Dynamic Programming

A method for solving complex problems by breaking them down into simpler subproblems and storing the results to avoid redundant calculations.

Key Principles

1. Optimal Substructure

An optimal solution to the problem contains optimal solutions to its subproblems. This allows us to build the solution to the original problem from the solutions of smaller subproblems.

2. Overlapping Subproblems

The same subproblems are solved multiple times during the computation. By storing solutions to these subproblems, we can avoid redundant calculations and improve efficiency.

Implementation Approaches

Top-Down (Memoization)

Start from the original problem and recursively break it down into smaller subproblems. Store the results of each subproblem to avoid recalculating them.

Bottom-Up (Tabulation)

Start by solving the smallest subproblems first and use their solutions to solve progressively larger problems until reaching the original problem.

Common Problems

Fibonacci Sequence

Calculates the nth Fibonacci number. The Fibonacci sequence is where each number is the sum of the two preceding ones, starting with 0 and 1.

function fibDP(n) {
  // Base case
  if (n <= 1) return n;
  
  // Initialize array to store values
  const dp = new Array(n + 1);
  dp[0] = 0;
  dp[1] = 1;
  
  // Build up solution using previous values
  for (let i = 2; i <= n; i++) {
    dp[i] = dp[i-1] + dp[i-2];
  }
  
  return dp[n];
}

Time Complexity

O(n)

Space Complexity

O(n)

Real-world Applications

Financial modeling, natural growth patterns, algorithmic trading strategies.

Interactive Demo: Fibonacci Calculation

Enter n:

Why Use Dynamic Programming?

Advantages

Drastically improves efficiency for problems with overlapping subproblems
Reduces time complexity from exponential to polynomial in many cases
Provides a systematic approach to solving complex optimization problems
Eliminates redundant calculations by storing intermediate results

Considerations

Requires identifying optimal substructure and overlapping subproblems
May need more memory for storing intermediate results
Implementation can be more complex than naive approaches
Not all problems can be solved using dynamic programming