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문제

n개의 행렬 M1, M2, ..., Mn 이 주어진다. 행렬 Mi (i = 1, 2, ..., n)은 R[i]개의 행, C[i]개의 열을 가지고 있다 (R[i] x C[i]).

정수 i(1 ≤ i ≤ n)에 대해서, 아래와 같이 정의되는 값 Vi를 계산하려고 한다.

먼저, i개의 행렬 M1, M2, ..., Mi을 모두 곱할 수 있는 순서가 있는지 알아본다. 참고로 행렬 곱셈 연산에서 RxC 행렬과 R'xC' 행렬을 곱하려면 C = R'이 성립하여야 하고 결과로 나오는 행렬은 RxC'이다. 만약 i개의 행렬을 (순서를 자유롭게 재배치 하더라도) 곱하는 것이 불가능하다면 Vi = 0 으로 정의된다. 만약 가능하다면, 가능한 모든 방법 중 최종 결과가 되는 행렬의 크기가 가장 커지는 방법을 찾아 그 때 결과로 나온 행렬의 크기 (행의 수 x 열의 수)가 Vi 값이 된다.

V1, V2, ..., Vn를 구해보자.

입력

첫째 줄에는 자연수 n이 주어진다 (1 ≤ n ≤ 1,000). 다음 n줄에는 각 줄에 정수 두 개가 주어지는데, 각 행렬의 행의 수 R[i]와 열의 수 C[i]이다. 각 행렬의 행의 수와 열의 수는 1 이상 1,000 이하이다.

출력

총 n줄을 출력해야 하고, 각 줄에는 V1 부터 Vn까지 Vi 값을 출력한다.

서브태스크 1 (5점)

  • 1 ≤ n ≤ 10
  • 1 ≤ R, C ≤ 1,000

서브태스크 2 (10점)

  • 1 ≤ n ≤ 1,000
  • 1 ≤ R, C ≤ 1,000

예제 입력 1

3
8 2
2 8
2 2

예제 출력 1

16
64
64

i = 1 일 때에는 8x2 행렬 (M1) 하나만 이용하여 원소의 수가 총 16개인 행렬을 만들 수 있다.

i = 2 일 때에는 M1 x M2 를 하여 8x8 행렬을 만들 수 있고, 원소의 수는 총 64개이다. 혹은 M2 x M1 을 하여 2x2 행렬을 만들 수 있지만, 원소의 수가 최대가 되는 방법은 상기한 방법이다.

i = 3 일 때에는 M1 x M3 x M2를 통해 8x8 행렬을 만들 수 있다.

예제 입력 2

3
4 9
9 1
9 9

예제 출력 2

36
4
4

예제 입력 3

3
4 10
10 1
10 6

예제 출력 3

40
4
0

i = 3일 때 어떤 순서로 3개의 행렬을 배치하여 곱하더라도 행렬 곱셈 연산을 완료할 수 없으므로 답이 0이다.

예제 입력 4

3
10 3
3 10
5 5

예제 출력 4

30
100
0

예제 입력 5

4
1 1
1 1
1 1
1 1

예제 출력 5

1
1
1
1
[{"problem_id":"15933","problem_lang":"0","title":"\ud589\ub82c \uacf1\uc148","description":"<p>n\uac1c\uc758 \ud589\ub82c M<sub>1<\/sub>, M<sub>2<\/sub>, ..., M<sub>n<\/sub> \uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ud589\ub82c M<sub>i<\/sub> (i = 1, 2, ..., n)\uc740 R[i]\uac1c\uc758 \ud589, C[i]\uac1c\uc758 \uc5f4\uc744 \uac00\uc9c0\uace0 \uc788\ub2e4 (R[i] x C[i]).<\/p>\r\n\r\n<p>\uc815\uc218 i(1 &le; i &le; n)\uc5d0 \ub300\ud574\uc11c, \uc544\ub798\uc640 \uac19\uc774 \uc815\uc758\ub418\ub294 \uac12 V<sub>i<\/sub>\ub97c \uacc4\uc0b0\ud558\ub824\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uba3c\uc800, i\uac1c\uc758 \ud589\ub82c M<sub>1<\/sub>, M<sub>2<\/sub>, ..., M<sub>i<\/sub>\uc744 \ubaa8\ub450 \uacf1\ud560 \uc218 \uc788\ub294 \uc21c\uc11c\uac00 \uc788\ub294\uc9c0 \uc54c\uc544\ubcf8\ub2e4. \ucc38\uace0\ub85c \ud589\ub82c \uacf1\uc148 \uc5f0\uc0b0\uc5d0\uc11c RxC \ud589\ub82c\uacfc R&#39;xC&#39; \ud589\ub82c\uc744 \uacf1\ud558\ub824\uba74 C = R&#39;\uc774 \uc131\ub9bd\ud558\uc5ec\uc57c \ud558\uace0 \uacb0\uacfc\ub85c \ub098\uc624\ub294 \ud589\ub82c\uc740 RxC&#39;\uc774\ub2e4. \ub9cc\uc57d i\uac1c\uc758 \ud589\ub82c\uc744 (\uc21c\uc11c\ub97c \uc790\uc720\ub86d\uac8c \uc7ac\ubc30\uce58 \ud558\ub354\ub77c\ub3c4) \uacf1\ud558\ub294 \uac83\uc774 \ubd88\uac00\ub2a5\ud558\ub2e4\uba74 V<sub>i<\/sub> = 0 \uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \ub9cc\uc57d \uac00\ub2a5\ud558\ub2e4\uba74, \uac00\ub2a5\ud55c \ubaa8\ub4e0 \ubc29\ubc95 \uc911 \ucd5c\uc885 \uacb0\uacfc\uac00 \ub418\ub294 \ud589\ub82c\uc758 \ud06c\uae30\uac00 \uac00\uc7a5 \ucee4\uc9c0\ub294 \ubc29\ubc95\uc744 \ucc3e\uc544 \uadf8 \ub54c \uacb0\uacfc\ub85c \ub098\uc628 \ud589\ub82c\uc758 \ud06c\uae30 (\ud589\uc758 \uc218 x \uc5f4\uc758 \uc218)\uac00 V<sub>i<\/sub> \uac12\uc774 \ub41c\ub2e4.<\/p>\r\n\r\n<p>V<sub>1<\/sub>, V<sub>2<\/sub>, ..., V<sub>n<\/sub>\ub97c \uad6c\ud574\ubcf4\uc790.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \uc790\uc5f0\uc218&nbsp;n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4 (1 &le; n&nbsp;&le;&nbsp;1,000). \ub2e4\uc74c n\uc904\uc5d0\ub294 \uac01 \uc904\uc5d0 \uc815\uc218 \ub450 \uac1c\uac00 \uc8fc\uc5b4\uc9c0\ub294\ub370, \uac01 \ud589\ub82c\uc758 \ud589\uc758 \uc218 R[i]\uc640 \uc5f4\uc758 \uc218 C[i]\uc774\ub2e4. \uac01 \ud589\ub82c\uc758 \ud589\uc758 \uc218\uc640 \uc5f4\uc758 \uc218\ub294 1 \uc774\uc0c1 1,000 \uc774\ud558\uc774\ub2e4.<\/p>\r\n","output":"<p>\ucd1d n\uc904\uc744 \ucd9c\ub825\ud574\uc57c \ud558\uace0, \uac01 \uc904\uc5d0\ub294 V<sub>1<\/sub> \ubd80\ud130 V<sub>n<\/sub>\uae4c\uc9c0 V<sub>i<\/sub> \uac12\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\ud55c\uad6d\uc5b4","subtask1":"<ul>\r\n\t<li>1 &le;&nbsp;n &le;&nbsp;10<\/li>\r\n\t<li>1 &le;&nbsp;R, C &le; 1,000<\/li>\r\n<\/ul>\r\n","subtask2":"<ul>\r\n\t<li>1 &le;&nbsp;n &le;&nbsp;1,000<\/li>\r\n\t<li>1 &le; R, C &le; 1,000<\/li>\r\n<\/ul>\r\n","sample_explain_1":"<p>i = 1 \uc77c \ub54c\uc5d0\ub294 8x2 \ud589\ub82c (M<sub>1<\/sub>) \ud558\ub098\ub9cc \uc774\uc6a9\ud558\uc5ec \uc6d0\uc18c\uc758 \uc218\uac00 \ucd1d 16\uac1c\uc778 \ud589\ub82c\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>i = 2 \uc77c \ub54c\uc5d0\ub294 M<sub>1<\/sub> x M<sub>2<\/sub> \ub97c \ud558\uc5ec 8x8 \ud589\ub82c\uc744 \ub9cc\ub4e4 \uc218 \uc788\uace0, \uc6d0\uc18c\uc758 \uc218\ub294 \ucd1d 64\uac1c\uc774\ub2e4. \ud639\uc740 M<sub>2<\/sub> x M<sub>1<\/sub> \uc744 \ud558\uc5ec 2x2 \ud589\ub82c\uc744 \ub9cc\ub4e4 \uc218 \uc788\uc9c0\ub9cc, \uc6d0\uc18c\uc758 \uc218\uac00 \ucd5c\ub300\uac00 \ub418\ub294 \ubc29\ubc95\uc740 \uc0c1\uae30\ud55c \ubc29\ubc95\uc774\ub2e4.<\/p>\r\n\r\n<p>i = 3 \uc77c \ub54c\uc5d0\ub294 M<sub>1<\/sub> x M<sub>3<\/sub> x M<sub>2<\/sub>\ub97c \ud1b5\ud574 8x8 \ud589\ub82c\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n","sample_explain_3":"<p>i = 3\uc77c \ub54c \uc5b4\ub5a4 \uc21c\uc11c\ub85c 3\uac1c\uc758 \ud589\ub82c\uc744 \ubc30\uce58\ud558\uc5ec \uacf1\ud558\ub354\ub77c\ub3c4 \ud589\ub82c \uacf1\uc148 \uc5f0\uc0b0\uc744 \uc644\ub8cc\ud560 \uc218 \uc5c6\uc73c\ubbc0\ub85c \ub2f5\uc774 0\uc774\ub2e4.<\/p>\r\n"},{"problem_id":"15933","problem_lang":"1","title":"Matrix Multiplication","description":"<p>You are given n matrices, labeled as M<sub>1<\/sub>, M<sub>2<\/sub>, ..., M<sub>n<\/sub>. Matrix M<sub>i<\/sub> (where i = 1, 2, ..., n) has R[i] rows and C[i] columns.&nbsp;<\/p>\r\n\r\n<p>For each i between 1 and n (inclusive), you want to compute a number (we call V<sub>i<\/sub>) as follows.<\/p>\r\n\r\n<p>Using the matrices M<sub>1<\/sub>, M<sub>2<\/sub>, ..., M<sub>i<\/sub>, you will first determine if you can find an ordering of these i matrices such that you can multiply all i of them. Note that two matrices of size RxC and R&#39;xC&#39; can be multiplied if and only if C = R&#39;, and the resulting matrix is of size RxC&#39;. If this is impossible, V<sub>i<\/sub> = 0. If this is possible, then V<sub>i<\/sub> is defined as the size of the largest matrix you can obtain in this manner. In other words, among all orderings of the i matrices, V<sub>i<\/sub> is the maximum number of elements found in the matrix that is the product of the i matrices.&nbsp;<\/p>\r\n\r\n<p>You are asked to compute the values V<sub>1<\/sub>, V<sub>2<\/sub>, ..., V<sub>n<\/sub>.<\/p>\r\n","input":"<p>The first line contains a positive integer n (1 &le;&nbsp;n &le; 1,000). Each of the following n lines contains two numbers each, denoting the number of rows and the number of columns of a matrix. &nbsp;The number of rows and the number of columns of each matrix is between 1 and 1,000, inclusive.&nbsp;<\/p>\r\n","output":"<p>You must output n lines where the i-th line contains the value V<sub>i<\/sub> described in the problem statement.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\uc601\uc5b4","subtask1":"<ul>\r\n\t<li>1 &le;&nbsp;n &le;&nbsp;10<\/li>\r\n\t<li>1 &le;&nbsp;R, C &le; 1,000<\/li>\r\n<\/ul>\r\n","subtask2":"<ul>\r\n\t<li>1 &le;&nbsp;n &le;&nbsp;1,000<\/li>\r\n\t<li>1 &le; R, C &le; 1,000<\/li>\r\n<\/ul>\r\n","sample_explain_1":"<p>When i = 1, the answer is trivially 16 as M<sub>1<\/sub> contains 16 elements.<\/p>\r\n\r\n<p>When i = 2, we can multiply them (M<sub>1<\/sub> x M<sub>2<\/sub>) to obtain a 8 x 8 matrix.<\/p>\r\n\r\n<p>Note that we can also multiply them in the other order (M<sub>2<\/sub> x M<sub>1<\/sub>) to obtain a 2 x 2 matrix, but it is not maximum.<\/p>\r\n\r\n<p>When i = 3, we can multiply the three matrices as M<sub>1<\/sub> x M<sub>3<\/sub> x M<sub>2<\/sub>, to obtain a 8 x 8 matrix.<\/p>\r\n","sample_explain_3":"<p>There is no ordering that results in a valid matrix multiplication of the three matrices.&nbsp;<\/p>\r\n"}]

출처

  • 문제를 만든 사람: haden
  • 데이터를 추가한 사람: jh05013