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문제

상근이는 DNA 수열을 연구하는 컴퓨터 과학자이다. 오늘은 두 문자열의 최대 공통 부분 수열을 구해보려고 한다.

알파벳 Σ으로 이루어진 단어 w = a1a2...ar (ai ∈ Σ for i = 1,2,...r)이 있다. w의 부분 수열은 1 ≤ i1 < i2 < .... < is ≤ r을 만족하는 x=ai1ai2...ais 이다. 모든 j = 1,2,...,s-1에 대해서 ij+1 = ij + 1을 만족하는 부분 수열 x를 w의 세그먼트라고 한다. 예를 들어, lovely는 ove의 세그먼트이다. 하지만, loly는 lovely의 부분 수열이지만, 세그먼트는 아니다.

두 단어 w1과 w2의 부분 수열이 되는 단어를 공통 부분 수열이라고 한다. 최대 공통 부분 수열은 w1와 w2의 공통 수열 중에서 길이가 가장 긴 것을 말한다. 예를 들어, w1 = lovxxelyxxxxx, w2 = xxxxxxxlovely인 경우에 w3 = lovely와 w4 = xxxxxxx는 w1와 w2의 공통 부분 수열이다. w4는 길이가 7이며, 두 단어 w1과 w2의 최대 공통 부분 수열이 된다. 길이가 0인 빈 단어는 항상 공통 부분 수열이다.

상근이는 부분 수열이 반드시 길이가 K이상인 공통 세그먼트를 포함해야 한다는 조건을 추가했다. 예를 들어, K=3인 경우에 lovxxelyxxxxx와 xxxxxxxlovely의 공통 부분 수열로 lovely는 가능하지만, xxxxxxx는 가능하지 않다. xxxxxxx에는 길이가 3 이상인 공통 세그먼트가 포함되어 있지 않기 때문이다.

두 단어가 주어졌을 때, 길이가 K 이상인 공통 세그먼트를 포함하는 최대 공통 부분 수열의 길이를 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다. 각 테스트 케이스의 첫째 줄에는 K가 주어진다. (1 ≤ K ≤ 100) 다음 두 줄에는 알파벳 소문자로만 이루어진 두 문자열이 주어진다. 문자열의 길이는 1보다 크거나 같으며, 103을 넘지 않는다.

입력의 마지막 줄에는 0이 하나 주어진다.

출력

각 테스트 케이스마다 길이가 K 이상인 공통 세그먼트를 포함한 최대 공통 부분 수열의 길이를 출력한다. 만약, 0보다 큰 공통 수열이 없는 경우에는 0을 출력한다.

예제 입력 1

3
lovxxelyxxxxx
xxxxxxxlovely
1
lovxxelyxxxxx
xxxxxxxlovely
3
lovxxxelxyxxxx
xxxlovelyxxxxxxx
4
lovxxxelyxxx
xxxxxxlovely
0

예제 출력 1

6
7
10
0
[{"problem_id":"5722","problem_lang":"0","title":"DNA \ubd80\ubd84 \uc218\uc5f4","description":"<p>\uc0c1\uadfc\uc774\ub294 DNA \uc218\uc5f4\uc744 \uc5f0\uad6c\ud558\ub294 \ucef4\ud4e8\ud130 \uacfc\ud559\uc790\uc774\ub2e4. \uc624\ub298\uc740 \ub450 \ubb38\uc790\uc5f4\uc758 \ucd5c\ub300 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc744 \uad6c\ud574\ubcf4\ub824\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc54c\ud30c\ubcb3 &Sigma;\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ub2e8\uc5b4 w = a<sub>1<\/sub>a<sub>2<\/sub>...a<sub>r<\/sub> (a<sub>i<\/sub> &isin; &Sigma; for i = 1,2,...r)\uc774 \uc788\ub2e4. w\uc758 \ubd80\ubd84 \uc218\uc5f4\uc740 1 &le; i<sub>1<\/sub> &lt; i<sub>2<\/sub> &lt; .... &lt; i<sub>s<\/sub> &le; r\uc744 \ub9cc\uc871\ud558\ub294 x=a<sub>i<sub>1<\/sub><\/sub>a<sub>i<sub>2<\/sub><\/sub>...a<sub>i<sub>s<\/sub><\/sub> \uc774\ub2e4. \ubaa8\ub4e0 j = 1,2,...,s-1\uc5d0 \ub300\ud574\uc11c i<sub>j+1<\/sub> = i<sub>j<\/sub> + 1\uc744 \ub9cc\uc871\ud558\ub294 \ubd80\ubd84 \uc218\uc5f4 x\ub97c w\uc758 \uc138\uadf8\uba3c\ud2b8\ub77c\uace0 \ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, lovely\ub294 ove\uc758 \uc138\uadf8\uba3c\ud2b8\uc774\ub2e4. \ud558\uc9c0\ub9cc, loly\ub294 lovely\uc758 \ubd80\ubd84 \uc218\uc5f4\uc774\uc9c0\ub9cc, \uc138\uadf8\uba3c\ud2b8\ub294 \uc544\ub2c8\ub2e4.<\/p>\r\n\r\n<p>\ub450 \ub2e8\uc5b4 w<sub>1<\/sub>\uacfc w<sub>2<\/sub>\uc758 \ubd80\ubd84 \uc218\uc5f4\uc774 \ub418\ub294 \ub2e8\uc5b4\ub97c \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc774\ub77c\uace0 \ud55c\ub2e4. \ucd5c\ub300 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc740 w<sub>1<\/sub>\uc640 w<sub>2<\/sub>\uc758 \uacf5\ud1b5 \uc218\uc5f4 \uc911\uc5d0\uc11c \uae38\uc774\uac00 \uac00\uc7a5 \uae34 \uac83\uc744 \ub9d0\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, w<sub>1<\/sub> = lovxxelyxxxxx, w<sub>2<\/sub> = xxxxxxxlovely\uc778 \uacbd\uc6b0\uc5d0 w<sub>3<\/sub> = lovely\uc640 w<sub>4<\/sub> = xxxxxxx\ub294 w<sub>1<\/sub>\uc640 w<sub>2<\/sub>\uc758 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc774\ub2e4. w<sub>4<\/sub>\ub294 \uae38\uc774\uac00 7\uc774\uba70, \ub450 \ub2e8\uc5b4 w<sub>1<\/sub>\uacfc w<sub>2<\/sub>\uc758 \ucd5c\ub300 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc774 \ub41c\ub2e4. \uae38\uc774\uac00 0\uc778 \ube48 \ub2e8\uc5b4\ub294 \ud56d\uc0c1 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc0c1\uadfc\uc774\ub294 \ubd80\ubd84 \uc218\uc5f4\uc774 \ubc18\ub4dc\uc2dc \uae38\uc774\uac00 K\uc774\uc0c1\uc778 \uacf5\ud1b5 \uc138\uadf8\uba3c\ud2b8\ub97c \ud3ec\ud568\ud574\uc57c \ud55c\ub2e4\ub294 \uc870\uac74\uc744 \ucd94\uac00\ud588\ub2e4. \uc608\ub97c \ub4e4\uc5b4, K=3\uc778 \uacbd\uc6b0\uc5d0 lovxxelyxxxxx\uc640 xxxxxxxlovely\uc758 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\ub85c lovely\ub294 \uac00\ub2a5\ud558\uc9c0\ub9cc, xxxxxxx\ub294 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4. xxxxxxx\uc5d0\ub294 \uae38\uc774\uac00 3 \uc774\uc0c1\uc778 \uacf5\ud1b5 \uc138\uadf8\uba3c\ud2b8\uac00 \ud3ec\ud568\ub418\uc5b4 \uc788\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub450 \ub2e8\uc5b4\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uae38\uc774\uac00 K \uc774\uc0c1\uc778 \uacf5\ud1b5 \uc138\uadf8\uba3c\ud2b8\ub97c \ud3ec\ud568\ud558\ub294 \ucd5c\ub300 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc758 \uae38\uc774\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 K\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; K &le; 100) \ub2e4\uc74c \ub450 \uc904\uc5d0\ub294 \uc54c\ud30c\ubcb3 \uc18c\ubb38\uc790\ub85c\ub9cc \uc774\ub8e8\uc5b4\uc9c4 \ub450 \ubb38\uc790\uc5f4\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ubb38\uc790\uc5f4\uc758 \uae38\uc774\ub294 1\ubcf4\ub2e4 \ud06c\uac70\ub098 \uac19\uc73c\uba70, 10<sup>3<\/sup>\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n\r\n<p>\uc785\ub825\uc758 \ub9c8\uc9c0\ub9c9 \uc904\uc5d0\ub294 0\uc774 \ud558\ub098 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \uae38\uc774\uac00 K \uc774\uc0c1\uc778 \uacf5\ud1b5 \uc138\uadf8\uba3c\ud2b8\ub97c \ud3ec\ud568\ud55c \ucd5c\ub300 \uacf5\ud1b5 \ubd80\ubd84 \uc218\uc5f4\uc758 \uae38\uc774\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d, 0\ubcf4\ub2e4 \ud070 \uacf5\ud1b5 \uc218\uc5f4\uc774 \uc5c6\ub294 \uacbd\uc6b0\uc5d0\ub294 0\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"5722","problem_lang":"1","title":"DNA Subsequences","description":"<p>Thomas, a computer scientist that works with DNA sequences, needs to compute longest common subsequences of given pairs of strings. Consider an alphabet &Sigma; of letters and a word w = a<sub>1<\/sub>a<sub>2<\/sub>...a<sub>r<\/sub>, where a<sub>i<\/sub> &isin; &Sigma;, foor i = 1, 2, ... , r. A subsequence of w is a word x = a<sub>i<sub>1<\/sub><\/sub>a<sub>i<sub>2<\/sub><\/sub>...a<sub>i<sub>s<\/sub><\/sub> such that 1 &le; i<sub>1<\/sub>&nbsp;&lt; i<sub>2<\/sub>&nbsp;&lt; .... &lt; i<sub>s<\/sub>&nbsp;&le; r. Subsequence x is a segment of w if&nbsp;i<sub>j+1<\/sub>&nbsp;= i<sub>j<\/sub>&nbsp;+ 1, for j = 1, 2, ..., s &minus; 1. For example the word ove is a segment of the word lovely, whereas the word loly is a subsequence of lovely, but not a segment.<\/p>\r\n\r\n<p>A word is a common subsequence of two words w<sub>1<\/sub> and w<sub>2<\/sub> if it is a subsequence of each of the two words. A longest common subsequence of w<sub>1<\/sub> and w<sub>2<\/sub> is a common subsequence of w<sub>1<\/sub> and w<sub>2<\/sub> having the largest possible length. For example, consider the words w<sub>1<\/sub> = lovxxelyxxxxx and w<sub>2<\/sub> = xxxxxxxlovely. The words w<sub>3<\/sub> =lovely and w<sub>4<\/sub> = xxxxxxx, the latter of length 7, are both common subsequences of w<sub>1<\/sub> and w<sub>2<\/sub>. In fact, w<sub>4<\/sub> is their longest common subsequence. Notice that the empty word, of length zero, is always a common subsequence, although not necessarily the longest.<\/p>\r\n\r\n<p>In the case of Thomas, there is an extra requirement: the subsequence must be formed from common segments having length K or more. For example, if Thomas decides that K = 3, then he considers lovely to be an acceptable common subsequence of lovxxelyxxxxx and xxxxxxxlovely, whereas xxxxxxx, which has length 7 and is also a common subsequence, is not acceptable. Can you help Thomas?<\/p>\r\n","input":"<p>The input contains several test cases. The \ufb01rst line of a test case contains an integer K representing the minimum length of common segments, where 1 &le; K &le; 100. The next two lines contain each a string on lowercase letters from the regular alphabet of 26 letters. The length l of each string satis\ufb01es the inequality 1 &le; l &le; 10<sup>3<\/sup>. There are no spaces on any line in the input. The end of the input is indicated by a line containing a zero.<\/p>\r\n","output":"<p>For each test case in the input, your program must print a single line, containing the length of the longest subsequence formed by consecutive segments of length at least K from both strings. If no such common subsequence of length greater than zero exists, then 0 must be printed.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

출처

ICPC > Regionals > Latin America > South America Regional Contests 2008 D번