My First USACO Problem Solution (not my first USACO problem lol):

This is my C++ solution to the infamous Lazy Cow problem in the 2024 Febuary Platinum Competition. I took approx. 2hrs to complete this solution, and here I will write a brief explanation.

If you do not know what the Lazy Cow Problem is, (shame) here is the link to the problem: https://usaco.org/index.php?page=viewproblem2&cpid=1404

Interpretation:

• Bessie can prepare test cases each minute, using powers of 3 as energy costs.
• She must meet D demands, each requiring at least b_i test cases within m_i minutes.
• We need to find the minimum energy expenditure for each demand.

Observations:

• Since b_i can be as large as 10^12, we should use large test case values first to minimize energy.
• The possible test cases per minute follow the pattern: 1, 3, 9, 27, ..., which are powers of 3.
• We need to distribute these optimally within m_i minutes. The obvious answer is to use a greedy algorithm.

Solution:

• Precompute powers of 3 up to 3^19, since 3^19 > 10^{12}.
• Process each demand by starting with the largest possible power of 3
  • For each power p = 3^a, we determine how many test cases are still needed: remaining = b_i - cases_done
  • Find the maximum times we can use this power p:
  • max_cases = remaining / p (How many times p fits into the remaining requirement)
  • use = min(max_cases, m_i) (We cannot exceed the available minutes)
• Reduce the remaining required test cases and continue with smaller powers by iterating
• Program keeps track of energy spent to process each demand

Avoiding Recalculations:

• Precomputing powers of 3 takes O(1), around 20 values
• Each demand is processed in O(20)=O(1), because we iterate over at most 20 powers
• The overall complexity is O(D) which is efficent for D<=2x10^5

Conclusion:

• This program utilizes iteration and a greedy approach.
• Iteration can be done a lot easier with python, but C++ is 10x faster than python, and there is a program time limit (2 sec.)

Key tricks:

• Lots of people overlook the observation step. This step is the most important step for a successful solution.
• Don't forget to simplify the question and what its asking. This step will allow you to see your solution more clearly.