Understanding Dynamic Programming: A Practical Guide with TypeScript

Dynamic programming (DP) is a powerful programming technique often used to solve complex problems by breaking them down into simpler subproblems. I recently revisited this concept during my daily studies through the book Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People. This book offers a fantastic refresher on key programming techniques, and I highly recommend it to any developer looking to sharpen their skills. In this article, we’ll cover a classic example of dynamic programming and demonstrate its implementation using TypeScript, making it accessible for modern developers.

What is Dynamic Programming?

Dynamic programming is an optimization technique used for problems that can be divided into overlapping subproblems. Unlike brute force methods, DP solves each subproblem only once and stores the result, thus avoiding redundant computations. It is widely used in problems involving sequences, arrays, or graphs.

Two key aspects of DP are:

1. Optimal Substructure: The solution to the overall problem can be constructed from solutions to its subproblems.
2. Overlapping Subproblems: The problem can be broken down into subproblems that recur multiple times.

Problem Example: Longest Substring Without Repeating Characters

Let's take a classic problem: finding the longest substring between two strings. This problem is effectively solved using a dynamic programming technique involving a two-dimensional table.

Problem Statement

Given two strings, find the longest substring they have in common.

Approach

We'll use a dynamic programming approach with a two-dimensional table. Each cell in the table represents the length of the common substring ending at the respective indices of the two strings. By iterating through the table and updating it based on character matches, we can efficiently track the longest common substring and its location.

Implementation in TypeScript

Below is the TypeScript implementation of the solution:

Explanation

1. Data Structures: A two-dimensional table is used to store the lengths of matching substrings at each combination of positions from the two input strings.
2. Iterative Comparison: The algorithm iterates through each character of the first word (rows) and compares it with each character of the second word (columns).
3. Table Updates: If the characters match, the current cell value is updated as one plus the value in the diagonally previous cell, representing an extension of the common substring.
4. Tracking the Longest Substring: Whenever a cell value exceeds the previously recorded maximum length, the algorithm updates the maximum length and the corresponding end index in the first word.
5. Extracting the Result: After completing the table, the longest common substring is extracted using the recorded end index and maximum length.

Why Use Dynamic Programming Here?

Dynamic programming is well-suited for this problem because:

• Optimal Substructure: The longest common substring at any position is determined by extending the common substrings already computed for previous positions.
• Overlapping Subproblems: The two-dimensional table ensures that each subproblem is solved once, storing the results for reuse in future computations.

This approach avoids the inefficiencies of recomputing substrings, achieving a time complexity of O(m × n), where m and n are the lengths of the two strings.

Conclusion

Dynamic programming is a versatile and efficient approach for solving many complex problems. By understanding its core principles and practicing with real-world examples, developers can leverage it to write optimal and elegant solutions. As demonstrated in this article, the two-dimensional table approach provides a clear and systematic way to solve problems like finding the longest common substring, emphasizing both clarity and performance.

Book link

Grokking Algorithms, Second Edition (English Edition)