F. Longest Rivers
The Chao Phraya River System is the main river system of Thailand. Its six longest rivers listed by decreasing length are: 1. Tha Chin (765 km) 2. Nan (740 km) 3. Yom (700 km) 4. Ping (658 km) 5. Pa Sak (513 km) 6. Wang (335 km) The Chao Phraya River System...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
The Chao Phraya River System is the main river system of Thailand. Its six longest rivers listed by decreasing length are:
- Tha Chin (765 km)
- Nan (740 km)
- Yom (700 km)
- Ping (658 km)
- Pa Sak (513 km)
- Wang (335 km) The Chao Phraya River System. Picture from Wikimedia Commons.
A simplified model of this river system is shown in Figure F.1, where the smaller red numbers indicate the lengths of various sections of each river. The point where two or more rivers meet as they flow downstream is called a confluence. Confluences are labeled with the larger black numbers. In this model, each river either ends at a confluence or flows into the sea, which is labeled with the special confluence number 0. When two or more rivers meet at a confluence (other than confluence 0), the resulting merged river takes the name of one of those rivers. For example, the Ping and the Wang meet at confluence 1 with the resulting merged river retaining the name Ping. With this naming, the Ping has length 658 km while the Wang is only 335 km. If instead the merged river had been named Wang, then the length of the Wang would be 688 km while the length of the Ping would be only 305 km.
Yom Nan
700
675
Ping 335 Pa Sak
305 1
Tha Chin 2 513
353 65
765
0Figure F.1: The river system in Sample Input 1. Same-colored edges indicate a river.The raised awareness of this phenomenon causes bitter rivalries among the towns along the rivers. For example, the townspeople along the Wang protest that maybe with a proper naming scheme, their river could actually be the longest, or maybe the second longest (or at least not last!). To end all the guessing, your task is to validate all such claims. The rank of a river is its position in a list of all rivers ordered by decreasing length, where the best rank is 1 for the longest river. For each river, determine its best possible rank over all naming schemes. At any confluence, the name of a new, larger river in any naming scheme must be one of the names of the smaller rivers which join at that confluence. If two or more rivers have equal lengths in a naming scheme, all the tied rivers are considered to have the best possible ranking. For example, if one river is the longest and all other rivers are equal, those rivers all have rank 2.
Input
The first line of input contains two integers n (1 ≤ n ≤ 500 000), which is the number of river sources in the system, and m (0 ≤ m ≤ n − 1), which is the number of confluences with positive labels. These confluences are numbered from 1 to m. The next n lines describe the rivers. Each of these lines consists of a string, which is the name of the river at the source, and two integers c (0 ≤ c ≤ m) and d (1 ≤ d ≤ 109 ), where c is the identifier of the nearest confluence downstream, and d is the distance from the source to that confluence in kilometers. River names use only lowercase and uppercase letters a–z, and consist of between 1 and 10 characters, inclusive. The final m lines describe confluences 1 to m in a similar fashion. The k th of these lines describes the confluence with identifier k and contains two integers: the identifier of the nearest confluence down- stream and the distance from confluence k to this confluence in kilometers. It is guaranteed that each confluence 1 through m appears as “the nearest downstream” at least twice, confluence 0 appears at least once, and all sources are connected to confluence 0.
Output
Display one river per line in the same order as in the input. On that line, display the name of the river and its best possible rank.
Sample Tests
6 2
PaSak 0 513
Nan 2 675
Yom 2 700
Wang 1 335
Ping 1 305
ThaChin 0 765
0 353
0 65PaSak 5
Nan 2
Yom 1
Wang 3
Ping 4
ThaChin 1Editorial
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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