Maximum Points in an Archery Competition

Problem Description

Alice and Bob are opponents in an archery competition. They each shoot a specified number of arrows at a target divided into scoring sections. The goal is for Bob to maximize the total points he can obtain after Alice has shot her arrows. Given the number of arrows and the distribution of Alice's shots, determine how many arrows Bob should shoot in each section to achieve the highest score.

Key Insights

• Bob must shoot more arrows than Alice in any section to claim the points for that section.
• Bob has a limited number of arrows, which means he must strategically choose which sections to target to maximize his score.
• The highest scoring sections should be prioritized since they provide more points.
• If Bob and Alice tie in a section (both shoot the same number of arrows), Bob does not gain points, so he needs to shoot more in order to score.

Space and Time Complexity

Time Complexity: O(1) - The algorithm operates on a fixed size of 12 scoring sections, leading to constant time complexity. Space Complexity: O(1) - The space used does not depend on the input size but is fixed due to the fixed number of scoring sections.

Solution

To solve the problem, we can use a greedy algorithmic approach. We will iterate through the scoring sections in descending order and decide how many arrows Bob should shoot in each section based on Alice's shots. The following steps summarize the approach:

1. Initialize an array for Bob's arrows with all zeros.
2. Start from the highest scoring section (11) and move down to 0.
3. If Bob can shoot more arrows than Alice in the current section, assign the necessary arrows to that section and update the remaining arrows.
4. Continue this process until all arrows are allocated or all sections are evaluated.
5. Ensure the total arrows used equals the given number of arrows.

This method ensures that Bob shoots strategically in the sections that yield the highest points.

Code Solutions