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ICPC 2016
ICPC 2016

C. Ceiling Function

Advanced Ceiling Manufacturers (ACM) is analyzing the properties of its new series of Incredibly Collapse-Proof Ceilings (ICPCs). An ICPC consists of n layers of material, each with a different value of collapse resistance (measured as a positive integer)....

Updated May 21, 2026
Track ICPC
Year 2016
Statement Text + PDF
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Problem statement

Problem Statement

Formatted from the contest statement text, with sample tests broken out into copyable blocks.

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Time limit 5 seconds

Advanced Ceiling Manufacturers (ACM) is analyzing the properties of its new series of Incredibly Collapse-Proof Ceilings (ICPCs). An ICPC consists of n layers of material, each with a different value of collapse resistance (measured as a positive integer). The analysis ACM wants to run will take the collapse-resistance values of the layers, store them in a binary search tree, and check whether the shape of this tree in any way correlates with the quality of the whole construction. Because, well, why should it not? To be precise, ACM takes the collapse-resistance values for the layers, ordered from the top layer to the bottom layer, and inserts them one-by-one into a tree. The rules for inserting a value v are:

  • If the tree is empty, make v the root of the tree.
  • If the tree is not empty, compare v with the root of the tree. If v is smaller, insert v into the left subtree of the root, otherwise insert v into the right subtree.

ACM has a set of ceiling prototypes it wants to analyze by trying to collapse them. It wants to take each group of ceiling prototypes that have trees of the same shape and analyze them together. For example, assume ACM is considering five ceiling prototypes with three layers each, as described by Sample Input 1 and shown in Figure C.1. Notice that the first prototype’s top layer has collapse- resistance value 2, the middle layer has value 7, and the bottom layer has value 1. The second prototype has layers with collapse-resistance values of 3, 1, and 4 – and yet these two prototypes induce the same tree shape, so ACM will analyze them together. Given a set of prototypes, your task is to determine how many different tree shapes they induce.

(2, 7, 1)            (3, 1, 4)            (1, 5, 9)         (2, 6, 5)         (9, 7, 3)
     2                    3              1                    2                         9
1          7         1           4                5                 6               7
9        5              3
Figure C.1: The four tree shapes induced by the ceiling prototypes in Sample Input 1.

Input

The first line of the input contains two integers n (1 ≤ n ≤ 50), which is the number of ceiling prototypes to analyze, and k (1 ≤ k ≤ 20), which is the number of layers in each of the prototypes. The next n lines describe the ceiling prototypes. Each of these lines contains k distinct integers (between 1 and 106 , inclusive), which are the collapse-resistance values of the layers in a ceiling prototype, ordered from top to bottom.

Output

Display the number of different tree shapes.

Sample Tests

Sample 1
Sample Input 
 5   3
 2   7   1
 3   1   4
 1   5   9
 2   6   5
 9   7   3
Sample Output 
4
Sample 2
Sample Input 
 3 4
 3 1 2 40000
 3 4 2 1
 33 42 17 23
Sample Output 
2
Editorial

Editorial

The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.

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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

  1. Describe the data structures and the state maintained by the algorithm.

  2. Explain the processing order and why it is sufficient.

  3. 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.

Implementation

Code

C++ solution used for this page.

C++

Clean code view with a raw-file link when you want the original source.

Raw file 
#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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