapsack Problem: Given a set of items, each with a weight and a value, and a knapsack with a maximum weight capacity, find the subset of items that maximizes the total value in the knapsack without exceeding the weight capacity.
- Shortest Common Supersequence (SCS): Given two strings, find the shortest common supersequence (SCS) that contains both strings as subsequences.
- Partition problem: Given a set of integers, determine whether the set can be partitioned into two subsets with equal sums.
- Rod Cutting: Given a rod of length n and a set of prices for different lengths of the rod, determine the optimal way to cut the rod to maximize the profit.
- Coin change problem: Given a set of coin denominations and a target amount, find the minimum number of coins required to make up the target amount, or determine that it is impossible.
- Word Break Problem: Given a dictionary of words and a string, determine whether the string can be segmented into a space-separated sequence of one or more dictionary words.
Beyond the Basics Expanding Your DP Arsenal
- Maximal Product when Cutting Rope: Given a rope of length n, find the maximum product you can obtain by cutting the rope into smaller pieces and multiplying their lengths.
- Dice Throw Problem: Given a dice with n sides, find the number of ways to get a sum of k after throwing the dice m times.
- Box Stacking: Given a set of boxes, each with a width, height, and depth, find the maximum height of a stack that can be formed by stacking the boxes.
- Egg Dropping Puzzle: Given an N-story building and an egg that can be dropped from any floor, find the minimum number of drops required to determine the highest floor from which the egg can be dropped without breaking.
Mastering the Art of DP Practice Makes Perfect
To truly master dynamic programming, consistent practice is key The problems listed above provide a solid foundation, but there are countless other DP problems waiting to be explored. Here are some resources to help you on your journey
- GeeksforGeeks: This website offers a comprehensive collection of dynamic programming problems with detailed solutions and explanations.
- LeetCode: This platform provides a wide range of coding challenges, including many dynamic programming problems, along with interactive coding environments and community discussions.
- HackerRank: Similar to LeetCode, HackerRank offers a variety of coding challenges, including dynamic programming problems, and allows you to track your progress and compete with other programmers.
You can get the skills and confidence you need to ace your next technical interview by using these resources and practicing solving dynamic programming problems over and over again. Don’t forget that the key to success is to understand the basic ideas, spot patterns, and use the right DP techniques to quickly solve issues. So, keep practicing, keep learning, and keep pushing your boundaries!.
Companies That Ask Dynamic Programming Questions
To quickly review, the Fibonacci Sequence is a set of numbers where each number is the sum of the two numbers before it:
We should also restate that the Fibonacci sequence is known to start with 0, 1. This represents the “base case” which will be covered in more detail later on.
Now, if you look from left to right, it’s pretty clear that each number and the one before it are needed to figure out the next number in the sequence. This is like a “bottom-up” method because it starts with the smallest example and works its way up to the final answer.
In other words, if you look at the sequence from right to left, you can figure out the next number by looking at the two numbers that came before it. This method is like a “top-down” approach because it starts with the end goal and works its way down to the first example.
It doesn’t matter how you look at it; the math can be written as F(n) = F(n-1) FC(n-2). This is known as a recurrence relation and is another important concept that we’ll get back to later.
Let’s now turn to some code and step through a simple top-down solution that makes use of recursion:
Consider what happens when invoked as fib(5) to determine the fourth Fibonacci number.
Since N – 1 = 4 and N – 2 = 3, this function calls back to itself many times to get fib(4) and fib(3). Notice that the invocation of fib(4) will in turn make recursive calls fib(3) and fib(2). This highlights the overlapping subproblems and where work is being repeated. The function calls can also be represented as a tree to visualize and identify the repeated work:
In the end, solving for the fifth Fibonacci number ends up solving for:
- The fourth Fibonacci number once
- The third number twice
- The second number three times
- The first number five times
This is where dynamic programming really shines because it keeps work from being done twice and gives a much better solution.
The recursive algorithm for this problem throws away right away the answer to fib(3) and all the other subproblems. What a waste!
We can use a hashmap to store the return value of fib(3) instead of doing it all over again. Then, when the same subproblem comes up again, we can quickly get the result with O(1) time complexity. For small N values, adding two numbers a few times might not seem like a big deal, but as N gets bigger, the problem gets repeated a lot more quickly. It is called “memorization” to store the result of a function call so that you don’t have to make the same call again eventually.
To do the same thing with this problem, you can solve it from the bottom up by computing and storing each number in the sequence as you go. This style of storing and reusing results is known as “tabulation”.
When solving for Fibonacci, an array can be used for tabulation, which is actually too much because you only need to see the last two results. However, this shows that the idea of using arrays in more than one dimension to solve more complicated problems often needs to be expanded.
Now that we’ve looked at Fibonacci from two different points of view and seen two different approaches, we’ve been introduced to all the main ideas behind dynamic programming:
- Recurrence relations are shown by base cases and the subproblems that happen over and over again.
- Top-down approaches which typically involve recursion and memoization
- Bottom-up approaches that are usually iterative and use tabulation
Let’s move on to look at when and how to apply Dynamic programming in interviews.
Optimal Substructure and Overlapping Subproblems
More formally, in order to apply dynamic programming to a problem two conditions must be present:
- Optimal substructure
- Overlapping subproblems
Optimal substructure requires that you can solve a problem based on the solutions of subproblems. So, if you want to find the fifth Fibonacci number, you can do so by computing fib(5) = fib(4) / fib(3). Beyond knowing how to solve those two subproblems, you don’t need to know anything else to figure out the solution.
A useful way to think about optimal substructure is whether a problem can be easily solved recursively. Recursive solutions inherently solve a problem by breaking it down into smaller subproblems. If you can solve a problem recursively, it most likely has an optimal substructure.
When you break a problem down into smaller problems, you may get the same smaller problem more than once. This is called “overlapping subproblems.” With the Fibonacci example, if we want to compute fib(5), we need to compute fib(4) and fib(3). However, to compute fib(4), we need to compute fib(3) again. This is a wasted effort, since we’ve already computed the value of fib(3).
Dynamic programming works by having subproblems that overlap, and it uses memory to store values that have already been calculated so that they don’t have to be calculated again. The more overlap there is, the more computational time is saved.
Find the first solution, analyze the solution, find the subproblems, and turn the solution around. This is what FAST stands for.
It’s not the only way to solve problems with dynamic programming, but it’s meant to be simple, easy to remember, and useful in a lot of situations.
Let’s break down each of these steps.
To use The FAST Method to solve a dynamic programming problem, the first step is to find a brute force recursive solution. Solve the problem without concern for efficiency, just as a starting point. Though there are a couple of constraints on how this brute force solution should look:
- Recursive functions should be self-contained. If you store results by changing global variables, you might not be able to add memoization later on. Make a function that depends only on its parameters and isn’t affected by anything else.
- Avoid unnecessary recursive function arguments. The arguments will eventually be used to remember the results of the subproblems; the fewer the better.
Analyze the initial brute force solution. This involves determining the time and space complexity and determining if there are any obvious areas for improvement.
As part of the analysis process, make sure that the first solution follows the rules for troubleshooting with dynamic programming:
- Does it have an optimal substructure?
- Are there overlapping subproblems?
If there is indeed a Dynamic programming solution, the appropriate subproblems can now be identified and coded. Apply memoization to avoid unnecessary repeated work. The problem has now been solved from the top down, which probably has the best level of complexity and doesn’t repeat any work.
Since we understand the problem well, we can go further. To do this, you have to code the different bottom-up approach that computes and stores the results of each subproblem in a table until the overall solution is found. Usually, you want to change the solution from top-down to bottom-up because it avoids problems with recursion and the call stack.
Top 5 Dynamic Programming Patterns for Coding Interviews – For Beginners
FAQ
How common are dynamic programming questions in interviews?
Are dynamic programming questions hard?
How does dynamic programming work?
Explore a range of commonly asked interview questions and their well-explained answers. Dynamic Programming (DP) is a powerful algorithmic paradigm that solves a complex problem by breaking it down into simpler subproblems and utilizes the fact that the optimal solution to the overall problem depends upon the optimal solution to its subproblems.
What are the easiest dynamic programming questions?
The Fibonacci problem [ Solution] This is one of the easiest dynamic programming questions and many of you have already solved it without even knowing that you are using Dynamic programming. This is also the most common example of DP and many instructors use Fibonacci numbers to teach Dynamic programming.
How many dynamic programming problems to test your skills?
Here are 20 Dynamic Programming problems to test your skills and prepare well. Hello guys, if you are preparing for coding interviews then there are two topics which I would say you must pay special attention, one is system Design and other is Dynamic Programming, both are difficult to master but extremely important for tech interviews.
What is dynamic programming (DP)?
Dynamic programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It’s particularly effective when these subproblems overlap, meaning the same subproblem needs to be solved multiple times. To implement DP in such a scenario, we follow these steps: 1.
