B. Buffed Buffet
You are buying lunch at a buffet. A number of different dishes are available, and you can mix and match them to your heart’s desire. Some of the dishes, such as dumplings and roasted potatoes, consist of pieces of roughly equal size, and you can pick an int...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
You are buying lunch at a buffet. A number of different dishes are available, and you can mix and
match them to your heart’s desire. Some of the dishes, such as dumplings and roasted potatoes, consist
of pieces of roughly equal size, and you can pick an integral number of such pieces (no splitting is
allowed). Refer to these as “discrete dishes.” Other dishes, such as tzatziki or mashed potatoes, are fluid
and you can pick an arbitrary real-valued amount of them. Refer to this second type as “continuous
dishes.”
Of course, you like some of the dishes more than others, but how much you like a dish also depends on
how much of it you have already eaten. For instance, even if you generally prefer dumplings to potatoes,
you might prefer a potato over a dumpling if you have already eaten ten dumplings. To model this, each
dish i has an initial tastiness ti , and a rate of decay of the tastiness ∆ti . For discrete dishes, the tastiness
you experience when eating the nth item of the dish is ti − (n − 1)∆ti . For continuous dishes, the
tastiness you experience when eating an infinitesimal amount dx grams of the dish after already having
eaten x grams is (ti − x∆ti )dx. In other words, the respective total amounts of tastiness you experience
when eating N items of a discrete dish or X grams of a continuous dish are as follows:
(ti − (n − 1)∆ti ) and (ti − x∆ti )dx
n=1 0For simplicity, do not take into account that different dishes may or may not go well together, so define the total tastiness that you experience from a meal as the sum of the total tastinesses of the individual dishes in the meal (and the same goes for the weight of a meal – there are no food antiparticles in the buffet!). You have spent days of painstaking research determining the numbers ti and ∆ti for each of the dishes in the buffet. All that remains is to compute the maximum possible total tastiness that can be achieved in a meal of weight w. Better hurry up, lunch is going to be served soon!
Input
The input consists of a single test case. The first line of input consists of two integers d and w (1 ≤ d ≤ 250 and 1 ≤ w ≤ 10 000), where d is the number of different dishes at the buffet and w is the desired total weight of your meal in grams. Then follow d lines, the ith of which describes the ith dish. Each dish description is in one of the following two forms:
- A description of the form “D wi ti ∆ti ” indicates that this is a discrete dish where each item weighs wi grams, with initial tastiness ti and decay of tastiness ∆ti .
- A description of the form “C ti ∆ti ” indicates that this is a continuous dish with initial tastiness ti and decay of tastiness ∆ti .
The numbers wi , ti , and ∆ti are integers satisfying 1 ≤ wi ≤ 10 000 and 0 ≤ ti , ∆ti ≤ 10 000.
Output
Display the maximum possible total tastiness of a meal of weight w based on the available dishes. Give the answer with a relative or absolute error of at most 10−6 . If it is impossible to make a meal of total weight exactly w based on the available dishes, display impossible.
Sample Tests
2 15
D 4 10 1
C 6 140.500000000 3 15
D 4 10 1
C 6 1
C 9 349.000000000 2 19
D 4 5 1
D 6 3 2impossibleEditorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
Key Observations
Write the structural observations that make the problem tractable.
State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
If the constraints matter, explain exactly which part of the solution they enable.
Algorithm
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
Mention corner cases explicitly if they affect the implementation.
Correctness Proof
We prove that the algorithm returns the correct answer.
Lemma 1.
State the first key claim.
Proof.
Provide a concise proof.
Lemma 2.
State the next claim if needed.
Proof.
Provide a concise proof.
Theorem.
The algorithm outputs the correct answer for every valid input.
Proof.
Combine the lemmas and finish the argument.
Complexity Analysis
State the running time and memory usage in terms of the input size.
Implementation Notes
Mention any non-obvious implementation detail that is easy to get wrong.
Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
Code
C++ solution used for this page.
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}
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