Dynamic Programming Visualization

Understand how dynamic programming solves complex problems by breaking them into simpler subproblems

DP Problems

Fibonacci Sequence

Classic recursive problem with overlapping subproblems

0/1 Knapsack

Optimization problem with capacity constraints

Longest Common Subsequence

Finding longest sequence common to both strings

Matrix Chain Multiplication

Minimizing scalar multiplications

Coin Change

Finding minimum coins for a given amount

Controls

Problem Parameters

Fibonacci Sequence Visualization

Time: O(n)
Space: O(n)

Computing Fibonacci(6)

n 0 1 2 3 4 5 6
fib(n) 0 1 1 2 3 5 8
fib(6) = fib(5) + fib(4) = 5 + 3 = 8

Recursion Tree Visualization

fib(6)
fib(5)
Calculated
fib(4)
Calculated
Using memoization to avoid redundant calculations

Current Step

Calculating fib(6) by adding previously computed values fib(5)=5 and fib(4)=3.

Memoized Fibonacci

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

Mathematical Analysis

\( F(n) = F(n-1) + F(n-2) \text{ for } n > 1 \)

The Fibonacci sequence exhibits optimal substructure and overlapping subproblems, making it ideal for dynamic programming optimization.

\( \text{Time Complexity: } O(n) \text{ with memoization} \)

Without memoization, the naive recursive approach has exponential time complexity \( O(\phi^n) \) where \( \phi \) is the golden ratio. Memoization reduces this to linear time by storing previously computed values.