시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

데이터를 압축하는 방법 중에서 상대적으로 간단한 방법은 허프만 트리를 이용하는 것이다. 허프만 트리를 이용하면 한 파일이 포함하고 있는 데이터를 압축하고 압축 해제하는 작업을 손쉽게 할 수 있다. 많은 응용프로그램은 바이너리 허프만 트리를 사용한다. (각 노드는 리프이거나 두 서브 노드를 갖는 경우) 이 문제에서는 임의의 서브 트리의 개수를 갖는 허프만 트리를 사용한다. (N-ary tree)

파일의 내용에 총 Z개의 서로 다른 글자가 있는 경우에 리프 노드의 개수는 Z개이다. 루트에서 리프로 향하는 경로 상에 써 있는 숫자가 그 글자를 어떻게 인코딩할 것인지를 나타낸다. 노드에 써 있는 숫자는 0부터 N-1까지이다.

자주 사용하는 글자를 루트와 가까운 곳에 놓고, 잘 사용하지 않는 글자를 먼 곳에 놓으면, 압축 효율을 늘릴 수 있다. 즉, 한 파일을 압축하는데 있어서 최소 개수의 N-ary 심볼을 사용한 트리를 허프만 트리라고 한다.

이 문제에서, 사용할 허프만 트리는 각 노드가 내부 노드이거나, 글자를 인코딩하는 리프 노드이다. 아무 글자도 인코딩하지 않는 dangling leave는 존재하지 않는다.

아래 그림은 N=3인 경우의 허프만 트리 예시이다. 'a'와 'e'는 심볼 1개로 인코딩 되며, 덜 사용하지 글자인 's'와 'p'는 심볼 2개, 드물게 사용하는 글자인 'x','q','y'는 세 개로 인코딩한다.

디코딩을 해야 원래 파일로 복원할 수 있기 때문에, 한 데이터를 인코딩하는데 사용한 트리를 기억하는 것이 중요하다. 트리를 저장하는 방법에는 여러가지가 있다. 이 문제에서는 모든 Z개의 문자가 사전순으로 하나씩 써있는 파일을 인코딩한 결과를 저장하는 방식으로 트리를 저장할 것이다.

N과 위에서 저장한 트리가 주어졌을 때, 각각의 글자가 어떤 심볼로 인코딩 되는지 구하는 프로그램을 작성하시오.

입력

첫째 줄에 테스트 케이스의 개수 T가 주어진다. 각 테스트 케이스는 아래와 같이 세 줄로 이루어져 있다.

  • 파일에 들어있는 서로 다른 글자의 개수 2 ≤ Z ≤ 20
  • 허프만 트리의 arity를 나타내는 2 ≤ N ≤ 10 (N진 트리로 생각하면 됩니다. N = 2인 경우엔 binary, 3인 경우엔 ternary)
  • 모든 글자가 사전 순으로 한 번씩 나타난 파일을 인코딩 한 결과. 길이는 200을 넘지 않으며, 항상 [0, (N-1)]에 들어있는 숫자로만 이루어져 있다.

Z=5, N=2인 경우에 010011101100은 여러 가지 정답이 나올 수 있다. 하지만, 이 문제에서는 답이 유일한 경우만 입력으로 주어진다.

출력

각각의 테스트 케이스에 대해서, Z줄을 출력해야 한다. 각 줄은 original->encoding 형식이고, original은 글자, encoding은 허프만 트리에 의해서 인코딩된 결과이다. 

예제 입력 1

3
3
2
10011
4
2
000111010
19
10
01234678950515253545556575859

예제 출력 1

0->10
1->0
2->11
0->00
1->011
2->1
3->010
0->0
1->1
2->2
3->3
4->4
5->6
6->7
7->8
8->9
9->50
10->51
11->52
12->53
13->54
14->55
15->56
16->57
17->58
18->59
[{"problem_id":"7904","problem_lang":"0","title":"\ud5c8\ud504\ub9cc \ud2b8\ub9ac","description":"<p>\ub370\uc774\ud130\ub97c \uc555\ucd95\ud558\ub294 \ubc29\ubc95 \uc911\uc5d0\uc11c \uc0c1\ub300\uc801\uc73c\ub85c \uac04\ub2e8\ud55c \ubc29\ubc95\uc740 \ud5c8\ud504\ub9cc \ud2b8\ub9ac\ub97c \uc774\uc6a9\ud558\ub294 \uac83\uc774\ub2e4. \ud5c8\ud504\ub9cc \ud2b8\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \ud55c \ud30c\uc77c\uc774 \ud3ec\ud568\ud558\uace0 \uc788\ub294 \ub370\uc774\ud130\ub97c \uc555\ucd95\ud558\uace0 \uc555\ucd95 \ud574\uc81c\ud558\ub294 \uc791\uc5c5\uc744 \uc190\uc27d\uac8c \ud560 \uc218 \uc788\ub2e4. \ub9ce\uc740 \uc751\uc6a9\ud504\ub85c\uadf8\ub7a8\uc740 \ubc14\uc774\ub108\ub9ac \ud5c8\ud504\ub9cc \ud2b8\ub9ac\ub97c \uc0ac\uc6a9\ud55c\ub2e4. (\uac01 \ub178\ub4dc\ub294 \ub9ac\ud504\uc774\uac70\ub098 \ub450 \uc11c\ube0c \ub178\ub4dc\ub97c \uac16\ub294 \uacbd\uc6b0) \uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 \uc784\uc758\uc758 \uc11c\ube0c \ud2b8\ub9ac\uc758 \uac1c\uc218\ub97c \uac16\ub294 \ud5c8\ud504\ub9cc \ud2b8\ub9ac\ub97c \uc0ac\uc6a9\ud55c\ub2e4. (N-ary tree)<\/p>\r\n\r\n<p>\ud30c\uc77c\uc758 \ub0b4\uc6a9\uc5d0 \ucd1d Z\uac1c\uc758 \uc11c\ub85c \ub2e4\ub978 \uae00\uc790\uac00 \uc788\ub294 \uacbd\uc6b0\uc5d0 \ub9ac\ud504 \ub178\ub4dc\uc758 \uac1c\uc218\ub294 Z\uac1c\uc774\ub2e4. \ub8e8\ud2b8\uc5d0\uc11c \ub9ac\ud504\ub85c \ud5a5\ud558\ub294 \uacbd\ub85c \uc0c1\uc5d0 \uc368 \uc788\ub294 \uc22b\uc790\uac00 \uadf8 \uae00\uc790\ub97c \uc5b4\ub5bb\uac8c \uc778\ucf54\ub529\ud560 \uac83\uc778\uc9c0\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ub178\ub4dc\uc5d0 \uc368 \uc788\ub294 \uc22b\uc790\ub294 0\ubd80\ud130 N-1\uae4c\uc9c0\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uae00\uc790\ub97c \ub8e8\ud2b8\uc640 \uac00\uae4c\uc6b4 \uacf3\uc5d0 \ub193\uace0, \uc798 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\ub294 \uae00\uc790\ub97c \uba3c \uacf3\uc5d0 \ub193\uc73c\uba74, \uc555\ucd95 \ud6a8\uc728\uc744 \ub298\ub9b4 \uc218 \uc788\ub2e4. \uc989, \ud55c \ud30c\uc77c\uc744 \uc555\ucd95\ud558\ub294\ub370 \uc788\uc5b4\uc11c \ucd5c\uc18c \uac1c\uc218\uc758 N-ary \uc2ec\ubcfc\uc744 \uc0ac\uc6a9\ud55c \ud2b8\ub9ac\ub97c \ud5c8\ud504\ub9cc \ud2b8\ub9ac\ub77c\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc774 \ubb38\uc81c\uc5d0\uc11c, \uc0ac\uc6a9\ud560 \ud5c8\ud504\ub9cc \ud2b8\ub9ac\ub294 \uac01 \ub178\ub4dc\uac00 \ub0b4\ubd80 \ub178\ub4dc\uc774\uac70\ub098, \uae00\uc790\ub97c \uc778\ucf54\ub529\ud558\ub294 \ub9ac\ud504 \ub178\ub4dc\uc774\ub2e4. \uc544\ubb34 \uae00\uc790\ub3c4 \uc778\ucf54\ub529\ud558\uc9c0 \uc54a\ub294 dangling leave\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n\r\n<p>\uc544\ub798 \uadf8\ub9bc\uc740 N=3\uc778 \uacbd\uc6b0\uc758 \ud5c8\ud504\ub9cc \ud2b8\ub9ac \uc608\uc2dc\uc774\ub2e4. &#39;a&#39;\uc640 &#39;e&#39;\ub294 \uc2ec\ubcfc 1\uac1c\ub85c \uc778\ucf54\ub529 \ub418\uba70, \ub35c \uc0ac\uc6a9\ud558\uc9c0 \uae00\uc790\uc778 &#39;s&#39;\uc640 &#39;p&#39;\ub294 \uc2ec\ubcfc 2\uac1c, \ub4dc\ubb3c\uac8c \uc0ac\uc6a9\ud558\ub294 \uae00\uc790\uc778 &#39;x&#39;,&#39;q&#39;,&#39;y&#39;\ub294 \uc138 \uac1c\ub85c \uc778\ucf54\ub529\ud55c\ub2e4.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/hmtree.png\" style=\"height:286px; width:291px\" \/><\/p>\r\n\r\n<p>\ub514\ucf54\ub529\uc744 \ud574\uc57c \uc6d0\ub798 \ud30c\uc77c\ub85c \ubcf5\uc6d0\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc5d0, \ud55c \ub370\uc774\ud130\ub97c \uc778\ucf54\ub529\ud558\ub294\ub370 \uc0ac\uc6a9\ud55c \ud2b8\ub9ac\ub97c \uae30\uc5b5\ud558\ub294 \uac83\uc774 \uc911\uc694\ud558\ub2e4. \ud2b8\ub9ac\ub97c \uc800\uc7a5\ud558\ub294 \ubc29\ubc95\uc5d0\ub294 \uc5ec\ub7ec\uac00\uc9c0\uac00 \uc788\ub2e4. \uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 \ubaa8\ub4e0 Z\uac1c\uc758 \ubb38\uc790\uac00 \uc0ac\uc804\uc21c\uc73c\ub85c \ud558\ub098\uc529 \uc368\uc788\ub294 \ud30c\uc77c\uc744 \uc778\ucf54\ub529\ud55c \uacb0\uacfc\ub97c \uc800\uc7a5\ud558\ub294 \ubc29\uc2dd\uc73c\ub85c \ud2b8\ub9ac\ub97c \uc800\uc7a5\ud560 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>N\uacfc \uc704\uc5d0\uc11c \uc800\uc7a5\ud55c \ud2b8\ub9ac\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uac01\uac01\uc758 \uae00\uc790\uac00 \uc5b4\ub5a4 \uc2ec\ubcfc\ub85c \uc778\ucf54\ub529 \ub418\ub294\uc9c0 \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\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \uc544\ub798\uc640 \uac19\uc774 \uc138 \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\ud30c\uc77c\uc5d0 \ub4e4\uc5b4\uc788\ub294 \uc11c\ub85c \ub2e4\ub978 \uae00\uc790\uc758 \uac1c\uc218 2 &le; Z &le; 20<\/li>\r\n\t<li>\ud5c8\ud504\ub9cc \ud2b8\ub9ac\uc758 arity\ub97c \ub098\ud0c0\ub0b4\ub294 2 &le; N &le; 10 (N\uc9c4 \ud2b8\ub9ac\ub85c \uc0dd\uac01\ud558\uba74 \ub429\ub2c8\ub2e4. N = 2\uc778 \uacbd\uc6b0\uc5d4 binary, 3\uc778 \uacbd\uc6b0\uc5d4 ternary)<\/li>\r\n\t<li>\ubaa8\ub4e0 \uae00\uc790\uac00 \uc0ac\uc804 \uc21c\uc73c\ub85c \ud55c \ubc88\uc529 \ub098\ud0c0\ub09c \ud30c\uc77c\uc744 \uc778\ucf54\ub529 \ud55c \uacb0\uacfc. \uae38\uc774\ub294 200\uc744 \ub118\uc9c0 \uc54a\uc73c\uba70, \ud56d\uc0c1 [0, (N-1)]\uc5d0 \ub4e4\uc5b4\uc788\ub294 \uc22b\uc790\ub85c\ub9cc \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>Z=5, N=2\uc778 \uacbd\uc6b0\uc5d0 010011101100\uc740 \uc5ec\ub7ec \uac00\uc9c0 \uc815\ub2f5\uc774 \ub098\uc62c \uc218 \uc788\ub2e4. \ud558\uc9c0\ub9cc, \uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 \ub2f5\uc774 \uc720\uc77c\ud55c \uacbd\uc6b0\ub9cc \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, Z\uc904\uc744 \ucd9c\ub825\ud574\uc57c \ud55c\ub2e4. \uac01 \uc904\uc740 original-&gt;encoding \ud615\uc2dd\uc774\uace0, original\uc740 \uae00\uc790, encoding\uc740 \ud5c8\ud504\ub9cc \ud2b8\ub9ac\uc5d0 \uc758\ud574\uc11c \uc778\ucf54\ub529\ub41c \uacb0\uacfc\uc774\ub2e4.&nbsp;<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"7904","problem_lang":"1","title":"Huffman Trees","description":"<p>A relatively simple method for compressing data works by creating a so-called Huffman tree for a \ufb01le and using it to compress and decompress the data it contains. For most applications, binary Huffman trees are used (i.e., each node is either a leaf or has exactly two sub-nodes). One can, however, construct Huffman trees with an arbitrary number of sub-trees (i.e, ternary or, in general, N-ary trees).<\/p>\r\n\r\n<p>A Huffman tree for a \ufb01le containing Z different characters has Z leaves. The path from the root to a leaf that represents a certain character determines its encoding; each step towards the leaf determines an encoding character (which can be 0, 1, ..., (N 1)). By placing often-occurring characters closer to the root, and less often occurring characters further away from the root, the desirable compression is achieved. Strictly speaking, such a tree is a Huffman tree only if the resulting encoding takes the minimal number ofN-ary symbols to encode the complete \ufb01le.<\/p>\r\n\r\n<p>For this problem, we only consider trees where each node is either an internal node, or a leaf encoding a character; there are no dangling leavesthat do not encode for a character.<\/p>\r\n\r\n<p>The \ufb01gure below shows a sample ternary Huffman tree; the characters &rsquo;a&rsquo; and &rsquo;e&rsquo; are encoded using one ternary symbol; the less frequently occurring characters &rsquo;s&rsquo; and &rsquo;p&rsquo; are encoded using two ternary symbols; and the very rare symbols &rsquo;x&rsquo;, &rsquo;q&rsquo; and &rsquo;y&rsquo; are encoded using three ternary symbols each.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/hmtree.png\" style=\"height:286px; width:291px\" \/><\/p>\r\n\r\n<p>Of course, if we want to decompress a list of N-ary symbols later on, it is important to know which tree was used to compress the data. This can be done in several ways. In this problem we use the following method: the stream of data will be preceded by a header consisting of the encoded versions of the Z characters occurring in the original \ufb01le, in lexicographical order.<\/p>\r\n\r\n<p>Given the number of source symbols Z, a value N denoting the &rsquo;arity&rsquo; of the Huffman tree, and a header, give the mapping from source symbols to encoded symbols.<\/p>\r\n","input":"<p>The input starts with a single integer T on a separate line, denoting the number of test cases that&nbsp;follow. Next, for each of the T test cases, three lines follow:<\/p>\r\n\r\n<ul>\r\n\t<li>The number of different characters in the \ufb01le (2 &le; Z &le; 20);<\/li>\r\n\t<li>The number N, denoting the &rsquo;arity&rsquo; of the Huffman tree (2 &le; N &le; 10);<\/li>\r\n\t<li>A string representing the header of the received message; each character will be a digit in the range [0, N-1]. This string will contain less than 200 characters.<\/li>\r\n<\/ul>\r\n\r\n<p>Note: Although rare, it is possible for a header to have multiple interpretations (e.g., the header &rsquo;010011101100&rsquo; with Z = 5 and N = 2). It is guaranteed that all cases in the test input have a unique solution.<\/p>\r\n","output":"<p>For each of the T test-cases, print Z lines giving the encoded version of each of the Z characters in the testcase in ascending order. Use the format original-&gt;encoding, where original is a decimal value in the range [0, (Z - 1)], and encoding is the string of encoding digits for this symbol (each digit is &ge; 0 and &lt; N).<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]