시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

이진 검색 트리는 이진 트리이다. 트리는 비어있을 수도 있다. 비어있지 않은 경우에는 다음과 같은 특징을 가진다.

  1. 모든 노드는 키를 가지고 있다. 두 노드가 같은 키를 가질 수 없다.
  2. 비어있지 않은 왼쪽 서브 트리에 있는 모든 키는 그 서브 트리의 루트의 키보다 작다.
  3. 비어있지 않은 오른쪽 서브 트리에 있는 모든 키는 그 서브 트리의 루트에 있는 키보다 크다.
  4. 왼쪽 서브 트리와 오른쪽 서브 트리도 이진 검색 트리이다.

아래 그림은 이진 검색 트리의 예이다.

이진 검색 트리 T에서 k가 키인 노드를 찾으려면, 먼저 루트에서 시작해야 한다. T가 비어있다면, T는 키를 가지고 있지 않기 때문에 검색은 실패한다. 비어있지 않은 경우에는 루트의 키와 k를 비교한다. 만약, k가 루트의 키와 같다면, 검색은 성공적으로 종료된다. k가 루트의 키보다 작은 경우에는 왼쪽 서브트리의 루트에서 검색을 시작한다. k가 루트의 키보드 큰 경우에는 오른쪽 서브 트리의 루트에서 검색을 시작한다. 같은 과정으로 T의 왼쪽 또는 오른쪽 서브 트리에서 검색을 수행한다.

이진 검색 트리 T에 T에 존재하지 않는 새로운 키 k를 삽입하려면, 먼저 T에서 k를 찾아야 한다. T에는 k가 없기 때문에, 검색은 실패로 끝날 것이고, 실패로 끝난 그 위치에 새로운 키 k를 삽입한다. 예를 들어, 위의 그림 (a)에 80을 삽입하는 경우를 살펴보자. 먼저, 트리에서 80을 찾아야 한다. 검색은 실패로 끝나고, 마지막으로 검사한 키는 40이다. 80을 그 노드의 오른쪽 자식으로 삽입하면, 이진 검색트리는 (b)와 같아진다.

이 문제에서는 1, 2, ..., N 총 N개의 키를 가지고 있는 이진 검색 트리를 이용한다. {1, 2, ..., N}의 순열 a1, a2, ..., aN에 대해서, a1, a2, ..., aN을 순서대로 비어있는 이진 검색 트리에 삽입한다면, 이진 검색 트리를 만들 수 있다. 예를 들어, 순열 2 1 4 3 5는 그림 (c)를 만들 수 있다. 또, 2 4 3 1 5도 같은 트리를 만든다. {1, 2, 3, 4, 5}의 순열 중에서 그림 (c)와 같은 이진 검색 트리를 만드는 순열의 개수는 8개이다.

N과 순열 P가 주어졌을 때, 그 순열과 같은 이진 검색 트리를 만드는 순열의 개수를 구하는 프로그램을 작성하시오.

입력

첫째 줄에 테스트 케이스의 개수 T (≤ 40,320)가 주어진다. 각 테스트 케이스의 첫째 줄에는 키의 수 N이 주어진다. (1 ≤ N ≤ 20) 다음 줄에는 길이가 N인 순열이 주어진다.

출력

각 테스트 케이스에 대해서, 입력으로 주어진 순열과 같은 트리를 만드는 순열의 개수를 9,999,991로 나눈 나머지를 출력한다.

예제 입력 1

3
5
2 1 4 3 5
4
2 4 1 3
12
1 2 3 4 5 6 7 8 9 10 11 12

예제 출력 1

8
3
1
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T\uac00 \ube44\uc5b4\uc788\ub2e4\uba74, T\ub294 \ud0a4\ub97c \uac00\uc9c0\uace0 \uc788\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc5d0 \uac80\uc0c9\uc740 \uc2e4\ud328\ud55c\ub2e4. \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uacbd\uc6b0\uc5d0\ub294 \ub8e8\ud2b8\uc758 \ud0a4\uc640 k\ub97c \ube44\uad50\ud55c\ub2e4. \ub9cc\uc57d, k\uac00 \ub8e8\ud2b8\uc758 \ud0a4\uc640 \uac19\ub2e4\uba74, \uac80\uc0c9\uc740 \uc131\uacf5\uc801\uc73c\ub85c \uc885\ub8cc\ub41c\ub2e4. k\uac00 \ub8e8\ud2b8\uc758 \ud0a4\ubcf4\ub2e4 \uc791\uc740 \uacbd\uc6b0\uc5d0\ub294 \uc67c\ucabd \uc11c\ube0c\ud2b8\ub9ac\uc758 \ub8e8\ud2b8\uc5d0\uc11c \uac80\uc0c9\uc744 \uc2dc\uc791\ud55c\ub2e4. k\uac00 \ub8e8\ud2b8\uc758 \ud0a4\ubcf4\ub4dc \ud070 \uacbd\uc6b0\uc5d0\ub294 \uc624\ub978\ucabd \uc11c\ube0c \ud2b8\ub9ac\uc758 \ub8e8\ud2b8\uc5d0\uc11c \uac80\uc0c9\uc744 \uc2dc\uc791\ud55c\ub2e4. \uac19\uc740 \uacfc\uc815\uc73c\ub85c T\uc758 \uc67c\ucabd \ub610\ub294 \uc624\ub978\ucabd \uc11c\ube0c \ud2b8\ub9ac\uc5d0\uc11c \uac80\uc0c9\uc744 \uc218\ud589\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc774\uc9c4 \uac80\uc0c9 \ud2b8\ub9ac T\uc5d0 T\uc5d0 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uc0c8\ub85c\uc6b4 \ud0a4 k\ub97c \uc0bd\uc785\ud558\ub824\uba74, \uba3c\uc800 T\uc5d0\uc11c k\ub97c \ucc3e\uc544\uc57c \ud55c\ub2e4. T\uc5d0\ub294 k\uac00 \uc5c6\uae30 \ub54c\ubb38\uc5d0, \uac80\uc0c9\uc740 \uc2e4\ud328\ub85c \ub05d\ub0a0 \uac83\uc774\uace0, \uc2e4\ud328\ub85c \ub05d\ub09c \uadf8 \uc704\uce58\uc5d0 \uc0c8\ub85c\uc6b4 \ud0a4 k\ub97c \uc0bd\uc785\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc704\uc758 \uadf8\ub9bc (a)\uc5d0 80\uc744 \uc0bd\uc785\ud558\ub294 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uba3c\uc800, \ud2b8\ub9ac\uc5d0\uc11c 80\uc744 \ucc3e\uc544\uc57c \ud55c\ub2e4. \uac80\uc0c9\uc740 \uc2e4\ud328\ub85c \ub05d\ub098\uace0, \ub9c8\uc9c0\ub9c9\uc73c\ub85c \uac80\uc0ac\ud55c \ud0a4\ub294 40\uc774\ub2e4. 80\uc744 \uadf8 \ub178\ub4dc\uc758 \uc624\ub978\ucabd \uc790\uc2dd\uc73c\ub85c \uc0bd\uc785\ud558\uba74, \uc774\uc9c4 \uac80\uc0c9\ud2b8\ub9ac\ub294 (b)\uc640 \uac19\uc544\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 1, 2, ..., N \ucd1d N\uac1c\uc758 \ud0a4\ub97c \uac00\uc9c0\uace0 \uc788\ub294 \uc774\uc9c4 \uac80\uc0c9 \ud2b8\ub9ac\ub97c \uc774\uc6a9\ud55c\ub2e4. {1, 2, ..., N}\uc758 \uc21c\uc5f4 a<sub>1<\/sub>, a<sub>2<\/sub>, ..., a<sub>N<\/sub>\uc5d0 \ub300\ud574\uc11c, a<sub>1<\/sub>, a<sub>2<\/sub>, ..., a<sub>N<\/sub>\uc744 \uc21c\uc11c\ub300\ub85c \ube44\uc5b4\uc788\ub294 \uc774\uc9c4 \uac80\uc0c9 \ud2b8\ub9ac\uc5d0 \uc0bd\uc785\ud55c\ub2e4\uba74, \uc774\uc9c4 \uac80\uc0c9 \ud2b8\ub9ac\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc21c\uc5f4 2 1 4 3 5\ub294 \uadf8\ub9bc (c)\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \ub610, 2 4 3 1 5\ub3c4 \uac19\uc740 \ud2b8\ub9ac\ub97c \ub9cc\ub4e0\ub2e4. {1, 2, 3, 4, 5}\uc758 \uc21c\uc5f4 \uc911\uc5d0\uc11c \uadf8\ub9bc (c)\uc640 \uac19\uc740 \uc774\uc9c4 \uac80\uc0c9 \ud2b8\ub9ac\ub97c \ub9cc\ub4dc\ub294 \uc21c\uc5f4\uc758 \uac1c\uc218\ub294 8\uac1c\uc774\ub2e4.<\/p>\r\n\r\n<p>N\uacfc \uc21c\uc5f4 P\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uadf8 \uc21c\uc5f4\uacfc \uac19\uc740 \uc774\uc9c4 \uac80\uc0c9 \ud2b8\ub9ac\ub97c \ub9cc\ub4dc\ub294 \uc21c\uc5f4\uc758 \uac1c\uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 T (&le; 40,320)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \ud0a4\uc758 \uc218 N\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; N &le; 20) \ub2e4\uc74c \uc904\uc5d0\ub294 \uae38\uc774\uac00 N\uc778 \uc21c\uc5f4\uc774 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \uc21c\uc5f4\uacfc \uac19\uc740 \ud2b8\ub9ac\ub97c \ub9cc\ub4dc\ub294 \uc21c\uc5f4\uc758 \uac1c\uc218\ub97c 9,999,991\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"8916","problem_lang":"1","title":"Binary Search Tree","description":"<p>A binary search tree is a binary tree. It may be empty. If it is not empty, it satisfies the following properties:<\/p>\r\n\r\n<p>Every node has a key, and no two nodes have the same key.<br \/>\r\nThe keys in a nonempty left subtree must be smaller than the key in the root of the subtree.<br \/>\r\nThe keys in a nonempty right subtree must be larger than the key in the root of the subtree.<br \/>\r\nThe left and right subtrees are also binary search trees.<\/p>\r\n\r\n<p>Sample binary search trees are shown in Figure 1.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/treeperm.png\" style=\"height:224px; width:526px\" \/><\/p>\r\n\r\n<p>Figure 1. binary search trees<\/p>\r\n\r\n<p>To search for a node with a key k in a binary search tree T , we begin at the root. If T is empty, T contains no keys and the search is unsuccessful. Otherwise, we compare k with the key in root. If k equals root&rsquo;s key, then the search terminates successfully. If k is less than root&rsquo;s key, we search the left subtree of the root. If kis larger than root&rsquo;s key, we search the right subtree of the root. In the same way, we can proceed the search in the left or right subree of T .&nbsp;<\/p>\r\n\r\n<p>To insert a new key k into a binary search tree T where k is different from those of existing keys in T , we first search the tree T . The search will be unsuccessful, then we insert the key at the point the search terminated. For instance, to insert a key 80 into the Figure 1(a), we first search the tree for 80. This search terminates unsuccessfully, and the last node examined has key 40. We insert a new node containing 80 as the right child of the node. The resulting search tree is shown in Figure 1(b).<\/p>\r\n\r\n<p>In this problem, we consider binary search trees with N keys 1, 2, ... , N . For a permutation a<sub>1<\/sub> a<sub>2<\/sub> ... a<sub>N<\/sub> of {1, 2, ... , N }, inserting a<sub>1<\/sub> a<sub>2<\/sub> ... a<sub>N<\/sub> successively into an initially empty binary search tree will produce a binary search tree. For instance, the permutation 2 1 4 3 5 will produce the tree in Figure 1(c). Also, 2 4 3 1 5 will produce the same tree. Actually, 8 permutations among all possible permutations of {1, 2, 3, 4, 5} will produce the same tree to the tree in Figure 1(c).&nbsp;<\/p>\r\n\r\n<p>We are interested in finding the number of permutations of {1, 2, &hellip; , N } such that all those permutations produce a binary search tree identical to the tree produced by a given permutation P . Given N and P , you are to write a program that calculates the number of permutations satisfying the above condition.<\/p>\r\n","input":"<p>Your program is to read from standard input. The input consists of T test cases. The number of test cases T is given in the first line. Each test case starts with a line containing an integer N representing the number of keys, 1 &le; N &le; 20 . In the next line, a permutation of length N is given. There is a single space between the integers representing keys in the permutation<\/p>\r\n","output":"<p>Your program is to write to standard output. Print exactly one line for each test case as follows: Let B be the number of permutations that produce the binary search tree identical to the tree produced by the input permutation. Print B mod 9,999,991 for each test case. For example, if B = 20,000,000 , the output should be 18 for that test case.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

ACM-ICPC > Regionals > Asia > Korea > Asia Regional - Daejeon 2010 E번