The "Maximum Coin Collection" problem is a quintessential dynamic programming challenge that focuses on pathfinding and value optimization. Usually set in a grid or an array, you are tasked with moving from a starting point to an endpoint while collecting coins along the way. Each cell or step has a specific number of coins (which could be positive or negative). The rules of movement are typically restricted—for example, you might only be able to move right or down. The goal is to identify the path that results in the highest possible total of collected coins.
The Maximum Coin Collection interview question is a staple in technical interviews because it perfectly illustrates the concept of "optimal substructure." Companies like Uber use it to see if a candidate can break down a large, intimidating problem into smaller, manageable sub-problems. It tests your ability to recognize that the best way to reach a certain point depends only on the best ways to reach the points immediately preceding it. This is a core skill for any software engineer working on optimization or logistics software.
The primary algorithmic pattern is Dynamic Programming (DP). In a 2D version, you would create a DP table where each entry dp[i][j] represents the maximum coins you can collect reaching cell (i, j). The value at dp[i][j] is calculated by taking the value of the current cell and adding the maximum of the DP values from the cells you could have come from (e.g., dp[i-1][j] or dp[i][j-1]). This "bottom-up" approach ensures that you calculate each value only once, leading to a much more efficient solution than a naive recursive one.
Consider a 2x2 grid: [1, 5] [3, 2] Starting at the top-left (0,0) and moving only right or down to the bottom-right (1,1):
A common error is not properly initializing the DP table, especially the first row and first column which often have only one possible predecessor. Another mistake is forgetting to handle negative coin values—if all paths lead to negative results, the "maximum" might still be a negative number, not zero. Candidates also sometimes struggle with space optimization; while a 2D table is intuitive, many grid problems can be solved using only one or two rows of space, which is a great way to impress an interviewer.
To get better at the dynamic programming interview pattern, always start by defining the "state" of your DP and the "transition" formula. Write these down before you start coding. If you're stuck, try solving the problem for a very small input (like a 2x2 grid) by hand. This often reveals the pattern you need to implement.
| Title | Difficulty | Topics | LeetCode |
|---|---|---|---|
| Largest Plus Sign | Medium | Solve | |
| Minimum Score Triangulation of Polygon | Medium | Solve | |
| Find the Maximum Length of Valid Subsequence II | Medium | Solve | |
| Paint House | Medium | Solve | |
| Minimum Cost For Tickets | Medium | Solve |
