Maximum Coins From K Consecutive Bags

Maximum Coins From K Consecutive Bags

What is this problem about?

The "Maximum Coins From K Consecutive Bags" is an intriguing problem that blends interval math with optimization. You are given a series of "bags" or segments placed on a number line. Each segment has a start position, an end position, and a specific number of coins per unit of length (often just one coin per integer position). Your task is to find a continuous range of length $k$ on this number line that covers the maximum possible number of coins. Unlike simple array problems, the segments might have gaps between them, and your range of length $k$ can start and end anywhere, potentially covering only parts of some segments.

Why is this asked in interviews?

This Maximum Coins From K Consecutive Bags interview question is highly regarded because it tests multiple skills: sorting, the sliding window technique, and sometimes binary search. It requires a candidate to handle "continuous" data rather than just discrete array indices. Tech giants like Google and Amazon use it to see if a developer can handle edge cases—specifically, what happens when the window of length $k$ partially overlaps with a segment at the beginning or the end.

Algorithmic pattern used?

The most efficient approach involves the Sliding Window pattern combined with Sorting. First, you sort the segments by their starting positions. Then, you use a window of length $k$ and slide it across the segments. To efficiently calculate the coins within the window, you might use a Prefix Sum array of the total coins in the segments. For each segment, you can consider two scenarios: the window starts at the beginning of the segment or the window ends at the end of the segment. This "greedy" observation—that the optimal window likely aligns with one of the segment boundaries—is key to the solution.

Example explanation?

Suppose you have two segments: [1, 4] and [7, 10], each containing 1 coin per unit. Total length of $k = 5$.

• If you start your window at 1, it covers [1, 6]. It gets 4 coins from the first segment and 0 from the second. Total = 4.
• If you start your window at 3, it covers [3, 8]. It gets 2 coins from the first ([3,4]) and 2 coins from the second ([7,8]). Total = 4.
• If you end your window at 10, it covers [5, 10]. It gets 0 coins from the first and 4 from the second. Total = 4. In the Maximum Coins From K Consecutive Bags coding problem, the goal is to shift this window to find the sweet spot where the coverage is maximized.

Common mistakes candidates make?

Many candidates fail to realize that the window can start in the middle of a gap or a segment. They might only check cases where the window aligns perfectly with the start of a segment. Another mistake is using a simple sliding window on individual units (1, 2, 3...), which is too slow if the positions are large (e.g., up to $10^9$). You must slide the window across the segments themselves. Forgetting to sort the segments initially is another common error that makes the sliding window logic impossible to implement.

Interview preparation tip

When you see problems involving ranges and a fixed-length window, think about the sliding window interview pattern. If the positions can be very large, don't use the units as your indices; use the segments. Practice how to calculate the intersection of two intervals, as this is a recurring sub-problem in many advanced coding challenges.