Dynamic programming (DP) stands as one of the most frequently tested algorithmic paradigms in technical interviews. This problem-solving approach breaks complex challenges into overlapping subproblems and builds optimal solutions through systematic computation. For candidates preparing for coding assessments, mastering core DP patterns significantly enhances problem-solving efficiency. Let's explore nine fundamental dynamic programming algorithms that regularly appear in interview scenarios.
1. Fibonacci Sequence Optimization The Fibonacci problem demonstrates DP's core principle of memoization. While the naive recursive approach exhibits exponential time complexity, DP reduces this to linear time through stored intermediate results:
def fibonacci(n):
dp = [0, 1] + [0]*(n-1)
for i in range(2, n+1):
dp[i] = dp[i-1] + dp[i-2]
return dp[n]
2. 0-1 Knapsack Problem This resource allocation challenge requires selecting items with maximum value without exceeding weight capacity. The solution employs a 2D array where dp[i][w] represents the maximum value using the first i items with weight limit w:
def knapsack(values, weights, capacity):
n = len(values)
dp = [[0]*(capacity+1) for _ in range(n+1)]
for i in range(1, n+1):
for w in range(1, capacity+1):
if weights[i-1] > w:
dp[i][w] = dp[i-1][w]
else:
dp[i][w] = max(dp[i-1][w], values[i-1] + dp[i-1][w-weights[i-1
