시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

지금껏 문제를 풀면서 시간 초과 메시지를 본 적이 있을 것이다.

시간 초과의 이유엔 여러 가지가 있지만, 주로 제출한 코드의 시간복잡도가 문제에서 요구하는 최대 한도의 시간복잡도를 상회했을 때 발생한다.

이번 문제를 풀면서 시간복잡도에 대한 이해도를 높이고 시간 초과 메시지를 덜 받도록 해보자.

어떤 소스 코드가 주어지면 이 소스 코드의 시간복잡도를 계산하면 된다.

물론 그냥 계산하기엔 너무 어려우므로, 다음과 같은 간소화된 방식으로 계산해보도록 하자.

프로그램은 단 네 개의 명령어만을 가진다고 가정한다. 목록은 아래와 같다.

  • basic : 사칙연산 혹은 값 할당 등의 기초적인 코드
  • loop : 반복문의 시작
  • endloop : 반복문의 끝
  • endprogram : 프로그램의 종료

프로그램 분석은 아래의 규칙에 따라 시행한다.

  • 위에 주어진 네가지 명령어만이 프로그램에 존재한다.
  • loop문은 항상 하나의 전달 인자를 가지며, 대응되는 하나의 endloop문(loop의 시작 이후로 만나는 첫 endloop문)과 묶여 하나의 반복문이 된다.
  • loop의 전달 인자는 x, y, 또는 양의 정수가 될 수 있으며, x, y는 상수이다. 프로그램 실행 도중 재할당되지 않는다. 이 전달인자만큼 루프를 반복하게 된다.
  • 만일 어떤 loop문 내에 basic 코드가 하나도 존재하지 않는다면 의미없는 루프이므로 전달인자의 값에 관계없이 그 loop를 즉시 종료하게 된다.
  • 만일 어떤 loop문 내에 basic 코드가 여러 개 존재하더라도 한 개의 basic 코드가 있는 것과 같이 생각해도 된다.
  • basic 코드는 실행에 상수 시간이 걸린다.

시간복잡도는 basic 구문의 실행 횟수를 함수로 나타낸 뒤 빅오 표기법을 사용하여 분석한다.

빅오 표기법의 정의는 다음과 같다.

어떤 임의의 양의 상수 c와 d를 잘 골라, 1 이상인 모든 x에 대하여 c*g(x) ≤ f(x) ≤ d*g(x)를 만족하게 할 수 있다면 f(x)의 빅오 표기는 O(g)이다.

좀 더 쉽게 설명하면, 모든 상수부는 떼어내버릴 수 있다는 의미이다.

예를 들면 4x 3의 빅오 표기는 x3가 된다.

또한, 어떤 항보다 낮은 차수를 가지는 항들은 통째로 제거하는 것이 가능하다.

예를 들어, x 3 + x2의 빅오 표기는 x3이 되며, x2 + 7의 빅오 표기는 x2가 된다.

하지만 여러 서로 다른 변수가 섞여있다면 다른 항보다 작다고 확신할 수 없는 항들은 모두 표기해야 한다.

예를 들어, x 2y + y2x + xy + x2 의 빅오 표기는 x2y + y2x가 되며, x2 + 17xy + y2 의 빅오 표기는 x2 + y2이 된다.

입력

첫 줄에 테스트 케이스의 수 K가 주어진다.

각 테스트 케이스는 빈 줄로 구분한다.

프로그램은 문제에서 설명한 네 가지의 명령어로만 구성되어 있으며, endprogram은 항상 가장 바깥의 loop문보다 더 아래에 단 하나 존재한다.

loop와 전달인자 x 사이에는 단 한개의 공백이 있으며, 이 경우를 제외하고는 프로그램 내에 어떤 공백이나 이상한 문자가 주어지는 경우는 없다.

loop문의 최대 중첩 횟수는 50회이다.

출력

각 테스트 케이스마다 Data Set K: 를 출력한 뒤, 입력으로 주어진 프로그램의 시간복잡도를 출력한다.

각 항은 x의 차수가 높은 것부터, x의 차수가 같다면 y의 차수가 높은 것부터 차례대로 출력한다.

각 항을 출력할 땐 최대한 축약해야 한다. 예를 들어, x^1y^1을 출력하는 대신에 xy를 출력해야 하며, x^1y^0을 출력하는 대신에 x를 출력해야 한다.

그 외의 출력 형식에 대한 내용은 예제 출력을 보면 된다.

각 테스트 케이스의 사이엔 빈 줄을 하나 출력한다.

예제 입력 1

3
loop x
endloop
endprogram

basic
basic
endprogram

loop y
basic
loop y
basic
basic
endloop
endloop
loop x
loop y
basic
endloop
loop x
basic
endloop
endloop
endprogram

예제 출력 1

Data Set 1:
0

Data Set 2:
1

Data Set 3:
x^2 + y^2
[{"problem_id":"5178","problem_lang":"0","title":"\uc2dc\uac04 \ucd08\uacfc","description":"<p>\uc9c0\uae08\uaecf \ubb38\uc81c\ub97c \ud480\uba74\uc11c \uc2dc\uac04 \ucd08\uacfc \uba54\uc2dc\uc9c0\ub97c \ubcf8 \uc801\uc774 \uc788\uc744 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc2dc\uac04 \ucd08\uacfc\uc758 \uc774\uc720\uc5d4 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\uc9c0\ub9cc, \uc8fc\ub85c \uc81c\ucd9c\ud55c \ucf54\ub4dc\uc758 \uc2dc\uac04\ubcf5\uc7a1\ub3c4\uac00 \ubb38\uc81c\uc5d0\uc11c \uc694\uad6c\ud558\ub294 \ucd5c\ub300 \ud55c\ub3c4\uc758 \uc2dc\uac04\ubcf5\uc7a1\ub3c4\ub97c \uc0c1\ud68c\ud588\uc744 \ub54c \ubc1c\uc0dd\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc774\ubc88 \ubb38\uc81c\ub97c \ud480\uba74\uc11c \uc2dc\uac04\ubcf5\uc7a1\ub3c4\uc5d0 \ub300\ud55c \uc774\ud574\ub3c4\ub97c \ub192\uc774\uace0 \uc2dc\uac04 \ucd08\uacfc \uba54\uc2dc\uc9c0\ub97c \ub35c \ubc1b\ub3c4\ub85d \ud574\ubcf4\uc790.<\/p>\r\n\r\n<p>\uc5b4\ub5a4 \uc18c\uc2a4 \ucf54\ub4dc\uac00 \uc8fc\uc5b4\uc9c0\uba74 \uc774 \uc18c\uc2a4 \ucf54\ub4dc\uc758 \uc2dc\uac04\ubcf5\uc7a1\ub3c4\ub97c \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ubb3c\ub860 \uadf8\ub0e5 \uacc4\uc0b0\ud558\uae30\uc5d4 \ub108\ubb34 \uc5b4\ub824\uc6b0\ubbc0\ub85c, \ub2e4\uc74c\uacfc \uac19\uc740 \uac04\uc18c\ud654\ub41c \ubc29\uc2dd\uc73c\ub85c \uacc4\uc0b0\ud574\ubcf4\ub3c4\ub85d \ud558\uc790.<\/p>\r\n\r\n<p>\ud504\ub85c\uadf8\ub7a8\uc740 \ub2e8 \ub124 \uac1c\uc758 \uba85\ub839\uc5b4\ub9cc\uc744 \uac00\uc9c4\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4. \ubaa9\ub85d\uc740 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>basic : \uc0ac\uce59\uc5f0\uc0b0 \ud639\uc740 \uac12 \ud560\ub2f9 \ub4f1\uc758 \uae30\ucd08\uc801\uc778 \ucf54\ub4dc<\/li>\r\n\t<li>loop : \ubc18\ubcf5\ubb38\uc758 \uc2dc\uc791<\/li>\r\n\t<li>endloop : \ubc18\ubcf5\ubb38\uc758 \ub05d<\/li>\r\n\t<li>endprogram : \ud504\ub85c\uadf8\ub7a8\uc758 \uc885\ub8cc<\/li>\r\n<\/ul>\r\n\r\n<p>\ud504\ub85c\uadf8\ub7a8 \ubd84\uc11d\uc740 \uc544\ub798\uc758 \uaddc\uce59\uc5d0 \ub530\ub77c \uc2dc\ud589\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc704\uc5d0 \uc8fc\uc5b4\uc9c4 \ub124\uac00\uc9c0 \uba85\ub839\uc5b4\ub9cc\uc774 \ud504\ub85c\uadf8\ub7a8\uc5d0 \uc874\uc7ac\ud55c\ub2e4.<\/li>\r\n\t<li>loop\ubb38\uc740 \ud56d\uc0c1 \ud558\ub098\uc758 \uc804\ub2ec \uc778\uc790\ub97c \uac00\uc9c0\uba70, \ub300\uc751\ub418\ub294 \ud558\ub098\uc758 endloop\ubb38(loop\uc758 \uc2dc\uc791 \uc774\ud6c4\ub85c \ub9cc\ub098\ub294 \uccab endloop\ubb38)\uacfc \ubb36\uc5ec \ud558\ub098\uc758 \ubc18\ubcf5\ubb38\uc774 \ub41c\ub2e4.<\/li>\r\n\t<li>loop\uc758 \uc804\ub2ec \uc778\uc790\ub294 x, y, \ub610\ub294 \uc591\uc758 \uc815\uc218\uac00 \ub420 \uc218 \uc788\uc73c\uba70, x, y\ub294 \uc0c1\uc218\uc774\ub2e4. \ud504\ub85c\uadf8\ub7a8 \uc2e4\ud589 \ub3c4\uc911 \uc7ac\ud560\ub2f9\ub418\uc9c0 \uc54a\ub294\ub2e4. \uc774 \uc804\ub2ec\uc778\uc790\ub9cc\ud07c \ub8e8\ud504\ub97c \ubc18\ubcf5\ud558\uac8c \ub41c\ub2e4.<\/li>\r\n\t<li>\ub9cc\uc77c \uc5b4\ub5a4 loop\ubb38 \ub0b4\uc5d0 basic \ucf54\ub4dc\uac00 \ud558\ub098\ub3c4 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \uc758\ubbf8\uc5c6\ub294 \ub8e8\ud504\uc774\ubbc0\ub85c \uc804\ub2ec\uc778\uc790\uc758 \uac12\uc5d0 \uad00\uacc4\uc5c6\uc774 \uadf8 loop\ub97c \uc989\uc2dc \uc885\ub8cc\ud558\uac8c \ub41c\ub2e4.<\/li>\r\n\t<li>\ub9cc\uc77c \uc5b4\ub5a4 loop\ubb38 \ub0b4\uc5d0 basic \ucf54\ub4dc\uac00 \uc5ec\ub7ec \uac1c \uc874\uc7ac\ud558\ub354\ub77c\ub3c4 \ud55c \uac1c\uc758 basic \ucf54\ub4dc\uac00 \uc788\ub294 \uac83\uacfc \uac19\uc774 \uc0dd\uac01\ud574\ub3c4 \ub41c\ub2e4.<\/li>\r\n\t<li>basic \ucf54\ub4dc\ub294 \uc2e4\ud589\uc5d0 \uc0c1\uc218 \uc2dc\uac04\uc774 \uac78\ub9b0\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc2dc\uac04\ubcf5\uc7a1\ub3c4\ub294 basic \uad6c\ubb38\uc758 \uc2e4\ud589 \ud69f\uc218\ub97c \ud568\uc218\ub85c \ub098\ud0c0\ub0b8 \ub4a4 \ube45\uc624 \ud45c\uae30\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ubd84\uc11d\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ube45\uc624 \ud45c\uae30\ubc95\uc758 \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>\uc5b4\ub5a4 \uc784\uc758\uc758 \uc591\uc758 \uc0c1\uc218 c\uc640 d\ub97c \uc798 \uace8\ub77c, 1 \uc774\uc0c1\uc778 \ubaa8\ub4e0 x\uc5d0 \ub300\ud558\uc5ec c*g(x) &le; f(x) &le; d*g(x)\ub97c \ub9cc\uc871\ud558\uac8c \ud560 \uc218 \uc788\ub2e4\uba74 f(x)\uc758 \ube45\uc624 \ud45c\uae30\ub294 O(g)\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc880 \ub354 \uc27d\uac8c \uc124\uba85\ud558\uba74, \ubaa8\ub4e0 \uc0c1\uc218\ubd80\ub294 \ub5bc\uc5b4\ub0b4\ubc84\ub9b4 \uc218 \uc788\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uba74 4x <sup>3<\/sup>\uc758 \ube45\uc624 \ud45c\uae30\ub294 x<sup>3<\/sup>\uac00 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ub610\ud55c, \uc5b4\ub5a4 \ud56d\ubcf4\ub2e4 \ub0ae\uc740 \ucc28\uc218\ub97c \uac00\uc9c0\ub294 \ud56d\ub4e4\uc740 \ud1b5\uc9f8\ub85c \uc81c\uac70\ud558\ub294 \uac83\uc774 \uac00\ub2a5\ud558\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, x <sup>3<\/sup> + x<sup>2<\/sup>\uc758 \ube45\uc624 \ud45c\uae30\ub294 x<sup>3<\/sup>\uc774 \ub418\uba70, x<sup>2<\/sup> + 7\uc758 \ube45\uc624 \ud45c\uae30\ub294 x<sup>2<\/sup>\uac00 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ud558\uc9c0\ub9cc \uc5ec\ub7ec \uc11c\ub85c \ub2e4\ub978 \ubcc0\uc218\uac00 \uc11e\uc5ec\uc788\ub2e4\uba74 \ub2e4\ub978 \ud56d\ubcf4\ub2e4 \uc791\ub2e4\uace0 \ud655\uc2e0\ud560 \uc218 \uc5c6\ub294 \ud56d\ub4e4\uc740 \ubaa8\ub450 \ud45c\uae30\ud574\uc57c \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, x <sup>2<\/sup>y + y<sup>2<\/sup>x + xy + x<sup>2<\/sup> \uc758 \ube45\uc624 \ud45c\uae30\ub294 x<sup>2<\/sup>y + y<sup>2<\/sup>x\uac00 \ub418\uba70, x<sup>2<\/sup> + 17xy + y<sup>2<\/sup> \uc758 \ube45\uc624 \ud45c\uae30\ub294 x<sup>2<\/sup> + y<sup>2<\/sup>\uc774 \ub41c\ub2e4.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 K\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ube48 \uc904\ub85c \uad6c\ubd84\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ud504\ub85c\uadf8\ub7a8\uc740 \ubb38\uc81c\uc5d0\uc11c \uc124\uba85\ud55c \ub124 \uac00\uc9c0\uc758 \uba85\ub839\uc5b4\ub85c\ub9cc \uad6c\uc131\ub418\uc5b4 \uc788\uc73c\uba70, endprogram\uc740 \ud56d\uc0c1 \uac00\uc7a5 \ubc14\uae65\uc758 loop\ubb38\ubcf4\ub2e4 \ub354 \uc544\ub798\uc5d0 \ub2e8 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4.<\/p>\r\n\r\n<p>loop\uc640 \uc804\ub2ec\uc778\uc790 x \uc0ac\uc774\uc5d0\ub294 \ub2e8 \ud55c\uac1c\uc758 \uacf5\ubc31\uc774 \uc788\uc73c\uba70, \uc774 \uacbd\uc6b0\ub97c \uc81c\uc678\ud558\uace0\ub294 \ud504\ub85c\uadf8\ub7a8 \ub0b4\uc5d0 \uc5b4\ub5a4 \uacf5\ubc31\uc774\ub098 \uc774\uc0c1\ud55c \ubb38\uc790\uac00 \uc8fc\uc5b4\uc9c0\ub294 \uacbd\uc6b0\ub294 \uc5c6\ub2e4.<\/p>\r\n\r\n<p>loop\ubb38\uc758 \ucd5c\ub300 \uc911\ucca9 \ud69f\uc218\ub294 50\ud68c\uc774\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 Data Set K: \ub97c \ucd9c\ub825\ud55c \ub4a4, \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ud504\ub85c\uadf8\ub7a8\uc758 \uc2dc\uac04\ubcf5\uc7a1\ub3c4\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud56d\uc740 x\uc758 \ucc28\uc218\uac00 \ub192\uc740 \uac83\ubd80\ud130, x\uc758 \ucc28\uc218\uac00 \uac19\ub2e4\uba74 y\uc758 \ucc28\uc218\uac00 \ub192\uc740 \uac83\ubd80\ud130 \ucc28\ub840\ub300\ub85c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud56d\uc744 \ucd9c\ub825\ud560 \ub550 \ucd5c\ub300\ud55c \ucd95\uc57d\ud574\uc57c \ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, x^1y^1\uc744 \ucd9c\ub825\ud558\ub294 \ub300\uc2e0\uc5d0 xy\ub97c \ucd9c\ub825\ud574\uc57c \ud558\uba70, x^1y^0\uc744 \ucd9c\ub825\ud558\ub294 \ub300\uc2e0\uc5d0 x\ub97c \ucd9c\ub825\ud574\uc57c \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uadf8 \uc678\uc758 \ucd9c\ub825 \ud615\uc2dd\uc5d0 \ub300\ud55c \ub0b4\uc6a9\uc740 \uc608\uc81c \ucd9c\ub825\uc744 \ubcf4\uba74 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc0ac\uc774\uc5d4 \ube48 \uc904\uc744 \ud558\ub098 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"5178","problem_lang":"1","title":"Time Limit Exceeded","description":"<p>&ldquo;Time limit exceeded&rdquo; is another favorite error message. Most frequently, the cause is either an infinite loop somewhere in your program, or &mdash; more often &mdash; a very slow brute-force solution to a problem that should be solved more elegantly. The problem is not that we terminate the program after 1 minute; your solutions would probably run 10<sup>100<\/sup> years if they use brute-force for a problem where they shouldn&rsquo;t.<\/p>\r\n\r\n<p>The whole thing might be a little easier if you had a little program to figure out the runtime complexity of your code. In general, this is an undecidable problem (i.e., there cannot be any computer program always getting it right, one of the cool results you learn in CSCI 303). So instead, we will restrict our attention to a severely restricted &ldquo;programming language&rdquo; for which the problem can be solved again. Programs only consist of four kinds of statements, namely <code>basic<\/code>, <code>loop<\/code>, <code>endloop<\/code> and <code>endprogram<\/code>. Each statement is on a line by itself, there are no semicolons or any other punctuation, and loop is the only command with a parameter. That parameter is either a positive integer number, or one of the two variables &lsquo;x&rsquo; and &lsquo;y&rsquo; (our simplified programming language has only two variables), denoting that the body between the <code>loop<\/code> and the corresponding endloop is executed that many times.<\/p>\r\n\r\n<p><code>basic<\/code> denotes a basic statement (like assignment, addition, or some other stuff that does not matter for complexity analysis), and <code>endprogram<\/code> is the end of the program. Notice that the number of times a loop is executed is always a constant (either the variable or a numerical one), since variables cannot be assigned values. Also, we assume that our compiler is smart enough to prune away loops with no <code>basic<\/code> statement anywhere in their bodies.<\/p>\r\n\r\n<p>You are to determine, in big-Oh (O) notation, the complexity of the program, i.e. the total number of times that any basic statement is executed. Your program should output an asymptotically tight bound of the form &Theta;(. . .).<\/p>\r\n\r\n<p>We say that a function f is &Theta;(g) for another function g, if there are positive constants c and d with c&middot;g(x) &le; f(x) &le; d&middot;g(x) for all x &ge; 1. In practice, that means that you can drop all constant factors (e.g. 4x<sup>3<\/sup> is &Theta;(x<sup>3<\/sup>)), and all terms that have smaller exponents for every variable (e.g. x<sup>3<\/sup> + x<sup>2<\/sup> is &Theta;(x<sup>3<\/sup>), because x<sup>3<\/sup> &le; x<sup>3<\/sup> + x<sup>2<\/sup> &le; 2x<sup>3<\/sup> for all x &ge; 1). For some more examples, x<sup>2<\/sup> + 7 is &Theta;(x<sup>2<\/sup>), and x<sup>2<\/sup>y + y<sup>2<\/sup>x + xy + x<sup>2<\/sup> is &Theta;(x<sup>2<\/sup>y + y<sup>2<\/sup>x).<\/p>\r\n\r\n<p>A slightly less obvious consequence is that certain mixed terms can be omitted as well. For instance, x<sup>2<\/sup> + 17xy + y<sup>2<\/sup> is &Theta;(x<sup>2<\/sup> + y<sup>2<\/sup>), because either x &le; y, which implies that x<sup>2<\/sup> + 17xy + y<sup>2<\/sup> &le; y<sup>2<\/sup> + 17y<sup>2<\/sup> + y<sup>2<\/sup> &le; 19(x<sup>2<\/sup> + y<sup>2<\/sup>), or x &ge; y, which implies that x<sup>2<\/sup> + 17xy + y<sup>2<\/sup> &le; x<sup>2<\/sup> + 17x<sup>2<\/sup> + y<sup>2<\/sup> &le; 19(x<sup>2<\/sup> + y<sup>2<\/sup>). If you have questions about the notation, please ask for assistance.<\/p>\r\n","input":"<p>The first line is the number K of input data sets, followed by K data sets, each being one program (and each followed by an empty line).<\/p>\r\n\r\n<p>A program consists of only the four commands above, and will always be syntactically correct (e.g. the loop and endloop commands match, there is exactly one <code>endprogram<\/code> outside the outermost <code>loop<\/code> and so forth). Each command will be on a line by itself. There will be exactly one whitespace between the string <code>loop<\/code> and its parameter, and no tabulator or whitespace characters anywhere else (besides the line breaks). The nesting depth of <code>loop<\/code> statements will be at most 50 in all examples.<\/p>\r\n","output":"<p>For each data set, output &ldquo;Data Set x:&rdquo; on a line by itself, where x is its number. On the next line, output the asymptotic complexity of the program. The order of the terms should be by non-increasing exponents of x, breaking ties by non-increasing exponents of y. Make your output look &ldquo;mathematically neat&rdquo;, e.g. output xy instead of x^1y^1 and x instead of x^1y^0. The output for each data set should be followed by an empty line.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

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University > The USC Programming Contest > Fall 2007 D번