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문제

Hazel the botanist visited a special exhibition in the Singapore Botanical Gardens. In this exhibition, $n$ plants of distinct heights are placed in a circle. These plants are labelled from $0$ to $n-1$ in clockwise order, with plant $n-1$ beside plant $0$.

For each plant $i$ ($0 \leq i \leq n-1$), Hazel compared plant $i$ to each of the next $k-1$ plants in clockwise order, and wrote down the number $r[i]$ denoting how many of these $k-1$ plants are taller than plant $i$. Thus, each value $r[i]$ depends on the relative heights of some $k$ consecutive plants.

For example, suppose $n=5$, $k=3$ and $i=3$. The next $k-1 = 2$ plants in clockwise order from plant $i = 3$ would be plant $4$ and plant $0$. If plant $4$ was taller than plant $3$ and plant $0$ was shorter than plant $3$, Hazel would write down $r[3] = 1$.

You may assume that Hazel recorded the values $r[i]$ correctly. Thus, there is at least one configuration of distinct heights of plants consistent with these values.

You were asked to compare the heights of $q$ pairs of plants. Sadly, you do not have access to the exhibition. Your only source of information is Hazel's notebook with the value $k$ and the sequence of values $r[0], \ldots, r[n-1]$.

For each pair of different plants $x$ and $y$ that need to be compared, determine which of the three following situations occurs:

  • Plant $x$ is definitely taller than plant $y$: in any configuration of distinct heights $h[0], \ldots, h[n - 1]$ consistent with the array $r$ we have $h[x] > h[y]$.
  • Plant $x$ is definitely shorter than plant $y$: in any configuration of distinct heights $h[0], \ldots, h[n - 1]$ consistent with the array $r$ we have $h[x] < h[y]$.
  • The comparison is inconclusive: neither of the previous two cases applies.

구현

You should implement the following procedures:

void init(int k, int[] r)
  • $k$: the number of consecutive plants whose heights determine each individual value $r[i]$.
  • $r$: an array of size $n$, where $r[i]$ is the number of plants taller than plant $i$ among the next $k-1$ plants in clockwise order.
  • This procedure is called exactly once, before any calls to compare_plants.
int compare_plants(int x, int y)
  • $x$, $y$: labels of the plants to be compared.
  • This procedure should return:
    • $1$ if plant $x$ is definitely taller than plant $y$,
    • $-1$ if plant $x$ is definitely shorter than plant $y$,
    • $0$ if the comparison is inconclusive.
  • This procedure is called exactly $q$ times.

제한

  • $2 \leq k \leq n \leq 200\;000$
  • $1 \leq q \leq 200\;000$
  • $0 \leq r[i] \leq k - 1$ (for all $0 \leq i \leq n - 1$)
  • $0 \leq x < y \leq n - 1$
  • There exists one or more configurations of distinct heightsof plants consistent with the array $r$.

예시 1

Consider the following call:

init(3, [0, 1, 1, 2])

Let's say the grader calls compare_plants(0, 2). Since $r[0] = 0$ we can immediately infer that plant $2$ is not taller than plant $0$. Therefore, the call should return $1$.

Let's say the grader calls compare_plants(1, 2) next. For all possible configurations of heights that fit the constraints above, plant $1$ is shorter than plant $2$. Therefore, the call should return $-1$.

예시 2

Consider the following call:

init(2, [0, 1, 0, 1])

Let's say the grader calls compare_plants(0, 3). Since $r[3] = 1$, we know that plant $0$ is taller than plant $3$. Therefore, the call should return $1$.

Let's say the grader calls compare_plants(1, 3) next. Two configurations of heights $[3,1,4,2]$ and $[3,2,4,1]$ are both consistent with Hazel's measurements. Since plant $1$ is shorter than plant $3$ in one configuration and taller than plant $3$ in the other, this call should return $0$.

서브태스크

번호 배점 제한
1 5

$k = 2$ 

2 14

$n \leq 5000$, $2\cdot k > n$

3 13

$2\cdot k > n$

4 17

The correct answer to each call of compare_plants is $1$ or $-1$.

5 11

$n \leq 300, q \leq \frac{n\cdot (n-1)}{2}$

6 15

$x=0$ for each call of compare_plants.

7 25

No additional constraints.

첨부

출처

Olympiad > International Olympiad in Informatics > IOI 2020 > Day 1 1번

  • 문제를 만든 사람: ainta

제출할 수 있는 언어

C++17, C++14, C++20, C++14 (Clang), C++17 (Clang), C++20 (Clang)

채점 및 기타 정보

  • 예제는 채점하지 않는다.