Ever found yourself doing the same math problem over and over again? That’s exactly what computers do when solving complex problems – unless we teach them dynamic programming.
What is dynamic programming? Simply put, it’s a problem-solving technique that saves time by remembering solutions to smaller problems. Instead of recalculating the same thing repeatedly, dynamic programming stores these answers and reuses them.
Think of it like taking notes during a lecture. You write down important points so you don’t have to remember everything in your head. Dynamic programming does the same thing for algorithms.
Why Dynamic Programming Matters
Dynamic programming isn’t just academic jargon. It powers the apps you use daily.
Netflix uses it for recommendation algorithms. Google Maps uses it for finding the shortest routes. Even your smartphone’s autocorrect relies on dynamic programming techniques.
The beauty lies in its simplicity: solve small problems first, then combine their solutions to tackle bigger challenges.
The Classic Example: Fibonacci Numbers
Let’s start with everyone’s favorite example – Fibonacci numbers.
Without dynamic programming, calculating Fibonacci numbers is painfully slow:
python
def fibonacci_slow(n):
if n <= 1:
return n
return fibonacci_slow(n-1) + fibonacci_slow(n-2)
# This takes forever for large numbers
print(fibonacci_slow(40)) # Takes several secondsWhy is this slow? Because it recalculates the same values repeatedly. For fibonacci_slow(5), it calculates fibonacci_slow(3) multiple times.
Here’s the dynamic programming solution:
python
def fibonacci_fast(n):
if n <= 1:
return n
# Store previously calculated values
dp = [0] * (n + 1)
dp[0] = 0
dp[1] = 1
for i in range(2, n + 1):
dp[i] = dp[i-1] + dp[i-2]
return dp[n]
# Lightning fast, even for large numbers
print(fibonacci_fast(40)) # Instant resultThe magic happens here: We store each calculated value in the dp array. When we need Fibonacci(3), we don’t recalculate it – we just look it up.
Two Approaches to Dynamic Programming
1. Top-Down Approach (Memoisation)
Start with the big problem and break it down:
python
def fibonacci_memo(n, memo={}):
if n in memo:
return memo[n] # Already calculated
if n <= 1:
return n
# Calculate and store the result
memo[n] = fibonacci_memo(n-1, memo) + fibonacci_memo(n-2, memo)
return memo[n]This approach feels natural because you start with what you want and work backwards.
2. Bottom-Up Approach (Tabulation)
Start with the smallest problems and build up:
python
def fibonacci_bottom_up(n):
if n <= 1:
return n
# Start from the bottom
prev2 = 0 # fibonacci(0)
prev1 = 1 # fibonacci(1)
for i in range(2, n + 1):
current = prev1 + prev2
prev2 = prev1
prev1 = current
return prev1This approach is usually more memory-efficient.
Real-World Example: Coin Change Problem
Imagine you’re building a payment system. You need to find the minimum number of coins to make change.
Problem: Make change for $11 using coins of $1, $4, and $5.
Without dynamic programming: Try every possible combination. Extremely slow.
With dynamic programming:
python
def min_coins(amount, coins):
# dp[i] = minimum coins needed for amount i
dp = [float('inf')] * (amount + 1)
dp[0] = 0 # Zero coins needed for amount 0
for i in range(1, amount + 1):
for coin in coins:
if coin <= i:
dp[i] = min(dp[i], dp[i - coin] + 1)
return dp[amount] if dp[amount] != float('inf') else -1
# Find minimum coins for $11
coins = [1, 4, 5]
result = min_coins(11, coins)
print(f"Minimum coins needed: {result}") # Output: 3 coins (5+5+1)
How it works:
● dp[0] = 0: No coins needed for $0
● dp[1] = 1: One $1 coin for $1
● dp[4] = 1: One $4 coin for $4
● dp[5] = 1: One $5 coin for $5
● dp[11] = 3: Three coins (5+5+1) for $11When to Use Dynamic Programming
Dynamic programming works best when problems have:
1. Overlapping Subproblems: The same smaller problems appear multiple times.
2. Optimal Substructure: The optimal solution contains optimal solutions to subproblems.
Common applications:
- Pathfinding in games
- Text editing (spell checkers)
- Resource allocation
- Stock trading algorithms
- Image processing
Common Mistakes to Avoid
Mistake 1: Using dynamic programming for every problem. Not every problem needs it. Simple recursive problems might be fine as-is.
Mistake 2: Forgetting base cases. Always define what happens with the smallest input.
Mistake 3: Using too much memory. Sometimes you only need the last few values, not the entire array.
The Bottom Line
What is dynamic programming? It’s a smart way to avoid redundant calculations by storing and reusing solutions.
Think of it as the difference between:
- Bad approach: Asking the same question repeatedly
- Dynamic programming: Writing down answers and referring to your notes
Dynamic programming transforms impossibly slow algorithms into lightning-fast solutions. It’s not magic – it’s just organised problem-solving.
The next time you face a complex problem, ask yourself: “Am I solving the same smaller problems repeatedly?” If yes, dynamic programming might be your answer.
Start with simple examples like Fibonacci numbers. Once you understand the pattern, you’ll see dynamic programming opportunities everywhere.
Remember: Dynamic programming is about being lazy in a smart way. Instead of doing the same work twice, do it once and remember the result.
Here are a few links to some of the important data structure topics:
Understanding Array vs Linked List Time Complexity
A Guide to Decision Tree Classifier Hyperparameter Tuning
How to Reverse a Linked List Iteratively: A Step-by-Step Guide
How to Implement Graph in Python: A Clear and Engaging Guide
How to Solve Two Sum Problem: The Ultimate Guide