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![? クエリーと聞いて前処理が頭をよぎるか
も知れませんが,O(NQ)なら間に合いそう
? O(QNlogN)はO(10^9)ぐらいになるのでか
なり危険???(実際通りません)
? それぞれのクエリーについて独立に考え
てみます
? 妥当にDPを考えてみましょう
? dp[i]:=列の最後の色がiとなるような部
分列の最大価値](https://image.slidesharecdn.com/choosingballs-130121044018-phpapp02/85/Choosing-balls-4-320.jpg)
![? i番目のボールの色をColor[i],価値を
Value[i]とおくと,
? dp[Color[i]] を,
①dp[Color[i]]+a*Value[i] か
②dp[j]+b*Value[i] (1<=j<=N,j!=Color[i])
に更新することができます
? ボールを順に見てこの更新を繰り返す
? ナイーブな実装で、O(N^2)](https://image.slidesharecdn.com/choosingballs-130121044018-phpapp02/85/Choosing-balls-5-320.jpg)
![? ①のパターンはO(1)で楽勝
? ②は、よくあるパターンとしてColor[i]
の左、右で分けると、それぞれの区間の
最大値を求め、最大値に+Value[i]*bす
ればOK
? Range Minimum Query&更新!
? Segment Tree!!!](https://image.slidesharecdn.com/choosingballs-130121044018-phpapp02/85/Choosing-balls-6-320.jpg)
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Choosing balls
- 1. Codeforces #162 Div1-C,Div2-E Choosing Balls Writer: hogloid improvement: snuke cooperation: rng_58,degwer
- 2. ? N個のボールが一列に並んでいます ? それぞれのボールには,色?価値が定めら れています ? あるボールの列の価値は以下のように定 められます ? あるボールが列の先頭,もしくは一つ前の ボールと色が違うなら,(ボールの価値)*b ? そうでないなら,(ボールの価値)*a ? 列の価値はボールはこれらの数の和
- 3. ? N個のボールの中から,相対的な順序を変えず に一部のボールを取り出します(部分列を取 ります) ? (a,b)のペアがQ個与えられるので,それぞれ の(a,b)について,選べるボールの部分列の 価値の最大値を答えなさい ただし,空の列の 価値は0とします。 ? 1<=N<=100000 1<=Q<=500 ? -100000<=a,b,ボールの価値<=100000 ? 1<=ボールの色<=N
- 4. ? クエリーと聞いて前処理が頭をよぎるか も知れませんが,O(NQ)なら間に合いそう ? O(QNlogN)はO(10^9)ぐらいになるのでか なり危険???(実際通りません) ? それぞれのクエリーについて独立に考え てみます ? 妥当にDPを考えてみましょう ? dp[i]:=列の最後の色がiとなるような部 分列の最大価値
- 5. ? i番目のボールの色をColor[i],価値を Value[i]とおくと, ? dp[Color[i]] を, ①dp[Color[i]]+a*Value[i] か ②dp[j]+b*Value[i] (1<=j<=N,j!=Color[i]) に更新することができます ? ボールを順に見てこの更新を繰り返す ? ナイーブな実装で、O(N^2)
- 6. ? ①のパターンはO(1)で楽勝 ? ②は、よくあるパターンとしてColor[i] の左、右で分けると、それぞれの区間の 最大値を求め、最大値に+Value[i]*bす ればOK ? Range Minimum Query&更新! ? Segment Tree!!!
- 7. ? しかし、更新/RangeMinimum Queryを どちらもO(1)で解くのはセグメント木で は無理そう??? ? logが一つかかると、TimeLimitにも間に 合いそうにない ? 別のO(1)で更新する方法を考えよう
- 8. ? dp配列の上位二つの(最後の色、価値)を 持っていれば、あらゆる色でも、違う色 の最大値がどちらかにはある ? dp配列の値の更新は常に増加するので、 消去などの面倒なことが起こらず、更新 はどれもO(1)で可能 ? 少し場合分けが必要です ここでのWAがか なり多かった
- 9. ? First Acceptance: KADR 25m24s ? Japanese First Acceptance: Komaki 27m50s ? Accepted User/All Rated User/All Submission:236/566/916 ? 普段のCよりかなり解かれました ? 赤族やそれに準ずる人はCを解けないと厳 しかったかもしれません ? バグで詰まってる人が多くて悲しかった
- 10. ? First Acceptance: Moor 70m52s ? Japanese First Acceptance: ikouki 108m6s ? Accepted User/All Rated User: 12/1526 ? Div2-Eとしては理想的な解かれ具合だと 思います ? これが解ける人は早くdiv1に来ましょう!
- 11. ? いつかまた問題を作ったときはよろしく お願いします ? Codeforcesの トップページを飾った 名誉なLissくん→