H. Qanat
A qanat is an irrigation system widely used to deliver water in hot, arid climates. The technology was originally developed by Persians over 2000 years ago. In Morocco, qanats are known as khettara and are still used today in the southern part of the countr...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
A qanat is an irrigation system widely used to deliver water in hot, arid climates. The technology was originally developed by Persians over 2000 years ago. In Morocco, qanats are known as khettara and are still used today in the southern part of the country. The basic feature of a qanat is an essentially horizontal channel that brings water from an underground water source to an outlet near a civilization. There is also a shaft known as a mother well that rises vertically from the underground water source to the surface of a mountain or hill. Creating such a system is extremely expensive, and was especially so in ancient times, since all of the materials excavated from the channel and mother well must be carried above ground, either through the channel outlet or the top of the mother well. To aid in the construction, there are often one or more additional vertical shafts placed at strategic locations above the underground channel. Although these shafts must also be excavated, they provide a means for lifting additional dirt from the horizontal channel as illustrated in Figure H.1.
Figure H.1: An illustration of a qanat.For this problem, model the cross-section of a qanat as shown in Figure H.2, with the channel outlet at (0, 0), the water source at (w, 0), and the top of the mother well at (w, h) with w > h. The surface of the mountain extends along a straight line from (w, h) to (0, 0).
(w, h)(0, 0) (w, 0)Figure H.2: A simplified model of a qanat cross-section.Every qanat must have a vertical mother well from the water source to the mountain surface above, along
with n additional vertical shafts. The channel and all shafts are modeled as line segments. Your goal
is to determine the placement for those additional shafts so as to minimize the overall excavation cost.
This cost is equal to the sum of the distances that each piece of excavated dirt must be transported to
reach the surface (using any combination of horizontal and vertical movement). For example, the cost
of excavating a continuous section
R` of dirt starting from the surface and going along a path of length `
(possibly including turns) is 0 x dx = 21 `2 .Input
The input consists of a single line containing three integers w (1 ≤ w ≤ 10 000), h (1 ≤ h < w), and n (1 ≤ n ≤ 1 000). The value w is the horizontal distance from the water source to the qanat outlet. The value h is the vertical distance from the water source to the mountain surface. The value n is the number of vertical shafts that must be used in addition to the mother well.
Output
First, display the minimum overall excavation cost. Next, display the x-coordinates, in increasing order, for n optimally placed vertical shafts. If n > 10, display only the first 10 x-coordinates. Answers within an absolute or relative error of 10−4 will be accepted. You may assume that there is a unique solution. No test case will result in a shaft within 0.001 units from the outlet of the qanat channel or from another shaft.
Sample Tests
8 4 131.500000
3.000000 195 65 212220.000000
48.000000
108.000000 10000 1 100030141.885677
9.956721
19.913443
29.870164
39.826887
49.783610
59.740334
69.697060
79.653786
89.610515
99.567245Editorial
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;
}
Source Files and Assets
Raw files are still available here when you want the original TeX, C++, or statement assets.
Show raw files
competitive_programming/icpc/2015/H-qanat/solution.texC++ implementationcompetitive_programming/icpc/2015/H-qanat/solution.cppStatement textcompetitive_programming/icpc/2015/H-qanat/statement.txtStatement PDFcompetitive_programming/icpc/2015/H-qanat/statement.pdfMetadatacompetitive_programming/icpc/2015/H-qanat/meta.jsonYear packetcompetitive_programming/icpc/2015/contest_problems.pdf