这是笔者的第一篇博客。为了能让更多的人看懂。所以本文的叙述的节奏相对很缓慢，有基础的同学可以选择性地跳读。
作为一个十八线的蒟蒻
，文中可能出现各种BUG和表达歧义的情况，同时，笔者对
的编辑器使用并不是特别熟练，望各位能给我提出意见，帮助我做得更好，谢谢！ （土下座）
原题题面
Rikka is interested in computer science, and she has been practicing coding for two years and a half. Today, she wants to do a simple summary of the algorithms she has learned.
One of the most important algorithms is Quicksort. Though its idea is quite simple, Rikka remembers that it took her a while to prove the time complexity. Let f(n) be the expected number of comparisons required by Quicksort on a sequence with length n. Then
follows the following equations:
After some simple derivations, Rikka finishes the proof and obtains the result that f(n)=O(nlogn): As an outstanding undergraduate student, this problem is just a piece of cake for her.
To make the task more challenging, Rikka asks Yuta, her boyfriend, to set several exercises for her. The following is the hardest one of them:
Consider a modified version of Quicksort: the recursive process terminates once the length of the interval is less than m. At this time, the expected number of comparisons gm(n) can be described with the following equations:
Now, Yuta shows the value of n,m, and he wants Rikka to calculate
. It is generally known that Rikka is not good at math. Therefore she wants you to help her calculate the answer.
Input
The first line is an integer
(
), the number of test cases.
For each test case, the input contains a single line with two positive integers
(
).
Output
For each test case, output a single line with a single number, the value of
.
Clearly, gm(n) is a rational number. Therefore, you are required to print
mod
, i.e., print
mod
where
is the irreducible fraction representation of
.
Sample Input
5
3 1
5 1
5 3
10 5
1000 500
Sample Output
666666674
800000013
400000008
308730177
3107840
题面分析
给你
，让你求
首先我们观察一下这个
内的两个项
和
在
内枚举
,我们会发现：
可以发现这其实是同一个东西。
所以其实原式可转化为↓↓↓
但这个式子计算起来还是很复杂，那怎么办呢？
注意到↓↓↓
我们可以很容易地 想到去消掉
.
于是，构造新式子
我们把不是
的统统移项,得到以下两个式子：
再相减，整理，得到
出于个人偏好我把
换成了
。
估计大部分人都能推到这个式子，
但这个式子怎么求解呢？
或许大部分能推到这里的人都已经被劝退了…
但聪明如你，很快就能找到突破口。
如果有良好的数学基础或参加过数学竞赛，你就会发现这其实是个经典题型
我们把两边同除以
，可以得到
我们把gm(n)/(n+1)看成an,把gm(n-1)/n看做an-1，得到
那就变成了一个累加问题，累加的起点是多少呢？
其实并不是
，而是
。
因为第
项是第一个非0项，且容易得出这一项为
。
所以上式变为
又一个麻烦事出现了：后面这一块儿怎么搞？？？
别急,先试着求下面这个
求出了
~
的
，那
~
其实就是
~
减去
~
。
首先，被高考数列百般折磨过的 我们很快能发现分母可以裂项。
我们把这一大块
列出几项，可以得到：
仔细观察就能发现，除了开头第一项的两个
以外，之后的每一项通过加减抵消，都是两个
（除了最后一项）
ok，那我们回到原来的题目，稍加整理可以得到：
建议各位读者务必自己推一遍来加深印象！
最后，就是对答案求逆元即可。
可能有些读者会不知道逆元是个啥，这里不再赘述，请自行查询。
简单地说呢，
对于
的逆元=(
)*(
对于
的逆元)
看到这里，那就恭喜你AC本题啦！
呵呵，才怪。你会TLE，不停地TLE。
为什么呢？
因为这里的
复杂度是
,而
又非常大(只要出题人够恶，最大
)。
那
应该如何快速计算呢？
也许有人会想到用
（
为欧拉常数）的公式去逼近答案，但是 分数的逆元需要知道其分子与分母的精确值 ，因此该方案可以就地枪毙。
这时候我们就要介绍一种巧妙的方法：分块（暴力）
分块
顾名思义，分块就是把区间分成若干块来进行处理，是个用代码长度换时间效率的操作。
比如说今天
，我们可以把
分成
个
，把这每一个
的和事先算出结果并保存下来。
那对于一个数
,那前
其实是提前处理好的（利用前缀和实现），我们只要暴力去求后面的56即可。
很显然，块分的越多效率越高，但这也会导致事先存下来的结果越来越多，可能导致代码过长而无法提交。
（只要代码长度和内存需求不限，我能打1e9)
这个题给出了
，其实时间是相当充足的 我试着选择分成了
个块(每个块
) （虽然标程则是分成了2000块，但我为了贪HDU的榜就在代码长度的极限边缘疯狂试探,各位可以自行选择合适的区间进行分块） 。
首先可以用这个代码得到每个块的和：
#include<bits/stdc++.h>
using namespace std;
const long long mod=1000000007;
const long long step=200000;
long long qsm(long long a,long long b,long long c)
{
long long ans=1,base=a;
while(b!=0)
{
if(b&1)
ans=ans*base%c;
base=base*base%c;
b>>=1;
}
return ans%c;
}
int main()
{
for(int i=1;i<=5000;i++)
{
//printf("%lld %lld->",(i-1)*step+1,i*step);
long long ans=0;
for(int j=(i-1)*step+1;j<=i*step;j++)
ans=(ans+qsm(j,mod-2,mod)%mod)%mod;
//这一步是统计逆元求和（采用费马小定理，不懂的请回去查查逆元）
printf("%lld,",ans);
}
}
运行大约
左右会得到这一堆数据…
请大胆，并且勇敢地把它们复制到你的代码里…然后统计前缀和即可。
（事实上，连前缀和你都可以事先算好）
AC代码
在经历了让人骂娘的数学推导和分块技巧后，我们就可以得到最终的代码。
AC代码如下：
#include<bits/stdc++.h>
using namespace std;
const long long mod=1000000007;
const long long step=200000;
//以下是大片的分块
long long a[5001]={0,118091052,131785116,352153536,850862847,428991732,414611584,583182491,896174392,124990201,323589505,206931761,988983654,858305161,425217713,413698619,414715665,478330630,532001401,40597178,482759685,232548225,560217862,630719995,160786774,867224271,754345375,576119455,843824329,49857979,817377043,877933418,47756642,342578027,586345914,446442141,749363516,771591947,343217317,32629120,398935802,68857293,968437606,441777156,93204835,182454693,196734065,701619129,768503428,470912794,72014525,573194313,738867983,285235222,132817554,304163043,35285861,250861922,516139955,379179065,183692795,389689611,543506390,769763854,31052462,373670191,840857983,367020146,54098258,781412962,174227848,104648028,599713848,993149165,704685486,307090292,469396434,67700252,659789365,448472765,456643961,47455224,273593776,96158151,866612273,751077136,990010787,151551952,86872827,817727400,102820197,655694974,28185869,893496173,427196953,543956351,369115458,511875420,703187315,367623814,189435206,822865222,367568985,982567104,204894272,989468986,813291513,532238400,850856025,353723944,646422091,169078682,389332165,199016196,257134087,89401236,969152043,255020093,898253226,274836613,699071793,827249742,247052267,587240731,281418569,980927262,197837567,667869657,639806788,134558025,134239210,337418715,403345377,578501799,100623182,196155066,603661715,462457033,503291518,620969426,328569085,633470314,283177385,721937275,689807350,149992644,204440151,559376136,775588884,934012348,795790096,781025085,677557667,372473316,887235233,102452577,828513970,105905930,689010280,425213825,558196091,224390157,283927529,717859527,955209097,950804136,363393780,722182354,899772563,814056465,351264506,715604119,969997742,9908974,65756994,357593050,28699444,37465593,650590847,853427962,23083244,569668678,378502735,778943897,890627771,411546796,513858302,982781493,671691476,62358974,934744009,327886125,890702901,256960121,712408225,783211611,204794185,521545503,419628499,792012729,985968231,226260985,397316849,867169170,661425854,333043889,120332596,880797499,652197633,290649867,544490628,118674346,58796548,858227890,211054111,385174173,612397191,209647448,659802176,914281364,936024665,916859008,731598650,690190169,471209919,666401078,884842952,76906371,693773595,185282570,915161368,774980465,422045434,479175425,948262056,222136099,114510549,935939795,987114971,221123203,885344792,258892876,51816306,353968481,444212249,118902244,216265929,936656056,422103538,282977247,665915027,625327049,676838306,595001633,582079856,693728179,126805007,200740079,579241904,259940450,764551612,374681877,649459652,29860243,956298180,966501167,332299133,377496463,138535479,973454198,338402575,549998562,463566624,683393756,435363108,701178337,742601738,386600011,782103008,899704980,43249130,933551108,749087341,518744012,820468954,641650320,803852494,858515125,64250767,339758727,760370368,395998646,82392807,371144566,840917056,648203573,217850784,270518674,158333836,456541281,615568434,303962314,653688076,496830008,114340000,726317863,792034163,279780217,78145485,775906948,898963757,298704831,298475456,89552584,9361075,498217473,271157956,355284752,932511891,346897655,275181087,871319715,805220866,603948208,212907660,388158757,606719565,992301868,282623520,272206076,32176290,337928137,328345548,385601531,21411627,740739850,89949909,874279910,214334658,721313407,842842872,700176444,924833221,380473937,240954799,489897412,856144573,999612030,643033151,464528272,522017862,51650593,367794839,218450551,129579181,581022501,447000608,750678394,112690503,721378979,6314451,154276233,302351311,366114374,638441816,349532389,337582189,446647611,970404432,49341525,272027785,880897112,854027573,631695800,902620258,285475053,429010688,688158583,264794462,486740329,120580617,698145367,770823936,739567288,277610613,461491198,243849878,24815707,237676884,286878021,63956382,757812480,365765156,610087711,549949010,909715282,82065213,71347275,711316316,704974366,486901062,588226877,603586096,446113263,592704334,561721545,661325950,933849411,45970546,2793756,823457921,721192327,581960715,533832813,43086126,63965068,639134744,922642952,834344221,236953710,876590700,112183037,479560418,743692132,613072198,767683245,274711143,562899998,893286299,700370167,581105403,605115604,489965267,667395305,662684595,774797253,272462255,698822391,859154730,314742526,173026752,607672027,958722126,724148786,364353395,394325096,23864281,819579337,159354481,831637081,514207796,541262854,419114703,547679135,519893462,805392010,330107744,466997310,985041438,840690617,978826950,919433703,346535455,69898686,582329781,23312128,255257792,255256645,847446258,224727034,685187932,391435997,719458091,245853096,940736872,131739081,632804317,447566131,905871735,860826585,449481106,485223822,97625822,212336462,773552418,676493393,221000168,926282633,242255317,534007544,62453652,349622713,150952250,411236816,588351154,456599721,464623868,363806407,242071425,456164018,965145919,488491869,701723857,319107790,859801440,2129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42291555,710818570,874576302,607544995,556507447,528901146,474674223,619929507,140502899,462394188,211660059,60274720,599672724,681109035,400630278,634103707,818278669,986563752,737098594,30959850,512173300,463274803,410630894,509300222,408329899,799563341,155721609,318164454,418812648,934087783,116713615,666481567,431343799,7765918,516863257,589075183,938895774,770823941,501410792,240746363,187736516,303682542,476653520,432628920,413919792,546634569,299087952,601753051,618843058,333244378,315049483,345900801,793784926,175543627,364098872,727679176,384949256,696170517,417433074,146235752,526258066,686521706,739963263,593982149,591050999,254348542,702914471,938714392,10579607,515732742,867708561,410278879,924595477,233707226,587900132,939699047,197137829,246932522,391551149,703355036,617878629,186740161,564002463,231977898,493887163,116115677,280218466,170413184,218779233,849109004,142921081,161145534,20361462,145432232,591906246,27630750,170389662,783817058,175611384,967338446,651880158,284733303,918438880,652869738,711927730,619022677,448280825,482596142,430193068,641796499,346138718,391663479,758048688,296839193,769468997,820212560,591200966,836350810,265308204,43311053,784766846,109916784,631440516,394998453,603915060,856392761,971209718,781892395,64737279,683131145,696665912,84403801,430621575,933874640,685958907,426998517,624664852,764606474,978289939,623058228,147314235,294244999,941833369,65651773,458138780,19178082,303398474,220369148,610345760,67654989,806893283,450916831,13820299,701289333,792933344,848484656,511170270,815633807,616844299,138151979,67594832,71229122,469465289,316968771,759047640,42401900,903444717,219572045,725793780,769398288,139784574,199037145,610972207,522546426,568351822,625677225,17435162,944944563,711996688,710225955,476276232,689487794,854085530,803234105,297639697,588307026,86238001,647486615,778649943,871280098,77846006,11376644,182966598,268688235,844671948,522355042,173255656,206238651,546568969,49471069,230426072,647123113,709990339,33042983,337734887,68497698,546877891,129799519,53982182,200318095,909988866,527321362,971986196,80807364,251423598,39546868,613406257,92327817,386987580,586055717,666111336,851189999,670376812,89095542,533464096,401917649,382271108,745325510,583002609,817566165,493426521,847063805,796595243,136774500,793241230,903974966,779211342,911149954,542859178,978902351,961216414,512123029,990947044,354508222,766455328,488410785,998130413,573513932,908924850,983446082,210754086,562058698,813199343,226723730,724855541,614969533,496173178,958212524,746987768,771760418,105197521,255881430,838035952,573295480,200324207,219175086,800839085,176314033,909772918,936431088,152305306,356124530,361923812,787756165,729246228,650218986,777340073,870978930,965633591,653998898,30793684,660774681,61250610,165290102,67518349,742428555,85566649,710395153,154036379,991763157,874873738,289549620,55836750,599637459,573728321,827428017,528645275,204543345,743654603,861917126,317149718,953913460,774710426,600965476,120057746,714780047,49261505,166658,317938767,95258846,747303938,238798365,731399395,570774064,671411,135517135,565527392,374626710,494565983,236254410,905712596,909403036,561628438,21360582,907620390,168997845,270086756,343216410,487181198,286060871,501690421,459664866,978359592,721480066,953143708,757034901,698678547,5517104,64215904,393672024,198959833,743700673,398220125,610408506,479024100,196335629,542856706,573605979,891196010,395822946,571606184,971949655,885647588,91269188,338862875,810530865,640285464,650848102,447495438,495956559,493553694,350906975,261118770,594244212,662416839,720184520,747495615,42858689,783845377,798900390,604850678,535531762,355258850,943528802,68126466,953261638,400622470,962426382,310037247,498315705,336738137,513107116,695370797,659674271,293488843,184549278,455295213,857982171,37652734,246771599,470271469,398716573,957278417,371560645,499207120,537248582,509749007,861075452,235427582,352785947,319951370,641309758,773747213,674298920,312097232,574308079,903737535,243495620,165520987,303504215,806453512,323208615,495376785,994133291,19603880,624219271,620399574,715491697,8943876,405130267,29917935,627920072,170146410,454398479,818028434,4631296,191274509,99954875,581863674,203951551,530514707,320800218,69960623,608096023,194028586,173803073,657660899,464844015,591960575,68867489,318148430,350060072,847480571,856437366,930493866,756115653,365749873,461143255,695445211,543945870};
void init()
{
for(int i=2;i<=5000;i++)
a[i]=(a[i]+a[i-1])%mod;//处理前缀和
}
long long qsm(long long a,long long b,long long c)//快速幂
{
long long ans=1,base=a;
while(b!=0)
{
if(b&1)
ans=ans*base%c;
base=base*base%c;
b>>=1;
}
return ans%c;
}
long long sigma(long long n)//处理Σ
{
long long sum=n/step;
long long ans=a[sum];
for(int i=sum*step+1;i<=n;i++)
ans=(ans+qsm(i,mod-2,mod)+mod)%mod;
return ans%mod;
}
int main()
{
init();
int t;
scanf("%d",&t);
while(t--)
{
long long m,n;
scanf("%lld%lld",&n,&m);
if(n==m)//特殊情况，因为题目允许二者相等
{
printf("0\n");
continue;
}
if(n==m+1)//我们推出的式子是必要满足n≥m+2的，因此要特判
{
printf("%lld\n",m);
continue;
}
long long ans=(((m*(n+1)%mod)*qsm(m+2,mod-2,mod)%mod)+2*(n+1)*((sigma(n)-sigma(m+1)+mod)%mod)%mod-((2*n-2)%mod)+((2*m*(n+1)%mod)*qsm(m+2,mod-2,mod)%mod)+mod+mod+mod)%mod;
//长长的求和，最后那边3个mod是怕出现负数随便写的
printf("%lld\n",ans%mod);
}
}
因为分了5000块，所以跑出来的耗时相当短，并成功地骗到榜首。
相对应的代码量就上去了…
后记
其实昨天比赛的时候不会分块，推导了半天最后卡在
的算法一下午…
分块与其说是技巧，倒不如说是为了卡时间的“歪门邪道”。
最后，感谢各位能看到这里，若还有什么不懂的或是文本内容有误，可以在评论区留言，我尽快处理 （咕咕咕） 。
Dr.Gilbert 2019.8.20
顺带一提，数据出的还挺弱的（手动滑稽）