Description：
设深度为 k的满二叉树的\(\mathrm{Pr\ddot{u}fer}\)编码序列为\(P=\{p_i\}\)
\(m\)次询问，对于每次询问给出 a d m，求\(\sum\limits_{i=0}^{m-1}p_{a+i\times d}\)
\(2\leq k \leq 30 , 1 \leq q \leq 300\)
Examples:
| input1 | output1 |
| 3 5 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 | 2 2 1 3 3 |
| input2 | output2 |
| 4 4 2 1 5 4 4 3 4 8 1 10 3 2 | 18 15 5 13 |
| input3 | output3 |
| 7 1 1 1 125 | 4031 |
Solution：
太神了！
考虑暴力搜索，搜索状态为 search(x, k)，表示以\(x\)为根深度为\(k\)的子树的\(\mathrm{Pr\ddot{u}fer}\)编码中对答案的贡献。为了方便顺便维护两个值 i m分别表示在删除这个子树之前已经删了多少个点，在这棵子树的\(\mathrm{Pr\ddot{u}fer}\)编码中有多少点属于等差数列。
这样 search(x, k)可以递归到 search(2 * x, k - 1), search(2 * x + 1, k - 1)，还要考虑下根节点是否属于等差数列，并且根节点，左子树，右子树的删除顺序和根节点是否有父边也有关系，需要处理一下。
但是这样复杂度会是\(\mathcal{O}(2^k)\)，需要优化。
然后很神的就是将搜索状态设为 solve(k, i, m, hasfa)， hasfa表示根节点是否有父边， ret表示根节点为\(x\)，深度为\(k\)的满二叉树，之前已有\(i\)位\(\mathrm{Pr\ddot{u}fer}\)编码，子树的编码中接下来还有\(m\)个所需的位的答案为\(ret.a\times x+ret.b+c\times \left \lfloor \frac{x}{2} \right \rfloor \)，这样统计答案的好处在于答案的值是关于根节点的函数，答案的值只与所需点的相对位置有关，而与根节点无关。在不影响左右子树答案的合并的前提下，有利于之后的记忆化搜索。
设当前节点(根节点为x)目前答案为\(ret.a, ret.b, ret.c\)左子树(根节点为2x+1)答案为\(son.a, son.b, son.c\)，根据定义合并后\(\begin{split}ans&=ret.a\times x+ret.b+ret.c\times \left \lfloor \frac{x}{2} \right \rfloor + son.a\times (2x)+son.b+son.c\times \left \lfloor \frac{2x}{2} \right \rfloor \\&=(ret.a+2\times son.a+son.c)\times x+(ret.b+son.b)+ret.c\times \left \lfloor \frac{x}{2} \right \rfloor\end{split}\)
右子树同理。
接下来则是记忆化需要记录的状态，首先，只有当该子树对应的编码区间属于\([a,a+(m-1)\times d]\)，且该子树有父边才进行记忆化。于是，记录的状态为 g[k][i], 表示根节点有父边，根节点为\(x\)，深度为\(k\)的满二叉树，且任意\(p’_{i+j*d}\)均为所需点的答案为\(g[k][i].a\times x+g[k][i].b+g[k][i].c \times \left \lfloor \frac{x}{2} \right \rfloor (P’=\{p’_i\}为该子树的编码序列)\)，因为所有有父边深度为\(k\)的满二叉树删除顺序是一定的，每次询问\(d\)又是固定的，所以选取的相对位置的集合（即答案）只与该子树编码中第一个被选取的点的 offset有关。
为了保证复杂度，只记录\(k\leq 15\)的状态。状态总数最多为\(15 \times (2^{15}-1)\)
然后考虑每次搜索时间复杂度，
- 当\(15<k\leq 30\)， 最多只会搜索\(2^{15}\)次
- 当\(1\leq k \leq 15\)，不能记忆化的区间只有\(\mathcal{O}(k)\)个，没有父边的区间最多也只有\(\mathcal{O}(k)\)个
所以单次询问复杂度为\(\mathcal{O}(2^{\frac{k}{2}})\)
Code：
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#include <cstdio> #include <algorithm> #define FOR(a,b,c) for(int a=b;a<=c;a++) using namespace std; struct rec { long long a, b, c; }ans; void addls(rec& x, rec y) { x.a += 2 * y.a + y.c; x.b += y.b; } void addrs(rec& x, rec y) { x.a += 2 * y.a + y.c; x.b += y.a + y.b; } int k, q, tim, a, d, m, lastp; int bl[16][33000]; rec g[16][33000]; //g[k][i]表示根节点有父边，根节点为x，深度为k的满二叉树，且任意px_{i+j*d}均为所需点的答案为g[k][i].a*x+g[k][i].b+g[k][i].c*(x/2); rec solve(int k, int i, int m, int hasfa) { //表示根节点是否有父边，根节点为x，深度为k的满二叉树，之前已有i位prufer编码，接下来还有m个所需的位的答案为ret.a*x+ret.b+ret.c*(x/2); if (m == 0) return {0, 0, 0}; int nextp = ((a - i) % d + d) % d; if (nextp == 0) nextp = d; if (nextp + i < a) nextp = a - i; if (nextp > (1 << k) - 1) return {0, 0, 0}; //nextp为使得i+nextp=a+j*d的最小的nextp //保证该子树中有所需的位 if (k == 1) return {0, 0, 1}; if (k <= 15 && hasfa && i + 1 >= a && i + (1 << k) - 1 <= lastp) { if (bl[k][nextp] == tim) return g[k][nextp]; } //如果完全包含则记忆化 rec ret = {0, 0, 0}; if (nextp <= (1 << (k - 1)) - 1) { int newm = (min(lastp - i, (1 << (k - 1)) - 1) - nextp) / d + 1; addls(ret, solve(k - 1, i, newm, 1)); m -= newm; } //走左子树 if (!hasfa) { int num = i + (1 << (k - 1)); if ((num - a) % d == 0 && num >= a && num <= lastp) { ret.a += 2; ret.b += 1; m--; } } //判断根节点是否为所需位 if (m > 0) { int num = i + (1 << k) - 1; int newm = m - (hasfa && (num - a) % d == 0 && num >= a && num <= lastp); addrs(ret, solve(k - 1, i + (1 << (k - 1)) - hasfa, newm, hasfa)); //注意这里右子树是否有父边应与当前节点一样 } //走右子树 if (hasfa) { int num = i + (1 << k) - 1; if ((num - a) % d == 0 && num >= a && num <= lastp) { ret.c += 1; } } //判断根节点是否为所需位 if (k <= 15 && hasfa && i + 1 >= a && i + (1 << k) - 1 <= lastp) { bl[k][nextp] = tim; g[k][nextp] = ret; } //更新记忆化 return ret; } int main() { scanf("%d%d", &k, &q); while (q--) { tim++; scanf("%d%d%d", &a, &d, &m); lastp = a + (m - 1) * d; ans = solve(k, 0, m, 0); printf("%I64d\n", ans.a + ans.b); } return 0; } |
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Search engines like yahoo, and Yhoo and google in particular, have good-tuned their algorithms to identify websites with highly higher website link process from four weeks to month.
As opposed to incteasing from the Yahoo and google standings, these internet
sites are penalized ffor competitive action and fall from eyesight.
Experts propose that a modest speed, say 15Percent growth in backlinks/month will assist you
to expand steadily and keep you of issues.
If you’re an authority on a matter, pertmit readers recognize that aat the start.
Followers are much molre likely for taking what you say significantly and invest their time into reading through what you’ve written on the topic if you’re an authority.
Don’t talk directly to them, but don’t cover up your
expertise both.
Remove your expression checking computer software.
Emphasis your creating on thorough information and information and which include just the content material you see
thhe most significant. In case your report is lengthier or shorter than you actually designed, whicxh is allowed.
Contijue to keep information heavy, plus your expression count will never issue all
the.
Once you have exprienced some exercise at writing, you wull
have created a significant amount of articles which is spread throughout the internet.
Produce an e-book that contains your very best content articles and then sell on it orr apply it as being a free of charge marketing too the consumers.
When you have a quality eBook, it will likely be sshared by many of
the followers and provide far more organization and targeted traffic aimed at
your website.
It is crucial that you keep to the matter that was requested.
Numerous websites do not desire to use a post that rants so on regafding a totally different article.
Should you be undecided about what thinngs to write about a
particular matter, you could question the requester to fihd out more.
To produce earnings in your published content, you need to make sure
you hae your own personal website. It’s okay to possess
a 1000 word article, provided that yyou keep the content that
youu just submit to article submission sites about 500 or
700 words. You can also have gives and internet affiliate backlinks
on your own site tto get your reader’s attention so aas to make your web site more important in your buyers.
You’re probably going tto convey more queries thn you’re able to
find solutions to, when you startt out with artijcle
promotion. It’s crucial that you just maintain yojr nose on the grindstone.
Once you learn the fundamentals, the veil
is taken off and you will begin to figure a variety of itt out all bby yourself without
having the help of others.
Sometimes when one is searching for a innovative or new way so they can market
place through an report utilizing ann post that is certainly not blatantly advertising anything could be the best way.
By producing a write-up that will not immediately shove an item in someones deal with but quietly shows it oone can acquire more
achievement.
Knowing the significance of information could keep you very competitive.
Evaluate your posts frequently annd make guaranteed to
upgrade it with fresh ideas and knowledge. Keep in mind 2 kinds of tageted traffic to your blog those who are there any
initially and those thbat have been there 100. Teach and seize with information to maintain your
web site feasible.
If someone is hosting a good cause function chances are they should think about employing marketing with
articles to increase the volume oof visitors they may
have participate inn that exct occasion that you can be internet hosting.
Articles conveyimg tthe particular occaxion occurring
and why everyone ought to participate in can consideerably support ones
cause.
As being an poswt internet marketer, you could find you need tto spend a little funds, but what you
will need to make investments can be your time. So
unless you have sufficient leisure time on thhe haands and wrists to commit to ensuring wwork is excellent, the chances are yoou shouldn’t get yourself started on marketing to start with.
Make the topic of the items that you subgmit to article submission sites relevant to the information on your
internet site. Readers, whho would lioe to try comparable content seen oon your web site, will probably
click on through to the web page the place you pitch your product or service.
For instance, if your wegsite is about games, then publish your article about
toys, to bring in your potential audience.
It iis vital that the data on your own web site basically perrtains to your web page.
Article promotion only works when the marrket that you
are currently targeting has an interest ineide the info that yyou offer.
The simplest way understand what your viewers wants
to knnow is to know your audience. When you
know wwho your target audience iss yyou can give them content they are looking for.
When composing for article marketing, mke sure to continue to bbe organized.
Consider your subject matter thoroughly before beginning creating.
Identify your assets and plan on keywords. Determine the time you
wish to pay for composing and adhsre to it. Don’t be derailed by other on the
internet enticements if you look at your references. Stay
focused!
A good article marketing hint iss always to placed
our content constantly. Visitors will receive irritated if they’re following your content and you’re getting irregular.
Try too shoot ffor a minimum of a number of content articles a week.
This will likely make your followers delighted and
wijll allso helkp insure your success.
Concentrating your content creation endeavours on the requirements
of your readers is a confident-fireplace approac to jumpstart an advertising
and marketing program. Thiss implies finding facinating subject
areas, eeping your posts pertinent andd clean, and listening to opinions.
Haave the pleasure of your viewdrs your number
one priority and also you are not able to get it wrong.
Listed here is a hjnt for article promotion! Whhen producing articles, make sure you locate ways to improve your creative process.
Article describes, brainstorming classes, ass well aas other easy artistic resources can assist you make much better articles easily.
Although it is wise to reserve time for good quality control,
having the capability to – generate content easily – twnds to make your career less
difficult!
You can’t compose and you don’t know a lot about ading content
on the internet. Properly, the greatest thing you could potentially do in order to make use of article promotion and its particular great advfertising and marketing power, iis for more information and
you can do so beginning with the tips presented.
Start making a huge aaffect online, just by begibning with distincttive posts, that say it all for you. https://Www.Toto-Mt.com/bbs/board.php?bo_table=6&wr_id=6
WOW just wuat I was searching for. Came here by searching for nicholas https://www.tosureinfor.com/
Hey there just wanted to give you a quick heads up. The text in your content seem to be running off
the screen in Ie. I’m not sure if this is a format
issue or something to do with internet browser compatibility but I figured I’d post to let you
know. The style and design look great though! Hope you get the
issue solved soon. Kudos https://socialbots.xyz/
I would like to say thank you for your materials on youtube channel that really help me to improve my English skills. And then, may I have a question? Do you have any classes in Vietnam? Thank you!
You’re very welcome, Vinh! I’m glad you enjoyed the lessons. I am sorry, right now the English classes in Vietnam have not been opened yet. I’ll inform you whenever they are ready.
Thank you Ms Michelle Tran, I am looking forward to your classes. And I think, just my own opinion, it is better to establish a club as a team on this website every English learner can joint, study and share experiences together by signing up!
All the pdf files are not available currently, Abshirmahad. I’ll inform you right when they’re ready. Now I suggest that you learn the lessons online. Gook luck in your learning journey.
You can totally spend time on the lessons, Carmen. Let me know how they work out for you.
You’re very welcome, Venacinna! I’m glad you enjoyed the lessons. Right now you can learn the lessons online. I’ll inform you whenever they are ready to share for learning offline.
Your method of telling the whole thing iin this paragraph is actually pleasant,every one be capable oof easily understand it, Thanks a lot.
always i used to read smaller posts that as well clear their motive, and that is also happening with this piece of writing which I am reading now.
thank you for your lessons, it’s very useful for the learner and also easy for them to follow each instruction once again much appreciated with that.
Thank you for your kind words. The Grammar lessons will be coming soon. Check them out.
Thanks for providing such a great collection of ques. & answer. Basically, I am a teacher of English language. This information will really help to my students.
It’s my pleasure, Manish. Thanks for sharing the material with your students. Good luck on your teaching journey.
It’s my pleasure, Manish. Thanks for sharing the material with your students. Good luck on your teaching journey.
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