2022.10.28 YY

今日份巨大多歌词环节：

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Keven_He

最终祈愿

虚无洁白

像漂浮着的尘埃

失去光

也失去所有色彩

枷锁挣不开

坠入深渊的无奈

谁还在能够听到我的

求救哭喊

命运安排还是意外

至今仍没想明白

一切都发生的太快

眼睁开世界已崩坏

毁灭已扑面而来

无法躲开

如今的我已经被困在无边的深渊

越用力挣扎却越陷越向下

你听得见吗我残破的世界正崩塌

拜托了

有谁在谁还在能听到吗

拜托了

留下来留下来别丢我一人离开

拜托了

请将我的心葬孤独深渊外

拜托了

就这一次能否请你别袖手旁观

宿命了断恩怨纠缠

全都已与我无关消散

迷雾中终于传来你对我的不断呼唤

信念永不会退散

Cause I know You are my answer

Cause I know You are my Polaris

Oh Just give me a sign

恍然间我已经安然地来到你身旁

如暗夜灯塔指引流星归家

孤单的小船在漫长漂泊后终于靠岸

拜托了

请让我将未完的事了结

拜托了

请让我持利剑把孽缘全部斩断

拜托了

请让我把悲惨宿命亲手掩埋

拜托了

与我同在将这故事画上终结

将命运轮回解开

拜托了请帮我

能够让万物都倒回无关爱恨岁月

lalala lalala lalalala

请实现请实现

这唯一的最终祈愿

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APJifengc

SUPERSTAR

Stand clear of the closing doors
These old friends, they ain't friends of yours
I can feel a pain in my chest
Under house arrest
Isn't your family looking for you?
Did mommy and daddy try to warn you?
Lend me your eyes and voice
Soon we're gonna have to make a choice

Simon says, eyes on the prize
'Cause you won't get another try
We'll uncover every lie
And I promise you'll survive

Well done, superstar
Shine a light, we're fighting the dark
Well done, superstar!
It's my grand design
I'll never ever leave you behind
It's my grand design!

Livin' ain't complementary
Lights are off and your life's in jeopardy
Knockin' on heaven's door
Don't you give up, 'cause you can win
Hide away within me
Losing sight of what's in front of you (Oh-oh)
If I'm undone, put me back together soon
Racing through a neon void
Too bad they all must be destroyed

Simon says, I found myself
I cleaned the dust off of the shelf
What's your choice? In the end
I'll always be your friend

Well done, superstar
Shine a light, we're fighting the dark
Well done, superstar!
It's my grand design
I'll never ever leave you behind (Leave behind)
It's my grand design!

Superstar!

Well done, superstar
Well done, superstar
Well done, superstar
Well done, superstar

I See a Dreamer

As the wind whips 'round
I take a breath for victory
Wanna play tag or wave your white flag?
'Cause you'll never touch me
King of hearts, all in (All in)
It's not a sin to wanna win (Sin to wanna win)

Can't see me
Flyin' like a bee, black and yellow energy
Only me on my team, naturally

I see a dreamer over there by the water
But I got no, but I got no kakorrhaphiophobia
I see a dreamer and he's ripe for the slaughter
But I got no, but I got no kakorrhaphiophobia
Phobia, phobia, phobia (Oh)

I hear battalions sing of my demise
But I don't know the words (What?)
I take a road of my own making
On a journey, no returning
Woah-oh, woah-oh
That's how it goes
They've drawn the battlе line
And I see fire in their еyes
Na-na-na-na-na-na
I'm better off not listening
Na-na-na-na-na-na
I've got my own song to sing
You might also like

Flyin' like a bee, black and yellow energy
Only me on my team, naturally, woah

I see a dreamer over there by the water
But I got no, but I got no kakorrhaphiophobia
I see a dreamer and he's ripe for the slaughter
But I got no, but I got no kakorrhaphiophobia
Phobia, phobia, phobia (Phobia, oh)

Oh

My boat is full, why don't you swim?
Enjoy my fortress, I'll be right in
I stare a hole through danger's soul
We all know I can do this eyes closed

I refuse to fail
So heed this cautionary tale
You've got dragons, my little friend
You'll conquer them in the end
If you can (Ha-ha-ha-ha, jump in)

I see a dreamer over there by the water
Oh, but I got no, but I got no kakorrhaphiophobia
I see a dreamer and he's ripe for the slaughter
Oh, but I got no, but I got no kakorrhaphiophobia
Phobia, phobia, phobia (Oh)

(Uh, uh)
(Oh, Dream...)
Phobia, phobia, phobia

Supertitious Foundation

May I introduce you to
The world's most decorated zoo
For everything you wish was make-belive
Oh buddy did you know
You could shake hands, with a demon
If you wanna take that dare so be it
You could lose an arm or a leg so easy here
So easy here

So let me give you a scenario
And don't you tell a soul
Its only hypothetical

Imagine all your nightmares
Contain in a single glorified reformatory
Where nothing ever gets out
Straitjackets have been worn out
Good guys, bad guys accomodated
If it don't make any sense like its something in a story
We'll make sure it stays fiction
In our supertitious foundation

Medicine unlike any other
Coffe machine dispensing blood of a sucker
All that we display will never leave
But buddy don't you know
We don't wanna let armageddon
Rear its ugly head like a nuclear weapon
We secure, contain, protect
So you don't live in fear

Did I even metion
The monsters that preside
Where do I begin? Where do I begin?
'Cause there is more than 99

The vermin God
The anklebite
Brothers bride
Shy guy
Trapped in a typewriter
Snapper of neck
On a perfect day
Polaroid
Build bear
Hungry clay
Water orb
River cat
Draws you in the book
Whistlebones
Abandoned row house
Theres was a crooked man
Mans best friend
Hateful star
Desert in a can

And that, isn't half of it

So let me give you a scenario
And don't you tell a soul
Its only hypothetical
(Its only hypothetical)
So let me give you a scenario
But don't you go thinking its real

Imagine all your nightmares
Contain in a single glorified reformatory
Where nothing ever gets out
Straitjackets have been worn out
Good guys, bad guys accomodated
If it don't make any sense like its something in a story
We'll make sure it stays fiction
In our supertitious foundation

Supertitious foundation
Supertitious foundation

the ntf man

Hey, darling
Look and see
I just bought an NFT!

I'm starting down at the bottom
My mo-money in tow (Aight)
I know just what I'm doing
Got life savings to blow (Woah)
I found one, so unique
And I could never say no
Why you wanna say no
To the non-fungible? (Woah-woah-woah, woah)

Look at me
Spending all of our money
To get more, more (Woah, woah)
Gonna get so much more (Woah)
I don't care, if my wife has left me in pain
I'll meet her again at the top of the block chain

What torture, man (Woah)
20 seconds in, I've lost a quarter grand
Don't you disrespect my grindset (Woah, woah)
Less is more, my friend
'Till the tortures end, end
When will it end? (Woah)

Oh? I'm in severe debt?
Wait, who do you think you are?
"Hello, this is the bank
You overdrew your credit card" (Woah, woah)
But I don't really care, because this song, it goes hard
Don't feel free to screenshot
Or I'ma take you to the graveyard (Woah, woah)

They repossessed my bed and sheets (La)
Never been so poor, livin' on the streets (La-la)
I wish I coulda kept all my receipts (La-la)
Boutta make an NFT out of all these beats (La-la) (Woah, woah)
While ethereum is plummeting my 30 G's (La)
I'm just gonna keep on burning trees (La-la)
Don't care for the disappearing bees (La-la)
Just kidding, won't you quit your cryin' please? (La-la)
(Woah, woah)

I don't care, if my wife has left me in pain
I'll meet her again at the top of the block chain

What torture, man
20 seconds in, I've lost a quarter grand
Don't you disrespect my grindset (Woah, woah)
Less is more, my friend
'Till the tortures end, end
When will it end? (Woah, woah-woah)

Out of the blue
Could I ask you for some
Spare change? (Spare change)
Only a few
Dollars will do
I need spare change (Spare change, spare change)
Need a little cryptocurrency

What torture, man
40 minutes in, I had to sell my land (Woah)
But don't you disrespect my grindset (Woah)
Less is more, my friend (Woah)
'Till the tortures end, end (Woah)
It never ends!

Oh, I'm in severe debt? (Woah)
Wait, who do you think you are?
I'm in severe debt, oh well (Woah)
There goes my credit card
(Uh)

附

Strangest Things

(Three to the right, four to the left)

Every day we're in danger of stranger events (Yeah-yeah-yeah)
If I called up a ranger, would they comprehend? (Theoretically)
What if fear was a neighbor in favor with death?
World of horrors, so profound
This is the upside down!

In the eye of the storm
Where the enemy is born
We've seen everything and more
Isn't it the strangest?
Isn't it the strangеst thing?
(Yeah-yeah-yeah)

Giving into thе darkness, infecting your mind (Yeah-yeah-yeah)
Taken over by evil, blood frozen inside (Theoretically)
What if everyone near me, couldn't see me or hear me
Can you feel me in the light now?
Right side up or upside down?

In the eye of the storm
Where the enemy is born
We've seen everything and more
Isn't it the strangest?
Isn't it the strangest thing?

(Hey-ey-ey-ey-ey)

Step right up and come right on through
(Step right up) (Come right on through)
Let the night love and numb you (You)
The light is blinding
But it'll all be over soon, soon
(Woah-oh-oh-oh-oh-oh-oh)
Step right up and come right on through
(Step right up) (Come right on through)
Let the dark of midnight love and numb you (You, you)
The light is blinding
But it'll all be over soon
(Woah-oh-oh-oh-oh-oh-oh) (Oh, oh)

Every day we're in danger of stranger events (Yeah-yeah-yeah)
If I called up a ranger, would they comprehend? (Theoretically)
What if fear was a neighbor in favor with death?
World of horrors, so profound
This is the upside down!

In the eye of the storm
Where the enemy is born
We've seen everything and more
Isn't it the strangest?
In the eye of the storm
Where the enemy is born
We've seen everything and more
Isn't it the strangest?
Isn't it the strangest thing?

(Hey-ey-ey-ey-ey) (Yeah-yeah-yeah)
(Isn't it the strangest thing?) (Yeah-yeah-yeah)
(Yeah-yeah-yeah) (Oh)
(Theoretically)
Isn't it the strangest?

Lyin' 2 Me

Oops, they done did it again
Another mate dead, misplaced his head
Oh, who could it be?
I mean, it definitely wasn't me
No clue where that charlatan ran?
I've been around the Skeld, only know too well
Villain vampires want fresh blood
So now everyone's sus

Are you crawling 'round the vents
Or can you prove your innocence?

Don't go lyin' to me!
I saw a body lyin' there!
Don't go lyin' to me
Don't go lyin', don't go lyin'
I ain't gonna let there be
No more dyin', no more dyin'
Don't go lyin' to me!
I saw a body lying there!
It was right there!

So Jordan was in Medbay, Ellie in elec
Jerome was roaming through the hallways
Guess we gotta, gotta do a triple-check
It ain't the killer's holiday

Are you crawling 'round the vents
Or can you prove your innocence?

Don't go lyin' to me!
I saw a body lyin' there!
Don't go lyin' to me
Don't go lyin', don't go lyin'
I ain't gonna let there be
No more dyin', no more dyin'
Don't go lyin' to me!
I saw a body lying there!
It was right there!

(Don't go lyin' to me!)
(Do-do-don't go lyin' to me!)
It was right there!
It was right there!
(Don't go lyin' to me!)
(Do-do-don't go lyin' to me!)
It was right there!

Listen to me, listen now, lend me your ears
Let me lay it out easy so to lessen all your fears
Just living and loving, not about to let you down
Let's look at this logically, figure this out
We can likely verify each other's alibis
No need to allegate, litigate, itemize lies
I just wanna live free, liberty, let me be
And I won't be ejected, not a liability
Back off!

Don't go lyin' to me!
I saw a body lyin' there!
Don't go lyin' to me
Don't go lyin', don't go lyin'
I ain't gonna let there be
No more dyin', no more dyin'
Don't go lyin' to me!
I saw a body, a body!

Don't go lyin' to me!
I saw a body lyin' there!
Don't go lyin' to me
Don't go lyin', don't go lyin'
I ain't gonna let there be
No more dyin', no more dyin'
Don't go lyin' to me!
I saw a body lying there!
It was right there!

Opinions

纯音乐

Parappa plays funky music (with cg5)

Cha-cha, ayy. Cha-cha, ayy
Cha-cha, ayy. Cha-cha, ayy

So you wanna learn how to play like a pro
Guitar and the drums and the piano
If you get good, you could go on the radio
Drop the base on the stereo

Yeah, I wanna learn how to play like a pro
Guitar and the drums and the piano
If I get good, I could go on the radio
Drop the base on the stereo

Do you want to be a star?
Let's start with the guitar!
I want to be a star!
Let's start with the guitar!
And if you don't make it far
I'ma hit you with a crowbar!
And if I don't make it far
You'll hit me with a... what?

Gotta tune it up
Gotta tune it up
Don't screw it up
Don't screw it up
Fingers on the frets
Fingers on the frets
Don't you lеt it be a mess
I won't let it bе a mess

Strum a chord and let it linger!
Strum a chord and let it linger!
You're playing like a winner!
I'm playing like a winner!

Keep on going, or I'll shoot you with a bow!
Let's move on over to the piano!
Keep on going, or you'll-
Let's move on to the piano!

Play low on your left
Play low on my left
Play high on your right
Play high on my right
Treble and base clef
Treble and base clef
Let the melody fly
Let the melody fly

Don't look behind you
Don't look behind me
No fears to fall to
No fears to fall to
Unless you're talking about the big bazooka pointed at your head
I'm scared...

So you wanna learn how to play like a pro
Guitar and the drums and the piano
If you get good, you could go on the radio
Drop the base on the stereo
Yeah, I wanna learn how to play like a pro
Guitar and the drums and the piano
If I get good, I could go on the radio
Drop the base on the stereo

It's time to try the drums, then we're done
Just for you, I'll put down the gun
It's time to try the drums, then we're done
Thank you for putting away your gun

Psych. Oops

Rum-pa-pum-pum
Rum-pa-pum-pum
Boom-ba-bum-bum
Boom-ba-bum-bum
Go big or go home
Go big or go home
Or you'll die alone!
Or I'll die alone?

Come on, bop it, twist it, pull up the beats
Come on, bop it, twist it, pull up the beats
Pa-ra-pa-pa you're droppin' the heat!
I'm droppin' the heat! I'm droppin' the heat!

So you wanna learn how to play like a pro
Guitar and the drums and the piano
If you get good, you could go on the radio
Drop the base on the stereo

Yeah, I wanna learn how to play like a pro
Guitar and the drums and the piano
If I get good, I could go on the radio
Drop the base on the stereo

Cha-cha, ayy. Cha-cha, ayy
Cha-cha, ayy. Cha-cha, bark bark!

Vibrant Eyes

I'm not the one to trust
When I've lost everything that I've touched
I've built a land in the sky
But I can't remember why

Oh, don't tell them what you've done, tell them what you've done
The war's already won, why do we go on?
Don't tell them what you've done, tell them what you've done
Tell them what you've done
Tell them what you've done
Tell them what you've done

Scared of the water
Scared of the rain (The rain)
Scared of the memories
That make me insane (Someone help me)
I'm scared of the smile I
See on the page (Oh-oh-oh)
I'll cover these vibrant eyes
And forget the pain (The pain)

Little voice ruining my sanity
Evil dreams haunting me
Don't you ever look my way?
Little voice ruining my consciousness
Nothing left to reminisce
I would rather run away
Lessons learned from the page
Lore is written to the day
All this pain will make me suffocate
All my thoughts ever strange
Free me from the mental cage
How could I forget my name?

Scared of the water (So scared)
Scared of the rain (Someone help me)
Scared of the memories (Oh-oh)
That make me insane (Someone help me)
I'm scared of the smile I (Hey, hey)
See on the page (Oh, I see it on the page)
I'll cover these vibrant eyes
And forget the pain (Oh-oh-oh-oh-oh)

Don't tell them what you've done, tell them what you've done (Tell them what you've done)
The war's already won, why do we go on? (Why do we go on?)
Don't tell them what you've done, tell them what you've done
Tell them what you've done
Tell them what you've done
Tell them what you've done

---

SoyTony

不再犹豫

无聊望见了犹豫
达到理想不太易
即使有信心斗志却抑止

谁人定我去或留
定我心中的宇宙
只想靠两手 向理想挥手
问句天几高 心中志比天更高
自信打不死的心态活到老

WO-OH 我有我心底故事
亲手写上每段得失乐与悲与梦儿
WO-OH 纵有创伤不退避
梦想有日达成找到心底梦想的世界
终可见

谁人没试过犹豫
达到理想不太易
即使有信心斗志却抑止
叶世荣：
谁人定我去或留
定我心中的宇宙
只想靠两手 向理想挥手
问句天几高 心中志比天更高
自信打不死的心态活到老

WO-OH 我有我心底故事
亲手写上每段得失乐与悲与梦儿
WO-OH 纵有创伤不退避
梦想有日达成找到心底梦想的世界
终可见
WO-OH
亲手写上每段得失乐与悲与梦儿
WO-OH
梦想有日达成找到心底梦想的世界
终可见

---

今天因为 LemonLime 不会 -DONLINE_JUDGE 于是爆蛋，和 zero4338 同分（迫真

artalter 交错文件导致 rk1 -> rk47 /kx

YY 的 GCD

YY 的 GCD

\(q\) 组询问，求

\[\sum_{i=1}^n\sum_{j=1}^m[\gcd(i,j)\in\mathbb P] \]

其中 \(\mathbb P\) 是素数集 .

不妨令 \(n\le m\)，后复杂度分析假设 \(n,m\) 同阶 .

目录

• YY 的 GCD
  • Algorithm 1 - \(\Theta(n\log n+q\sqrt n)\)
  • Algorithm 2 - \(\Theta(n\log\log n+q\sqrt n)\)
  • Algorithm 3 - \(\Theta(n\log\log n+q\sqrt n)\)
  • Algorithm 4 - \(\Theta(n+q\sqrt n)\)
  • Algorithm 5 - \(\Theta(n+q\sqrt n)\)

前置：

\[\sum_{i=1}^n\dfrac 1i=\Theta(\log n) \]

\[\sum_{p\in[1,n]\cap\mathbb P}\dfrac 1p=\Theta(\log\log n) \]

应用于时间复杂度分析：

\[\sum_{i=1}^n\dfrac ni=\Theta(n\log n) \]

\[\sum_{p\in[1,n]\cap\mathbb P}\dfrac np=\Theta(n\log\log n) \]

Algorithm 1 - \(\Theta(n\log n+q\sqrt n)\)

令素数指示函数 \(\operatorname{pid}(x) = [x\in\mathbb P]\)，数论函数 \(f\) 满足 \(f*1=\operatorname{pid}\)，则显然有 \(f=\operatorname{pid}*\mu\) .

于是有

\[\begin{aligned}\sum_{i=1}^n\sum_{j=1}^mg(\gcd(i,j))&=\sum_{i=1}^n\sum_{j=1}^m\sum_{d\mid\gcd(i,j)}f(d)\\&=\sum_{d=1}^n\sum_{i=1}^{\lfloor\frac nd\rfloor}\sum_{j=1}^{\lfloor\frac md\rfloor}f(d)\\&=\sum_{d=1}^n\left\lfloor\dfrac nd\right\rfloor\left\lfloor\dfrac md\right\rfloor f(d)\end{aligned} \]

\(f\) 直接 \(\Theta(n\log n)\) 暴力卷一下，然后整除分块处理即可 .

Algorithm 2 - \(\Theta(n\log\log n+q\sqrt n)\)

沿用 Algorithm 1 的思路，瓶颈在 Dirichlet 卷积，于是考虑优化一下 . 因为 \(\operatorname{pid}\) 比较难写，后面我们就改成要求 \(f*\mu\) 了 .

rqy 指出，求一个函数 \(f\) 卷上 \(\mu\) 是可以做到 \(\Theta(n\log\log n)\) 的 . 具体的，就是群直积的卷积变换可以分解成独立的卷积变换 .

考虑一个 DP，令 \(\displaystyle dp_{i,n}=\sum_{d\mid n}\mu\left(\dfrac nd\right)f(d)\)，\(\displaystyle\sum\) 中额外要求 \(d\) 只有前 \(i\) 种素因子 .

则可以得到转移：

\[dp_{i,n}=\begin{cases}dp_{i-1,n}&p_i\nmid n\\dp_{i-1,n}-dp_{i-1,n/p_i}&p_i\mid n\end{cases} \]

其中 \(p_i\) 是第 \(i\) 个素数 .

这样枚举一个 \(p_i\) 然后累加贡献，时间复杂度就是 \(\Theta(n\log\log n)\) 的了 .

Algorithm 3 - \(\Theta(n\log\log n+q\sqrt n)\)

其实也不一定要卷，考虑规约到一个经典问题 \(\displaystyle\sum_{i=1}^n\sum_{j=1}^m[\gcd(i,j)=1]\) .

\[\begin{aligned}\sum_{i=1}^n\sum_{j=1}^m[\gcd(i,j)\in\mathbb{P}]&=\sum_{p\in[1,n]\cap\mathbb P}\sum_{i=1}^n\sum_{j=1}^m [\gcd(i,j)=p]\\&=\sum_{p\in[1,n]\cap\mathbb P}\sum_{i=1}^{\lfloor\frac{n}{p}\rfloor}\sum_{j=1}^{\lfloor\frac{m}{p}\rfloor}[\gcd(i,j)=1]\\&=\sum_{i=1}^n\left\lfloor\dfrac{n}{i}\right\rfloor\left\lfloor\dfrac{m}{i}\right\rfloor\sum_{d\mid i,d\in \mathbb{P}} \mu\left(\dfrac{i}{d}\right) \end{aligned} \]

最后一步是一个经典莫反 .

然后就只需要考虑如何处理这个 \(\displaystyle\sum_{d\mid i,d\in \mathbb{P}} \mu\left(\dfrac{i}{d}\right)\) .

最 naive 的想法就是，枚举每个素数 \(d\) 然后累加贡献，这样可以得到预处理是 \(\Theta(n\log\log n)\) 的，最后还要一次整除分块于是总时间复杂度就是 \(\Theta(n\log\log n+q\sqrt n)\) .

Algorithm 4 - \(\Theta(n+q\sqrt n)\)

沿用 Algorithm 3 的思路，考虑优化一下求 \(\displaystyle\sum_{d\mid i,d\in \mathbb{P}} \mu\left(\dfrac{i}{d}\right)\) 的过程 .

可以考虑一个线性筛，直接整一个素数进去的方法比较平凡，说一个 sxb 科技：

令 \(\displaystyle n=\prod_{i=1}^kp_i^{\alpha_i}\) 是 \(n\) 的唯一分解形式，则 \(\displaystyle f(n)=\sum_{d\mid i,d\in \mathbb{P}} \mu\left(\dfrac{i}{d}\right)\) 的取值可以表示为：

• 当 \(\displaystyle\max_{1\le i\le k}\{\alpha_i\}\ge 3\) 时，\(f(n)=0\) .
• 存在 \(i,j\) 使得 \(\alpha_i=\alpha_j=2\) 时，\(f(n)=0\) .
• 只存在一个 \(i\) 使得 \(\alpha_i=2\) 时，\(f(n)=(-1)^k\) .
• 所有 \(\alpha_i\) 都是 \(1\) 时，\(f(n)=k(-1)^{k-1}\) .

线性筛即可 \(\Theta(n)\) 处理 .

Algorithm 5 - \(\Theta(n+q\sqrt n)\)

也不一定要用前面几个比较常见的做法 .

熟知莫比乌斯反演的第二种形式：

\[f(n)=\sum_{n\mid d}g(d)\iff g(n)=\sum_{n\mid d}\mu\left(\dfrac dn\right)f(d) \]

考虑给它编一个意义出来，令 \(f(d)\) 表示 \(d\mid\gcd(x, y)\) 的对数，\(g(d)\) 表示 \(\gcd(x, y)=d\) 的对数 .

显然 \(\displaystyle f(d)=\left\lfloor\dfrac nd\right\rfloor\left\lfloor\dfrac md\right\rfloor\)，于是反演得

\[g(d)=\sum_{x\mid d}\mu\left(\dfrac dx\right)\left\lfloor\dfrac nd\right\rfloor\left\lfloor\dfrac md\right\rfloor \]

于是答案就是

\[\sum_{k\in[1,n]\cap\mathbb P}\sum_{d=1}^n\mu(d)\left\lfloor\dfrac n{kd}\right\rfloor\left\lfloor\dfrac m{kd}\right\rfloor \]

令 \(i=kd\)，则答案变为

\[\sum_{i=1}^n\left\lfloor\dfrac{n}{i}\right\rfloor\left\lfloor\dfrac{m}{i}\right\rfloor\sum_{d\mid i,d\in \mathbb{P}} \mu\left(\dfrac{i}{d}\right) \]

后面就和 Algorithm 4 一样了 . 时间复杂度就是 \(\Theta(n+q\sqrt n)\) .