What is Dynamic Programming?

Dynamic Programming (DP) is a problem-solving technique that breaks complex problems into simpler overlapping subproblems, solves each subproblem only once, and stores the results to avoid redundant computation.

Core Idea

“Those who cannot remember the past are condemned to repeat it.” — DP avoids repeating work by remembering past results.

Two Key Properties

For DP to apply, a problem must have:

1. Optimal Substructure — The optimal solution can be built from optimal solutions of its subproblems.
2. Overlapping Subproblems — The same subproblems are solved multiple times (unlike divide & conquer).

Two Approaches

1. Top-Down (Memoization) Start with the big problem, recurse down, and cache results as you go.

memo = {}
def fib(n):
    if n <= 1: return n
    if n in memo: return memo[n]
    memo[n] = fib(n-1) + fib(n-2)
    return memo[n]

2. Bottom-Up (Tabulation) Start with the smallest subproblems and build up to the answer iteratively.

def fib(n):
    dp = [0, 1]
    for i in range(2, n+1):
        dp.append(dp[i-1] + dp[i-2])
    return dp[n]

Why It Matters

Approach	Time Complexity (Fibonacci)
Naive recursion	O(2ⁿ)
Dynamic Programming	O(n)

That’s an enormous difference — DP turns exponential problems into polynomial ones.

Classic DP Problems

• Fibonacci sequence — the “hello world” of DP
• Knapsack problem — maximize value within a weight limit
• Longest Common Subsequence — used in diff tools and bioinformatics
• Shortest Path (Bellman-Ford, Floyd-Warshall) — used in GPS/networking
• Matrix Chain Multiplication — optimal parenthesization
• Coin Change — minimum coins to make an amount

Mental Model

Think of DP as filling in a table where each cell depends on previously filled cells. Once the table is complete, your answer is right there — no recomputation needed.

It’s one of the most powerful tools in algorithm design and appears heavily in competitive programming, technical interviews, and real-world systems like compilers, bioinformatics, and financial modeling.

Did this post help you better understand the concept? Let me know your thoughts in the comments below! Don’t forget to follow us on Instagram @coderz.py for more DP tips and solutions to ace your coding interviews.