문제
Hudson River Trading(HRT)은 뛰어난 수학, 기술적 역량을 활용하여 거래 전략을 설계하는 퀀트 트레이딩 회사이다. HRT는 엔지니어와 연구자들이 한 팀을 이뤄 어려운 문제를 해결하며, 글로벌 금융 시장에서 매일 수백만 주의 주식을 거래한다.
HRT의 신입 알고리즘 개발자는 새로 설계한 거래 전략을 따르는 시장조성(market-making) 엔진을 개발한 뒤, 그 엔진의 안정성을 평가하고자 한다. 시장조성 엔진은 주식을 보유하지 않은 가상의 계좌를 이용하여 거래를 시작하며, $N$개의 연속된 매매 시점(틱)에 아래와 같은 전략을 따라 거래한다.
- $i$번째 매매 시점에 엔진은 $a_i$ 이상 $b_i$ 이하의 정수 하나를 선택한다. 선택한 정수가 양수라면 그만큼 주식을 매수하고, 음수라면 선택한 정수의 절댓값만큼 주식을 매도하고, 0ドル$이라면 주식을 매매하지 않는다. 이 때 보유한 주식이 부족해도 매도가 가능하며, 이 경우 보유 주식의 수는 음수가 된다.
- $i$번째 매매 직후 보유 주식이 0ドル$주라면, 해당 전략이 포지션 노출을 최소화하여 위험성을 완화하고 시장 안정성에 기여했다고 평가하여 시장조성 엔진의 안정성이 $x_i$만큼 증가한다.
- 수수료 또는 슬리피지 등 다른 요소는 모두 무시한다.
총 $N$번의 매매를 모두 마쳤을 때, 신규 개발한 시장조성 엔진이 달성할 수 있는 안정성의 최댓값을 구해보자.
출력
총 $N$번의 매매를 모두 마쳤을 때, 신규 개발한 시장조성 엔진이 달성할 수 있는 안정성의 최댓값을 출력한다.
[{"problem_id":"34057","problem_lang":"0","title":"\uc2dc\uc7a5\uc870\uc131\ud558\uae30","description":"<p>Hudson River Trading(HRT)\uc740 \ub6f0\uc5b4\ub09c \uc218\ud559, \uae30\uc220\uc801 \uc5ed\ub7c9\uc744 \ud65c\uc6a9\ud558\uc5ec \uac70\ub798 \uc804\ub7b5\uc744 \uc124\uacc4\ud558\ub294 \ud000\ud2b8 \ud2b8\ub808\uc774\ub529 \ud68c\uc0ac\uc774\ub2e4. HRT\ub294 \uc5d4\uc9c0\ub2c8\uc5b4\uc640 \uc5f0\uad6c\uc790\ub4e4\uc774 \ud55c \ud300\uc744 \uc774\ub904 \uc5b4\ub824\uc6b4 \ubb38\uc81c\ub97c \ud574\uacb0\ud558\uba70, \uae00\ub85c\ubc8c \uae08\uc735 \uc2dc\uc7a5\uc5d0\uc11c \ub9e4\uc77c \uc218\ubc31\ub9cc \uc8fc\uc758 \uc8fc\uc2dd\uc744 \uac70\ub798\ud55c\ub2e4.<\/p>\r\n\r\n<p>HRT\uc758 \uc2e0\uc785 \uc54c\uace0\ub9ac\uc998 \uac1c\ubc1c\uc790\ub294 \uc0c8\ub85c \uc124\uacc4\ud55c \uac70\ub798 \uc804\ub7b5\uc744 \ub530\ub974\ub294 \uc2dc\uc7a5\uc870\uc131(market-making) \uc5d4\uc9c4\uc744 \uac1c\ubc1c\ud55c \ub4a4, \uadf8 \uc5d4\uc9c4\uc758 \uc548\uc815\uc131\uc744 \ud3c9\uac00\ud558\uace0\uc790 \ud55c\ub2e4. \uc2dc\uc7a5\uc870\uc131 \uc5d4\uc9c4\uc740 \uc8fc\uc2dd\uc744 \ubcf4\uc720\ud558\uc9c0 \uc54a\uc740 \uac00\uc0c1\uc758 \uacc4\uc88c\ub97c \uc774\uc6a9\ud558\uc5ec \uac70\ub798\ub97c \uc2dc\uc791\ud558\uba70, $N$\uac1c\uc758 \uc5f0\uc18d\ub41c \ub9e4\ub9e4 \uc2dc\uc810(\ud2f1)\uc5d0 \uc544\ub798\uc640 \uac19\uc740 \uc804\ub7b5\uc744 \ub530\ub77c \uac70\ub798\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>$i$\ubc88\uc9f8 \ub9e4\ub9e4 \uc2dc\uc810\uc5d0 \uc5d4\uc9c4\uc740 $a_i$ \uc774\uc0c1 $b_i$ \uc774\ud558\uc758 \uc815\uc218 \ud558\ub098\ub97c \uc120\ud0dd\ud55c\ub2e4. \uc120\ud0dd\ud55c \uc815\uc218\uac00 \uc591\uc218\ub77c\uba74 \uadf8\ub9cc\ud07c \uc8fc\uc2dd\uc744 \ub9e4\uc218\ud558\uace0, \uc74c\uc218\ub77c\uba74 \uc120\ud0dd\ud55c \uc815\uc218\uc758 \uc808\ub313\uac12\ub9cc\ud07c \uc8fc\uc2dd\uc744 \ub9e4\ub3c4\ud558\uace0, $0$\uc774\ub77c\uba74 \uc8fc\uc2dd\uc744 \ub9e4\ub9e4\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc774 \ub54c \ubcf4\uc720\ud55c \uc8fc\uc2dd\uc774 \ubd80\uc871\ud574\ub3c4 \ub9e4\ub3c4\uac00 \uac00\ub2a5\ud558\uba70, \uc774 \uacbd\uc6b0 \ubcf4\uc720 \uc8fc\uc2dd\uc758 \uc218\ub294 \uc74c\uc218\uac00 \ub41c\ub2e4.<\/li>\r\n\t<li>$i$\ubc88\uc9f8 \ub9e4\ub9e4 \uc9c1\ud6c4 \ubcf4\uc720 \uc8fc\uc2dd\uc774 $0$\uc8fc\ub77c\uba74, \ud574\ub2f9 \uc804\ub7b5\uc774 \ud3ec\uc9c0\uc158 \ub178\ucd9c\uc744 \ucd5c\uc18c\ud654\ud558\uc5ec \uc704\ud5d8\uc131\uc744 \uc644\ud654\ud558\uace0 \uc2dc\uc7a5 \uc548\uc815\uc131\uc5d0 \uae30\uc5ec\ud588\ub2e4\uace0 \ud3c9\uac00\ud558\uc5ec \uc2dc\uc7a5\uc870\uc131 \uc5d4\uc9c4\uc758 \uc548\uc815\uc131\uc774 $x_i$\ub9cc\ud07c \uc99d\uac00\ud55c\ub2e4.<\/li>\r\n\t<li>\uc218\uc218\ub8cc \ub610\ub294 \uc2ac\ub9ac\ud53c\uc9c0 \ub4f1 \ub2e4\ub978 \uc694\uc18c\ub294 \ubaa8\ub450 \ubb34\uc2dc\ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ucd1d $N$\ubc88\uc758 \ub9e4\ub9e4\ub97c \ubaa8\ub450 \ub9c8\ucce4\uc744 \ub54c, \uc2e0\uaddc \uac1c\ubc1c\ud55c \uc2dc\uc7a5\uc870\uc131 \uc5d4\uc9c4\uc774 \ub2ec\uc131\ud560 \uc218 \uc788\ub294 \uc548\uc815\uc131\uc758 \ucd5c\ub313\uac12\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \uc8fc\uc2dd\uc744 \ub9e4\ub9e4\ud55c \ud69f\uc218 $N$\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. ($1 \\leq N \\leq 1\\,000\\,000$)<\/p>\r\n\r\n<p>\ub2e4\uc74c $N$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0, \uadf8 \uc911 $i$\ubc88\uc9f8 \uc904\uc5d0 $i$\ubc88\uc9f8 \ub9e4\ub9e4\uc5d0 \ub300\ud55c \uc815\ubcf4\ub97c \uc758\ubbf8\ud558\ub294 \uc138 \uc815\uc218 $a_i, b_i, x_i$\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. ($-10^9 \\leq a_i \\leq b_i \\leq 10^9$; $1 \\leq x_i \\leq 10^9$)<\/p>\r\n","output":"<p>\ucd1d $N$\ubc88\uc758 \ub9e4\ub9e4\ub97c \ubaa8\ub450 \ub9c8\ucce4\uc744 \ub54c, \uc2e0\uaddc \uac1c\ubc1c\ud55c \uc2dc\uc7a5\uc870\uc131 \uc5d4\uc9c4\uc774 \ub2ec\uc131\ud560 \uc218 \uc788\ub294 \uc548\uc815\uc131\uc758 \ucd5c\ub313\uac12\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"34057","problem_lang":"1","title":"Market-Making","description":"<p>HRT is first and foremost a math and technology company. We are engineers and researchers working as one team to solve difficult problems, and trading millions of shares a day on the world&rsquo;s financial markets.<\/p>\r\n\r\n<p>A new employee at HRT has developed a market-making engine based on the newly designed trading scheme. The developer wishes to analyze the stability of the new engine. The engine uses a virtual account (initially holding no shares) for trading and will follow the strategy below for $N$ contiguous transaction periods (ticks).<\/p>\r\n\r\n<ul>\r\n\t<li>At the $i$-th transaction tick, the engine selects an integer between $a_i$ and $b_i$, inclusive. If the selected integer is positive, it buys an equal number of shares as the integer. If it is negative, the engine sells an equal number of shares as the absolute value of the integer. If it is $0$, the engine does not make a transaction. Shares can be sold even if the engine does not hold a sufficient amount, and in this scenario, the number of shares held becomes negative.<\/li>\r\n\t<li>If the number of shares held right after the $i$-th transaction is $0$, the trading scheme is considered to have minimized position exposure and thus contributed to market stability. In this case, the stability of the engine increases by $x_i$.<\/li>\r\n\t<li>All other factors, such as fees or slippage, are ignored.<\/li>\r\n<\/ul>\r\n\r\n<p>Find the maximum stability the new market-making engine can achieve after all $N$ transactions have been completed.<\/p>\r\n","input":"<p>The first line of input contains the number of transactions $N$. ($1 \\leq N \\leq 1\\,000\\,000$)<\/p>\r\n\r\n<p>The $i$-th of the following $N$ lines contains three space-separated integers $a_i, b_i, x_i$, denoting the information about the $i$-th transaction. ($-10^9 \\leq a_i \\leq b_i \\leq 10^9$; $1 \\leq x_i \\leq 10^9$)<\/p>\r\n","output":"<p>Print the maximum stability the new market-making engine can achieve after all $N$ transactions have been completed.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]