Introduction
For a software engineer, cracking interview problems is essential. The number of new problems coming out each year makes it challenging. However, it’s the dynamic programming problems that I despise the most - the type I struggle with. They say, teaching others what you learned help master it even more. This article will reiterate what dynamic programming is about and the common strategies to identify and solve them.
What is DP?
Dynamic programming is a technique, that solves problems by breaking them into smaller and more manageable pieces. These subproblems usually share similar structures, allowing a common solution pattern to be applied repeatedly to build the solution for the entire problem.
All dynamic programming problems can be solved using recursion because they can be broken down into smaller tasks that share a similar logic. A typical recursive function includes a base case—a logic that’s usually placed at the top of the function and checks a condition to determine when to stop. The base case acts as the termination point for the recursive call stack, ensuring the function eventually returns a result rather than letting it run an infinite loop.
def fib(n): if n <= 1: # base case return n return fib(n - 1) + fib(n - 2) # recursion calls In addition, some problems can be further optimized using memoization, which is essentially a form of caching. Instead of repeatedly performing the same recursive calls, we store the results of those calls in a cache. Then, if a recursive call needs a value we’ve already computed, we can simply return the cached result.
Memoization significantly improves runtime efficiency by avoiding redundant computations. It can also reduce the depth of the recursive call stack in some problems, helping to avoid issues like Python’s default maximum recursion depth of around 1000. However, memoization doesn’t completely eliminate recursion limits if the problem itself requires very deep recursive calls.
def fib(n, memo={}): if n <= 1: # base case return n if n in memo: # memoization check return memo[n] memo[n] = fib(n - 1, memo) + fib(n - 2, memo) # populate memo return memo[n] Tabulation: A Bottom-up Approach
The top-down approach to dynamic programming, using recursion and memoization, often feels more intuitive to write and understand. However, in some problems, the bottom-up tabulation method can be significantly faster because it avoids the overhead of recursive calls and handles the subproblems more efficiently.
def fib(self, n: int) -> int: if n < 1: return 0 fb = [0] * (n + 1) fb[1] = 1 if n > 1: for i in range(2, n+1): fb[i] = fb[i - 1] + fb[i - 2] return fb[n] A typical tabulation approach involves creating a DP table, often implemented as an array, to store the results of previously solved subproblems—which, in a way, acts similarly to memoization. Unlike the top-down method, tabulation relies on loops rather than recursive calls to fill in this table and build the solution from smaller subproblems upward.
Although every dynamic programming problem can be solved using either top-down or bottom-up techniques, recursion comes with practical limits, such as the maximum call stack depth, which can make top-down solutions unsuitable for problems requiring very deep recursion. Bottom-up tabulation avoids these issues by relying on loops rather than recursion, making it applicable to all DP problems. Still, there are cases where the top-down approach remains more efficient, especially when the DP state space is large and sparse, since it avoids computing unnecessary states.
Recognizing Dynamic Programming Problems
- Dynamic programming problems usually have optimal substructure, meaning the solution to the overall problem can be built from solutions to smaller subproblems. If you can describe your problem as “my answer depends on smaller answers,” it’s often a DP problem. For example, the shortest path to a cell in a grid depends on the shortest paths to the neighboring cells.
- Another key sign of DP is overlapping subproblems. This happens when the same smaller problems keep coming up repeatedly in different parts of your solution. A classic example is the naive recursive Fibonacci, where
fib(n-1)andfib(n-2)get recomputed many times. DP avoids this by storing results to prevent redundant calculations. - Many DP problems involve making choices at each step, like deciding whether to include or exclude an item, or choosing one path over another. For instance, in the Knapsack problem, you choose whether to add an item or leave it out. These decisions create different subproblems that DP can solve efficiently.
- Problems that ask you to find the minimum, maximum, longest, shortest, cheapest, or number of ways to do something often indicate DP. For example, finding the minimum cost to climb stairs or counting the number of unique paths in a grid both fall under DP because they involve accumulating or comparing results across subproblems.
Conclusion
Dynamic programming (DP) is a powerful problem-solving technique used to efficiently solve problems that can be broken into overlapping subproblems with optimal substructure. Instead of solving the same subproblems repeatedly, DP stores intermediate results either through recursion with memoization (top-down) or by filling in a table iteratively (bottom-up tabulation). Many DP problems involve finding the best, minimum, maximum, or counting the number of ways to achieve something. Recognizing DP problems often comes down to spotting repeated work, decision-making at each step, and defining states that capture the essence of subproblems. Although the top-down approach is often more intuitive, bottom-up solutions can avoid issues like stack overflows and may run faster in certain cases. Overall, DP is essential for tackling many complex problems efficiently, especially in coding interviews.
Top comments (0)