Dynamic Programming Examples: Solved Problems with Step-by-Step Explanations

Problem: Climbing Stairs

Description: Count ways to climb n stairs taking 1 or 2 steps at a time.

Approach: dp[i] = dp[i-1] + dp[i-2]. Base: dp[0]=1, dp[1]=1. Fibonacci-like recurrence.

Complexity: Time O(n²) typical, Space O(n) to O(n²)

Key Insight: This is a classic easy Dynamic Programming problem. Understanding this example helps you solve 8 similar problems.

Interview Tip: Explain your approach before coding. State the time and space complexity upfront.

Problem: Coin Change

Description: Find minimum coins to make a target amount.

Approach: dp[amount] = min(dp[amount], dp[amount-coin]+1) for each coin. Bottom-up tabulation.

Complexity: Time O(n²) typical, Space O(n) to O(n²)

Key Insight: This is a classic medium Dynamic Programming problem. Understanding this example helps you solve 8 similar problems.

Interview Tip: Explain your approach before coding. State the time and space complexity upfront.

Problem: Longest Increasing Subsequence

Description: Find the length of the longest strictly increasing subsequence.

Approach: O(n²) DP: dp[i] = max(dp[j]+1) for j < i where arr[j] < arr[i]. O(n log n) with binary search.

Complexity: Time O(n²) typical, Space O(n) to O(n²)

Key Insight: This is a classic hard Dynamic Programming problem. Understanding this example helps you solve 8 similar problems.

Interview Tip: Explain your approach before coding. State the time and space complexity upfront.

Each example was chosen because it:

Study the pattern, not just the solution. The goal is to recognize when to apply Dynamic Programming in new, unseen problems.

Before coding, ask yourself:

If 2+ answers are yes, try the Dynamic Programming approach.