IOI '18 P3 - Werewolf

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Points: 30 (partial)
Time limit: 1.8s
Memory limit: 512M

Problem type
Allowed languages
C++

There are N cities and M roads in Ibaraki Prefecture, Japan. Cities are numbered from 0 through N-1 in the increasing order of their population. Each road connects a pair of distinct cities, and can be traveled in both directions. You can travel from any city to any other city by using one or more of these roads.

You planned Q trips, numbered from 0 through Q-1. The trip i (0 \le i \le Q-1) is to travel from the city S_i to the city E_i.

You are a werewolf. You have two forms: human form and wolf form. At the beginning of each trip you are in human form. At the end of each trip, you must be in wolf form. During the trip you have to transform (change from human form to wolf form) exactly once. You can transform only when you are in some city (possibly S_i or E_i).

Living as a werewolf is not easy. You must avoid low-populated cities when you are in human form, and avoid highly-populated cities when you are in wolf form. For each trip i (0 \le i \le Q-1), there are two thresholds L_i and R_i (0 \le L_i \le R_i \le N-1) that indicate which cities must be avoided. More specifically, you must avoid the cities 0, 1, \dots, L_i-1 when you are in human form, and must avoid the cities R_i+1, R_i+2, \dots, N-1 when you are in wolf form. This means in the trip i, you can only transform in one of the cities L_i, L_i + 1, \dots, R_i.

Your task is to determine, for each trip, whether it is possible to travel from the city S_i to the city E_i in a way that satisfies the aforementioned constraints. The route you take can have an arbitrary length.

Implementation details

You should implement the following function:

std::vector<int> check_validity(int N, std::vector<int> X, std::vector<int> Y, std::vector<int> S, std::vector<int> E, std::vector<int> L, std::vector<int> R)
  • N: the number of cities.
  • X and Y: arrays of length M. For each j (0 \le j \le M-1), the city X[j] is directly connected to the city Y[j] by a road.
  • S, E, L, and R: arrays of length Q, representing the trips.

Note that the values of M and Q are the lengths of the arrays, and can be obtained as indicated in the implementation notice.

The function check_validity is called exactly once for each test case. This function should return an array A of integers of length Q. The value of A_i \space (0 \le i \le Q-1) must be 1 if the trip i is possible while satisfying the aforementioned conditions, or 0 otherwise.

Example

Let

\begin{align*} N &= 6 \\ M &= 6 \\ Q &= 3 \\ X &= [5, 1, 1, 3, 3, 5] \\ Y &= [1, 2, 3, 4, 0, 2] \\ S &= [4, 4, 5] \\ E &= [2, 2, 4] \\ L &= [1, 2, 3] \\ R &= [2, 2, 4] \end{align*}

For the trip 0, you can travel from the city 4 to the city 2 as follows:

  • Start at city 4 (You are in human form)
  • Move to the city 3 (You are in human form)
  • Move to city 1 (You are in human form)
  • Transform yourself into wolf form (You are in wolf form)
  • Move to the city 2 (You are in wolf form)

For the trips 1 and 2, you cannot travel between the given cities.

Hence, your program should return [1, 0, 0].

Constraints

  • 2 \le N \le 200\,000
  • N-1 \le M \le 400\,000
  • 1 \le Q \le 200\,000
  • For each 0 \le j \le M-1
    • 0 \le X_j \le N-1
    • 0 \le Y_j \le N-1
    • X_j \neq Y_j
  • You can travel from any city to any other city by using roads.
  • Each pair of cities are directly connected by at most one road. In other words, for all 0 \le j < k \le M-1, (X_j, Y_j) \neq (X_k, Y_k) and (Y_j, X_j) \neq (X_k, Y_k).
  • For each 0 \le i \le Q-1
    • 0 \le L_i \le S_i \le N-1
    • 0 \le E_i \le R_i \le N-1
    • S_i \neq E_i
    • L_i \le R_i

Subtasks

Subtask Points Constraints
1 7 points N \le 100, M \le 200, Q \le 100
2 8 points N \le 3\,000, M \le 6\,000, Q \le 3\,000
3 34 points M = N - 1 and each cities incident to at most 2 roads (the cities are connected in a line)
4 51 points No additional constraints

Sample Grader

The sample grader reads the input in the following format:

  • line 1: N \space M \space Q
  • line 2 + j (0 \le j \le M-1): X_j \space Y_j
  • line 2 + M + i (0 \le i \le Q-1): S_i \space E_i \space L_i \space R_i

The sample grader prints the return value of check_validity in the following format:

  • line 1 + i (0 \le i \le Q-1): A_i

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