시간 제한메모리 제한제출정답맞힌 사람정답 비율
2 초 128 MB114739627832.251%

문제

사과나무는 나무(tree)의 일종으로, 각각의 정점에 정확히 한 개의 사과가 있고, 모든 내부 정점(자식이 있는 정점)이 최소 두 개의 자식을 갖는 나무이다. 예를 들면 아래의 그림은 사과나무의 예이다. 나무같이 보이기 위해서 그림은 루트를 아래에 그린다.

이러한 사과나무에 서식하는 벌레를 생각해 보자. 이 벌레는 이 사과나무의 루트에서 DFS 순서로 탐색을 하게 된다. 자식이 여러 개인 경우에는 (뒤집혀진 그림에서) 왼쪽을 먼저 방문하게 된다. 이러한 탐색을 하면서, 새로운 노드를 방문할 때 0을, 모든 자식 노드를 방문하고 리턴할 때 1을 나열하면 하나의 이진 수열이 된다. 위의 그림으로 이진 수열을 만들면 다음과 같다.

0 0 0 1 0 1 1 0 1 1
a b c   d     e    
      c   d b   e a

이진수의 각 숫자들은 그 숫자가 0이든 1이든 하나의 정점에 대응되게 된다. 즉 0의 경우에는 새로 방문되는 정점에 대응되고, 1의 경우에는 리턴하기 전에 있었던 정점에 대응된다. 위의 표에서는 각 숫자에 대응되는 정점도 표시하였다.

이러한 사과나무에서 썩은 사과가 발견된 경우에는 가지를 잘라 내어야 한다. 만약 우리가 어떤 정점을 제거하면, 그 정점과 그 자손 정점들이 모두 제거되게 된다. 위의 예에서 b를 제거하면 b, c, d가 모두 제거되게 된다.

만약 한 개의 사과가 썩은 경우라면 그 사과를 제거하면 되지만, 두 개의 사과가 썩은 경우라면 문제가 복잡해진다. 사과나무의 성질을 유지하기 위해서, 우리는 오직 한 개의 사과만 제거할 수 있다. 이 경우 루트를 제거하면 되지만, 루트를 제거하게 되면 멀쩡한 사과들을 많이 잃게 된다(제거되는 것은 잃는 것). 따라서 우리는 한 개의 사과를 제거하되, 이를 통해 두 개(이하)의 썩은 사과를 함께 제거하고, 그러면서도 가장 적은 개수의 멀쩡한 사과를 잃도록 잘라야 한다. 위의 예에서 c, d의 사과가 썩은 경우에는 b를 제거하면 된다.

사과나무에 대한 정보가 주어졌을 때, 제거해야 하는 사과를 알아내는 프로그램을 작성하시오.

입력

첫째 줄에 정점의 개수 N(1 ≤ N ≤ 2,000)이 주어진다. 둘째 줄에는 벌레가 만드는 2×N자리의 이진수가 주어진다. 셋째 줄에는 썩은 사과의 위치를 나타내는 두 정수 X, Y가 주어진다. 이는 2×N자리의 이진수에서 X번째의 숫자에 대응되는 정점과, Y번째 숫자에 대응되는 정점에 있는 사과가 썩었다는 의미이다. 이때 두 정점이 서로 같을 수도 있다.

출력

첫째 줄에 제거해야 할 사과를 나타내는 두 정수 i, j를 출력한다. 제거해야 할 사과를 Z라고 했을 때, 이는 Z를 방문할 때의 0의 위치와 Z에서 리턴될 때의 1의 위치가 이진수에서 각각 i, j 번째임을 나타낸다.

예제 입력 1

5
0001011011
4 5

예제 출력 1

2 7
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style=\"width:3%;\">0<\/th>\r\n\t\t\t<th style=\"width:3%;\">1<\/th>\r\n\t\t\t<th style=\"width:3%;\">0<\/th>\r\n\t\t\t<th style=\"width:3%;\">1<\/th>\r\n\t\t\t<th style=\"width:3%;\">1<\/th>\r\n\t\t\t<th style=\"width:3%;\">0<\/th>\r\n\t\t\t<th style=\"width:3%;\">1<\/th>\r\n\t\t\t<th style=\"width:3%;\">1<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td>a<\/td>\r\n\t\t\t<td>b<\/td>\r\n\t\t\t<td>c<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>d<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>e<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>c<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>d<\/td>\r\n\t\t\t<td>b<\/td>\r\n\t\t\t<td>&nbsp;<\/td>\r\n\t\t\t<td>e<\/td>\r\n\t\t\t<td>a<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>\uc774\uc9c4\uc218\uc758 \uac01 \uc22b\uc790\ub4e4\uc740 \uadf8 \uc22b\uc790\uac00 0\uc774\ub4e0 1\uc774\ub4e0 \ud558\ub098\uc758 \uc815\uc810\uc5d0 \ub300\uc751\ub418\uac8c \ub41c\ub2e4. \uc989 0\uc758 \uacbd\uc6b0\uc5d0\ub294 \uc0c8\ub85c \ubc29\ubb38\ub418\ub294 \uc815\uc810\uc5d0 \ub300\uc751\ub418\uace0, 1\uc758 \uacbd\uc6b0\uc5d0\ub294 \ub9ac\ud134\ud558\uae30 \uc804\uc5d0 \uc788\uc5c8\ub358 \uc815\uc810\uc5d0 \ub300\uc751\ub41c\ub2e4. \uc704\uc758 \ud45c\uc5d0\uc11c\ub294 \uac01 \uc22b\uc790\uc5d0 \ub300\uc751\ub418\ub294 \uc815\uc810\ub3c4 \ud45c\uc2dc\ud558\uc600\ub2e4.<\/p>\r\n\r\n<p>\uc774\ub7ec\ud55c \uc0ac\uacfc\ub098\ubb34\uc5d0\uc11c \uc369\uc740 \uc0ac\uacfc\uac00 \ubc1c\uacac\ub41c \uacbd\uc6b0\uc5d0\ub294 \uac00\uc9c0\ub97c \uc798\ub77c \ub0b4\uc5b4\uc57c \ud55c\ub2e4. \ub9cc\uc57d \uc6b0\ub9ac\uac00 \uc5b4\ub5a4 \uc815\uc810\uc744 \uc81c\uac70\ud558\uba74, \uadf8 \uc815\uc810\uacfc \uadf8 \uc790\uc190 \uc815\uc810\ub4e4\uc774 \ubaa8\ub450 \uc81c\uac70\ub418\uac8c \ub41c\ub2e4. \uc704\uc758 \uc608\uc5d0\uc11c b\ub97c \uc81c\uac70\ud558\uba74 b, c, d\uac00 \ubaa8\ub450 \uc81c\uac70\ub418\uac8c \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d \ud55c \uac1c\uc758 \uc0ac\uacfc\uac00 \uc369\uc740 \uacbd\uc6b0\ub77c\uba74 \uadf8 \uc0ac\uacfc\ub97c \uc81c\uac70\ud558\uba74 \ub418\uc9c0\ub9cc, \ub450 \uac1c\uc758 \uc0ac\uacfc\uac00 \uc369\uc740 \uacbd\uc6b0\ub77c\uba74 \ubb38\uc81c\uac00 \ubcf5\uc7a1\ud574\uc9c4\ub2e4. \uc0ac\uacfc\ub098\ubb34\uc758 \uc131\uc9c8\uc744 \uc720\uc9c0\ud558\uae30 \uc704\ud574\uc11c, \uc6b0\ub9ac\ub294 \uc624\uc9c1 \ud55c \uac1c\uc758 \uc0ac\uacfc\ub9cc \uc81c\uac70\ud560 \uc218 \uc788\ub2e4. \uc774 \uacbd\uc6b0 \ub8e8\ud2b8\ub97c \uc81c\uac70\ud558\uba74 \ub418\uc9c0\ub9cc, \ub8e8\ud2b8\ub97c \uc81c\uac70\ud558\uac8c \ub418\uba74 \uba40\uca61\ud55c \uc0ac\uacfc\ub4e4\uc744 \ub9ce\uc774 \uc783\uac8c \ub41c\ub2e4(\uc81c\uac70\ub418\ub294 \uac83\uc740 \uc783\ub294 \uac83). \ub530\ub77c\uc11c \uc6b0\ub9ac\ub294 \ud55c \uac1c\uc758 \uc0ac\uacfc\ub97c \uc81c\uac70\ud558\ub418, \uc774\ub97c \ud1b5\ud574 \ub450 \uac1c(\uc774\ud558)\uc758 \uc369\uc740 \uc0ac\uacfc\ub97c \ud568\uaed8 \uc81c\uac70\ud558\uace0, \uadf8\ub7ec\uba74\uc11c\ub3c4 \uac00\uc7a5 \uc801\uc740 \uac1c\uc218\uc758 \uba40\uca61\ud55c \uc0ac\uacfc\ub97c \uc783\ub3c4\ub85d \uc798\ub77c\uc57c \ud55c\ub2e4. \uc704\uc758 \uc608\uc5d0\uc11c c, d\uc758 \uc0ac\uacfc\uac00 \uc369\uc740 \uacbd\uc6b0\uc5d0\ub294 b\ub97c \uc81c\uac70\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\uc0ac\uacfc\ub098\ubb34\uc5d0 \ub300\ud55c \uc815\ubcf4\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc81c\uac70\ud574\uc57c \ud558\ub294 \uc0ac\uacfc\ub97c \uc54c\uc544\ub0b4\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \uc815\uc810\uc758 \uac1c\uc218 N(1 &le; N &le; 2,000)\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ub458\uc9f8 \uc904\uc5d0\ub294 \ubc8c\ub808\uac00 \ub9cc\ub4dc\ub294 2&times;N\uc790\ub9ac\uc758 \uc774\uc9c4\uc218\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uc14b\uc9f8 \uc904\uc5d0\ub294 \uc369\uc740 \uc0ac\uacfc\uc758 \uc704\uce58\ub97c \ub098\ud0c0\ub0b4\ub294 \ub450 \uc815\uc218 X, Y\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\ub294 2&times;N\uc790\ub9ac\uc758 \uc774\uc9c4\uc218\uc5d0\uc11c X\ubc88\uc9f8\uc758 \uc22b\uc790\uc5d0 \ub300\uc751\ub418\ub294 \uc815\uc810\uacfc, Y\ubc88\uc9f8 \uc22b\uc790\uc5d0 \ub300\uc751\ub418\ub294 \uc815\uc810\uc5d0 \uc788\ub294 \uc0ac\uacfc\uac00 \uc369\uc5c8\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. \uc774\ub54c \ub450 \uc815\uc810\uc774 \uc11c\ub85c \uac19\uc744 \uc218\ub3c4 \uc788\ub2e4.<\/p>\r\n","output":"<p>\uccab\uc9f8 \uc904\uc5d0 \uc81c\uac70\ud574\uc57c \ud560 \uc0ac\uacfc\ub97c \ub098\ud0c0\ub0b4\ub294 \ub450 \uc815\uc218 i, j\ub97c \ucd9c\ub825\ud55c\ub2e4. \uc81c\uac70\ud574\uc57c \ud560 \uc0ac\uacfc\ub97c Z\ub77c\uace0 \ud588\uc744 \ub54c, \uc774\ub294 Z\ub97c \ubc29\ubb38\ud560 \ub54c\uc758 0\uc758 \uc704\uce58\uc640 Z\uc5d0\uc11c \ub9ac\ud134\ub420 \ub54c\uc758 1\uc758 \uc704\uce58\uac00 \uc774\uc9c4\uc218\uc5d0\uc11c \uac01\uac01 i, j \ubc88\uc9f8\uc784\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"2233","problem_lang":"1","title":"Apple Tree","description":"<p>Figure 1 shows an apple tree in which each node has exactly one apple and every internal node has at least two children nodes.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/appletree1.png\" style=\"height:250px; width:320px\" \/><\/p>\r\n\r\n<p>Figure 1<\/p>\r\n\r\n<p>Consider a worm in the apple tree. The worm visits apples (i.e., nodes) following the order of depth-first traversal from the root of the apple tree. (Notice that, the branching priority of an internal node is left-first order.) Figure 2 shows a binary representation of the depth-first traversal by a worm in the apple tree.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/appletree2.png\" style=\"height:176px; width:638px\" \/><\/p>\r\n\r\n<p>The binary representation is defined as follows: During the depth-first traversal,<\/p>\r\n\r\n<ul>\r\n\t<li>&ldquo;0&rdquo; is printed, if a new node is visited, and<\/li>\r\n\t<li>&ldquo;1&rdquo; is printed, if a node is returned after visiting all descendent nodes.<\/li>\r\n<\/ul>\r\n\r\n<p>The second line of the table shows the first visited node corresponding to each &ldquo;0&rdquo; and the third line shows the returned node corresponding to each &ldquo;1&rdquo;. For example, the apple tree in Figure 1 can be represented by the definition above.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/appletree3.png\" style=\"height:72px; width:556px\" \/><\/p>\r\n\r\n<p>If there are wormy apples in the apple tree, we should remove them. When we cut a node z, every apple in the subtree rooted at node z (including the apple of node z) is going down. For example, apples k, l, m, and n are going down by cutting node k and apples b, c, d, e, f, g and h are going down by cutting node b in the apple tree in Figure 1. In order to remove wormy apples, we should find the subtree rooted at a node in which every wormy apple exists, and then cut the node.<\/p>\r\n\r\n<p>Given two wormy apples (possibly same) in an apple tree, the problem is to find a node satisfying that two wormy apples must drop and the minimum number of normal apples drop simultaneously by cutting the node. For example, suppose that nodes c and f are given as wormy apples in the apple tree in Figure 1. By cutting nodes a or b , two wormy apples can be removed. However, the smaller number of normal apples drop by cutting node b than the case that we cut node a. So, we want to find the node b.<\/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 of the input. Each test case is composed of three lines. In first line, the number of nodes N ( 3 &le; N &le; 2000 ) is given. In next second line, 2N binary representation of an apple tree is given. There is no space between each representation. In last third line, two indices A and B ( 1 &le; A, B &le; 2N ) corresponding to two wormy apples in the binary representation are given. These two indices are not corresponding to node directly but index of binary representation. Notice that, the value of each index can be &ldquo;0&rdquo; or &ldquo;1&rdquo;.<\/p>\r\n","output":"<p>Your program is to write to standard output. Print exactly one line for each test case. For each test case, find a node satisfying that the wormy apples must be removed and the minimum number of normal apples are removed simultaneously by cutting the node, and then print an index of &ldquo;0&rdquo; where the node was first visited and the other index of &ldquo;1&rdquo; where the node was returned in the binary representation. Print exactly one line for each test case index of &ldquo;0&rdquo; first, and index of &ldquo;1&rdquo; later. And there must be a single space between two indices.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]