Welcome to the realm of Dynamic Programming, a powerful algorithmic design paradigm that helps us conquer complex problems by cleverly avoiding redundant computations. Imagine you're trying to solve a large, intricate puzzle. If you discover a smaller piece of that puzzle is identical to another piece you've already solved, would you re-solve it? Of course not! Dynamic Programming operates on a similar principle, storing the solutions to subproblems so they can be reused whenever needed.
At its core, Dynamic Programming is about breaking down a problem into smaller, overlapping subproblems. The key insight is that the optimal solution to the overall problem can be constructed from the optimal solutions to these subproblems. This paradigm is particularly effective for problems exhibiting two crucial properties: overlapping subproblems and optimal substructure.
Let's first understand 'overlapping subproblems'. This means that the same subproblems are encountered multiple times when solving the larger problem. A naive recursive solution might recompute the solution to these subproblems repeatedly, leading to exponential time complexity. Dynamic Programming addresses this by memoizing, or storing, the results of these subproblems.
The second property is 'optimal substructure'. This indicates that the optimal solution to the problem contains within it the optimal solutions to its subproblems. If we can find the best way to solve the smaller pieces, we can combine them to find the best way to solve the whole.
There are generally two main approaches to implementing Dynamic Programming: memoization (top-down) and tabulation (bottom-up).
Memoization, often implemented with recursion, starts with the main problem and recursively breaks it down. Before computing a subproblem's solution, it checks if it has already been computed and stored. If so, it returns the stored value; otherwise, it computes it, stores it, and then returns it. Think of it as asking "Have I solved this smaller piece before?"
function fibonacciMemo(n, memo = {}) {
if (n in memo) {
return memo[n];
}
if (n <= 1) {
return n;
}
memo[n] = fibonacciMemo(n - 1, memo) + fibonacciMemo(n - 2, memo);
return memo[n];
}