Unbounded Knapsack Explained: The DP Blueprint Behind Coin Change

Original author：The Logic Blueprint

Original source：https://www.youtube.com/embed/VrD_ma_wpms

Dynamic Programming fundamentals get much easier once you see what actually changes between 0/1 Knapsack, Unbounded Knapsack, Coin Change, and count-ways problems. In this The Logic Blueprint lesson, we build Unbounded Knapsack from first principles and show why one small recurrence change completely changes the problem. You’ll learn: • what “unbounded” really means • how to derive the recurrence step by step • why the take branch stays on the same row • how to collapse the 2D table into 1D DP • why loop direction is everything • how the same blueprint extends to count-ways and optimization problems This video is for students, interview prep learners, and competitive programmers who want more than a memorized formula. The goal is to give you a reusable mental model, not just one solved problem. If this helped, subscribe to The Logic Blueprint for deeper algorithm lessons that focus on intuition, structure, and transfer across problem types. Practice resources: • AtCoder Educational DP Contest: https://atcoder.jp/contests/dp • CSES Dynamic Programming section: https://cses.fi/problemset/ • CSES Minimizing Coins: https://cses.fi/problemset/task/1634 • CSES Coin Combinations I: https://cses.fi/problemset/task/1635 • CSES Coin Combinations II: https://cses.fi/problemset/task/1636 • USACO Guide, Introduction to DP: https://usaco.guide/gold/intro-dp?lang=cpp • USACO Guide, Knapsack DP: https://usaco.guide/gold/knapsack?lang=cpp • LeetCode Dynamic Programming problem list: https://leetcode.com/problem-list/dynamic-programming/ Suggested practice order: 1. Minimizing Coins 2. Coin Combinations I 3. Coin Combinations II 4. AtCoder DP Contest problems 5. USACO Guide knapsack module 6. LeetCode DP variants for interview-style reinforcement These are especially good for this video because they force you to reason about: • reuse vs use-once transitions • whether order matters • count vs min/max objectives • 2D DP vs 1D DP compression #dynamicprogramming #unboundedknapsack #coinchange #algorithms #competitiveprogramming