ICPC 2017 - J. Son of Pipe Stream
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
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competitive_programming/icpc/2017/J-son-of-pipe-stream/solution.tex to update the written solution and
competitive_programming/icpc/2017/J-son-of-pipe-stream/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Rapid City
event
sponsor
ICPC 2017
Problem J
Son of Pipe Stream
Time limit: 5 seconds
Two years ago, you helped install the nation’s very first Flubber pipe network in your hometown, to great
success. Polls show that everyone loves having their own Flubber dispenser in their kitchen, and now a
few enterprising citizens have discovered a use for it. Apparently Flubber, when mixed with water, can
help extinguish fires! This is a very timely discovery, as out-of-control fires have lately been surprisingly
common.
Your hometown’s city council would like to make use of this property of Flubber by creating the Flubber/wa-
ter mixture at a centrally located station. This station, which is called the Flubber Department (FD) will also
have specialized employees trained to travel to the locations of fires and make use of their processed Flubber
to control the blazes.
The pipes are already in place all around the city. You are given a layout of the pipes, and must determine
how to route Flubber from the Flubber factory and water from a local source through the pipes to the FD.
Note that both Flubber and water will be flowing through the same network of pipes, perhaps even the same
pipe. All pipes are bidirectional, but Flubber and water cannot move in opposite directions through the same
pipe. Furthermore, if both liquids are sent in the same direction through the same pipe, they will inevitably
mix. Therefore the nodes in the network have been equipped with special membranes and filters that enable
you to separate and reorganize all incoming mixtures as you see fit. The network is a closed system, so the
total rate of each fluid going into a node must equal the total rate of that fluid going out, except at the source
of that fluid and the destination (the FD).
Each pipe has a certain capacity. Flubber, being somewhat sluggish, has a viscosity value v, so a pipe that
can transport v liters/second of water can transport only 1 liter/second of Flubber. The pipe’s capacity scales
linearly for mixtures of the two. To be precise, if c denotes the water capacity of the pipe and f and w are
the rates of Flubber and water moving through the pipe (all measured in liters/second), then the capacity
constraint is given by the inequality v · f + w ≤ c.
Your main concern is balancing the mixture that reaches the FD. You would like as much total liquid as
possible, but you also need a sufficient amount of water – because undiluted Flubber is highly flammable
– and a sufficient amount of Flubber – because it would not be much of a “Flubber Department” without
Flubber! You have come up with a formula to measure the “value” of the final mixture: F a · W 1−a , where
F is the rate of incoming Flubber in liters/second, W is the rate of incoming water in liters/second, and a is
a given constant between 0 and 1.
Determine the maximum value of F a · W 1−a that can be achieved and how to route the Flubber and water
to achieve it.
Input
The input starts with a line containing the number of locations n (3 ≤ n ≤ 200), the number of pipes p
(n − 1 ≤ p ≤ 12 n(n − 1)), and the real values v (1 ≤ v ≤ 10) and a (0.01 ≤ a ≤ 0.99). Locations are
numbered from 1 to n; 1 is the Flubber factory, 2 is the water source, and 3 is the FD. The real values have
at most 10 digits after the decimal point.
Rapid City
event
sponsor
ICPC 2017
The following p lines each describe one pipe. Each line contains two integers j and k (1 ≤ j < k ≤ n),
giving the locations connected by the pipe, and an integer c (1 ≤ c ≤ 10), giving the water capacity of the
pipe in liters/second.
No two pipes connect the same pair of locations. Furthermore, it is guaranteed that the network is connected.
Output
First, for each pipe (in the order given in the input), display two values: the rate of Flubber moving through
it, and the rate of water moving through it (negative if the liquid is moving from k to j), such that F a · W 1−a
is maximized. Then display that maximum value accurate to within an absolute error of 10−4 .
If there are multiple solutions, any one will be accepted. All constraints (not sending Flubber and water
in opposite directions along the same pipe, flow conservation, pipe capacities, and consistency between the
constructed solution and its claimed value) must be satisfied within an absolute error of 10−4 .
Sample Input 1 Sample Output 1
6 6 3.0 0.66 0.000000000 1.360000000
2 4 8 0.000000000 1.000000000
4 6 1 0.000000000 -1.000000000
3 6 1 0.000000000 0.360000000
4 5 5 0.880000000 0.000000000
1 5 7 -0.880000000 -0.360000000
3 5 3 1.02037965897
Sample Input 2 Sample Output 2
5 5 1.0 0.5 5 0
1 2 10 5 5
2 3 10 4.2 3.14159
3 4 10 4.2 3.14159
4 5 10 -4.2 -3.14159
3 5 10 5
Editorial
Rendered from the copied solution.tex file. The original TeX source remains
available below.
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
Exact copied C++ implementation from solution.cpp.
#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;
}
Source Files
Exact copied source-of-truth files. Edit solution.tex for the write-up and solution.cpp for the implementation.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{enumitem}
\title{ICPC World Finals 2017\\J. Son of Pipe Stream}
\author{}
\date{}
\begin{document}
\maketitle
\section*{Problem Summary}
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
\section*{Key Observations}
\begin{itemize}[leftmargin=*]
\item Write the structural observations that make the problem tractable.
\item State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
\item If the constraints matter, explain exactly which part of the solution they enable.
\end{itemize}
\section*{Algorithm}
\begin{enumerate}[leftmargin=*]
\item Describe the data structures and the state maintained by the algorithm.
\item Explain the processing order and why it is sufficient.
\item Mention corner cases explicitly if they affect the implementation.
\end{enumerate}
\section*{Correctness Proof}
We prove that the algorithm returns the correct answer.
\paragraph{Lemma 1.}
State the first key claim.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Lemma 2.}
State the next claim if needed.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Theorem.}
The algorithm outputs the correct answer for every valid input.
\paragraph{Proof.}
Combine the lemmas and finish the argument.
\section*{Complexity Analysis}
State the running time and memory usage in terms of the input size.
\section*{Implementation Notes}
\begin{itemize}[leftmargin=*]
\item Mention any non-obvious implementation detail that is easy to get wrong.
\item Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
\end{itemize}
\end{document}
#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;
}