Maximum Coins From K Consecutive Bags

To solve this problem, we need to find the maximum sum of coins we can collect from k consecutive bags. A binary search approach over the valid answer space allows us to efficiently determine the maximum sum. This solution involves identifying an optimal range of bags and applying greedy or sliding window methods for efficient calculations.

Problem Statement

You are given a 2D array coins where each entry coins[i] = [li, ri, ci] represents a range of bags on a number line from li to ri, and the number of coins ci in all bags within this range. Your task is to select exactly k consecutive bags from any position on the number line and collect the maximum number of coins possible.

The segments in coins do not overlap, and there are an infinite number of bags on the number line. You are allowed to select bags from any point, and the goal is to determine the optimal set of k consecutive bags that yields the maximum coin count.

Examples

Example 1

Input: coins = [[8,10,1],[1,3,2],[5,6,4]], k = 4

Output: 10

Selecting bags at positions [3, 4, 5, 6] gives the maximum number of coins: 2 + 0 + 4 + 4 = 10 .

Example 2

Input: coins = [[1,10,3]], k = 2

Output: 6

Selecting bags at positions [1, 2] gives the maximum number of coins: 3 + 3 = 6 .

Constraints

• 1 <= coins.length <= 105
• 1 <= k <= 109
• coins[i] == [li, ri, ci]
• 1 <= li <= ri <= 109
• 1 <= ci <= 1000
• The given segments are non-overlapping.

Solution Approach

Binary Search on Coin Count

To find the maximum sum of coins, we apply binary search over the possible answer space for the sum of coins. The key observation is that the starting position for the k consecutive bags must be either the left endpoint li or the right endpoint ri - k + 1 of the given segments.

Sliding Window or Greedy Approach

Once the starting position is determined, the next step is to select k consecutive bags using a sliding window technique or a greedy approach. This helps efficiently calculate the sum of coins in a set of consecutive bags and check if this sum is the maximum.

Prefix Sum Optimization

Using prefix sums can optimize the calculation of the total coin count within any segment of bags. This method reduces redundant calculations when sliding through the number line.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the binary search procedure over the range of possible coin sums and the method used for summing the coins, such as sliding window or prefix sum. Typically, the time complexity will be O(log(max possible coin sum) * n), where n is the number of segments in coins. Space complexity depends on the storage of prefix sums or other optimization structures.

What Interviewers Usually Probe

• Ability to apply binary search on a non-continuous space.
• Efficient use of sliding window or greedy techniques for maximum sum calculations.
• Understanding of optimization techniques like prefix sum for fast computations.

Common Pitfalls or Variants

Common pitfalls

• Not considering the correct bounds for the k consecutive bags, such as starting from the right endpoint of a segment.
• Overcomplicating the solution by attempting brute force instead of leveraging binary search and sliding window.
• Missing edge cases where the number of bags in a segment is smaller than k.

Follow-up variants

• Increasing k and testing how the solution scales.
• Modifying the problem to allow overlapping segments and calculating coins over a more dynamic space.
• Expanding the problem to include negative coin values and adjusting the selection strategy accordingly.

FAQ

How do I use binary search in the Maximum Coins From K Consecutive Bags problem?

Binary search is used over the possible range of coin sums, narrowing down the best starting position for the k consecutive bags. The optimal solution lies within this range.

What is the role of sliding window in solving this problem?

Sliding window helps in efficiently calculating the sum of coins in a fixed-length window of k consecutive bags, ensuring that the sum is updated incrementally.

What is the key observation in the Maximum Coins From K Consecutive Bags problem?

The key observation is that the optimal starting position for the k consecutive bags will either be at the left endpoint li or right endpoint ri - k + 1 of the segments.

Can the solution be optimized using a prefix sum approach?

Yes, using a prefix sum allows for constant-time calculation of the total coin count within any segment of consecutive bags, making the solution more efficient.

What are the common mistakes when solving the Maximum Coins From K Consecutive Bags problem?

Common mistakes include failing to consider the correct starting points for the k consecutive bags, not applying binary search or sliding window efficiently, and overlooking edge cases.