尾递归与Continuation
关于递归操作,相信大家都已经不陌生。简单地说,一个函数直接或间接地调用自身,是为直接或间接递归。例如,我们可以使用递归来计算一个单向链表的长度:
public class Node
{
public Node(int value, Node next)
{
this.Value = value;
this.Next = next;
}
public int Value { get; private set; }
public Node Next { get; private set; }
}
编写一个递归的GetLength方法:
public static int GetLengthRecursively(Node head)
{
if (head == null) return 0;
return GetLengthRecursively(head.Next) + 1;
}
在调用时,GetLengthRecursively方法会不断调用自身,直至满足递归出口。对递归有些了解的朋友一定猜得到,如果单项链表十 分长,那么上面这个方法就可能会遇到栈溢出,也就是抛出StackOverflowException。这是由于每个线程在执行代码时,都会分配一定尺寸 的栈空间(Windows系统中为1M),每次方法调用时都会在栈里储存一定信息(如参数、局部变量、返回地址等等),这些信息再少也会占用一定空间,成 千上万个此类空间累积起来,自然就超过线程的栈空间了。不过这个问题并非无解,我们只需把递归改成如下形式即可(在这篇文章里我们不考虑非递归的解法):
public static int GetLengthTailRecursively(Node head, int acc)
{
if (head == null) return acc;
return GetLengthTailRecursively(head.Next, acc + 1);
}
GetLengthTailRecursively方法多了一个acc参数,acc的为accumulator(累加器)的缩写,它的功能是在 递归调用时“积累”之前调用的结果,并将其传入下一次递归调用中——这就是GetLengthTailRecursively方法与 GetLengthRecursively方法相比在递归方式上最大的区别:GetLengthRecursive方法在递归调用后还需要进行一次 “+1”,而GetLengthTailRecursively的递归调用属于方法的最后一个操作。这就是所谓的“尾递归”。与普通递归相 比,由于尾 递归的调用处于方法的最后,因此方法之前所积累下的各种状态对于递归调用结果已经没有任何意义,因此完全可以把本次方法中留在堆栈中的 数据完全清除,把空间让给最后的递归调用。这样的优化1便使得递归不会在调用堆栈上产生堆积,意味着即时是“无限”递归也不会让堆 栈溢出。这便是尾递归的优势。
有些朋友可能已经想到了,尾递归的本质,其实是将递归方法中的需要的“所有状态”通过方法的参数传入下一次调用中。对于 GetLengthTailRecursively方法,我们在调用时需要给出acc参数的初始值:
GetLengthTailRecursively(head, 0)
为了进一步熟悉尾递归的使用方式,我们再用著名的“菲波纳锲”数列作为一个例子。传统的递归方式如下:
public static int FibonacciRecursively(int n)
{
if (n < 2) return n;
return FibonacciRecursively(n - 1) + FibonacciRecursively(n - 2);
}
而改造成尾递归,我们则需要提供两个累加器:
public static int FibonacciTailRecursively(int n, int acc1, int acc2)
{
if (n == 0) return acc1;
return FibonacciTailRecursively(n - 1, acc2, acc1 + acc2);
}
于是在调用时,需要提供两个累加器的初始值:
FibonacciTailRecursively(10, 0, 1)
尾 递归与Continuation
Continuation, 即为“完成某件事情”之后“还需要做的事情”。例如,在.NET中标准的APM调用方式,便是由BeginXXX方法和EndXXX方法构成,这其实便是 一种Continuation:在完成了BeginXXX方法之后,还需要调用EndXXX方法。而这种做法,也可以体现在尾递归构 造中。例如以下为阶乘方法的传统递归定义:
public static int FactorialRecursively(int n)
{
if (n == 0) return 1;
return FactorialRecursively(n - 1) * n;
}
显然,这不是一个尾递归的方式,当然我们轻易将其转换为之前提到的尾递归调用方式。不过我们现在把它这样“理解”: 每次计算n的阶乘时,其实是“先获取n – 1的阶乘”之后再“与n相乘并返回”,于是我们的FactorialRecursively方法可以改造成:
public static int FactorialRecursively(int n)
{
return FactorialContinuation(n - 1, r => n * r);
}
// 6. FactorialContinuation(n, x => x)
public static int FactorialContinuation(int n, Func<int, int> continuation)
{
...
}
FactorialContinuation方法的含义是“计算n的阶乘,并将结果传入continuation方法,并返回其调用结果”。于 是,很容易得出,FactorialContinuation方法自身便是一个递归调用:
public static int FactorialContinuation(int n, Func<int, int> continuation)
{
return FactorialContinuation(n - 1,
r => continuation(n * r));
}
FactorialContinuation方法的实现可以这样表述:“计算n的阶乘,并将结果传入continuation方法并返回”,也就是“计算n – 1的阶乘,并将结果与n相乘,再调用continuation方法”。为了实现“并将结果与n相乘,再调用continuation方法”这个逻辑,代码 又构造了一个匿名方法,再次传入FactorialContinuation方法。当然,我们还需要为它补充递归的出口条件:
public static int FactorialContinuation(int n, Func<int, int> continuation)
{
if (n == 0) return continuation(1);
return FactorialContinuation(n - 1,
r => continuation(n * r));
}
很明显,FactorialContinuation实现了尾递归。如果要计算n的阶乘,我们需要如下调用 FactorialContinuation方法,表示“计算10的阶乘,并将结果直接返回”:
FactorialContinuation(10, x => x)
再加深一下印象,大家是否能够理解以下计算“菲波纳锲”数列第n项值的写法?
public static int FibonacciContinuation(int n, Func<int, int> continuation)
{
if (n < 2) return continuation(n);
return FibonacciContinuation(n - 1,
r1 => FibonacciContinuation(n - 2,
r2 => continuation(r1 + r2)));
}
在函数式编程中,此类调用方式便形成了“Continuation Passing Style(CPS)”。由于C#的Lambda表达式能够轻松构成一个匿名方法,我们也可以在C#中实现这样的调用方式。您 可能会想——汗,何必搞得这么复杂,计算阶乘和“菲波纳锲”数列不是一下子就能转换成尾递归形式的吗?不过,您试试看以下的例子呢?
对二叉树进行先序遍历(pre-order traversal)是典型的递归操作,假设有如下TreeNode类:
public class TreeNode
{
public TreeNode(int value, TreeNode left, TreeNode right)
{
this.Value = value;
this.Left = left;
this.Right = right;
}
public int Value { get; private set; }
public TreeNode Left { get; private set; }
public TreeNode Right { get; private set; }
}
于是我们来传统的先序遍历一下:
public static void PreOrderTraversal(TreeNode root)
{
if (root == null) return;
Console.WriteLine(root.Value);
PreOrderTraversal(root.Left);
PreOrderTraversal(root.Right);
}
您能用“普通”的方式将它转换为尾递归调用吗?这里先后调用了两次PreOrderTraversal,这意味着必然有一 次调用没法放在末尾。这时候便要利用到Continuation了:
public static void PreOrderTraversal(TreeNode root, Action<TreeNode> continuation)
{
if (root == null)
{
continuation(null);
return;
}
Console.WriteLine(root.Value);
PreOrderTraversal(root.Left,
left => PreOrderTraversal(root.Right,
right => continuation(right)));
}
我们现在把每次递归调用都作为代码的最后一次操作,把接下来的操作使用Continuation包装起来,这样就实现了尾递归, 避免了堆栈数据的堆积。可见,虽然使用Continuation是一个略有些“诡异”的使用方式,但是在某些时候它也是必不可少的使用技巧。
Continuation的改进
看看刚才的先序遍历实现,您有没有发现一个有些奇怪的地方?
PreOrderTraversal(root.Left,
left => PreOrderTraversal(root.Right,
right => continuation(right)));
关于最后一步,我们构造了一个匿名函数作为第二次PreOrderTraversal调用的Continuation,但是其内部直接调用了 continuation参数——那么我们为什么不直接把它交给第二次调用呢?如下:
PreOrderTraversal(root.Left,
left => PreOrderTraversal(root.Right, continuation));
我们使用Continuation实现了尾递归,其实是把原本应该分配在栈上的信息丢到了托管堆上。每个匿名方法其实都是托管堆上 的对象,虽然说这种生存周期短的对象不会对内存资源方面造成多大问题,但是尽可能减少此类对象,对于性能肯定是有帮助的。这里再举一个更为明显的例子,求 二叉树的大小(Size):
public static int GetSize(TreeNode root, Func<int, int> continuation)
{
if (root == null) return continuation(0);
return GetSize(root.Left,
leftSize => GetSize(root.Right,
rightSize => continuation(leftSize + rightSize + 1)));
}
GetSize方法使用了Continuation,它的理解方法是“获取root的大小,再将结果传入continuation,并返回其调 用结果”。我们可以将其进行改写,减少Continuation方法的构造次数:
public static int GetSize2(TreeNode root, int acc, Func<int, int> continuation)
{
if (root == null) return continuation(acc);
return GetSize2(root.Left, acc,
accLeftSize => GetSize2(root.Right, accLeftSize + 1, continuation));
}
GetSize2方法多了一个累加器参数,同时它的理解方式也有了变化:“将root的大小累加到acc上,再将结果传入 continuation,并返回其调用结果”。也就是说GetSize2返回的其实是一个累加值,而并非是root参数的实际尺寸。当然,我们在调用时 GetSize2时,只需将累加器置零便可:
GetSize2(root, 0, x => x)
不知您清楚了吗?
结束
在命令式编程中,我们解决一些问题往往可以使用循环来代替递归,这样便不会因为数据规模造成堆栈溢出。但是在函数式编程中,要实现“循环”的唯 一方法便是“递归”,因此尾递归和CPS对于函数式编程的意义非常重大。了解尾递归,对于编程思维也有很大帮助,因此大家不妨多 加思考和练习,让这样的方式为自己所用。
注1:事实上,在C#中,即使您实现了尾递归,编译器(包括C#编译器及JIT)也不会进行优化,也就是说还是无法避免 StackOverflowException。
源文档 <http://blog.201314.info/2010/05/24/%E5%B0%BE%E9%80%92%E5%BD%92%E4%B8%8Econtinuation.html>
Jan 14, 2023 01:04:01 AM
Thank you for this impressive report. I am refreshed following reading this. Thank you! 카지노사이트
Feb 12, 2023 09:39:16 PM
This is my first time i visit here. I found so many informative stuffs on your blog, especially its discussion. Form the tons of comments and posts, I guess I am not the only one having all enjoyment here. Keep up the good work. Macbook Pro
===========================
Hello i am so delighted I discovered your blog, I actually discovered you by error, while I was searching Yahoo for something else, Anyways I am here now and would just like to say thanks for a great blog posting and a all round absorbing blog (I also love the theme/design), I do not have time to read it all at the right now but I have bookmarked it and also added your RSS feeds, so when I have time I will be back to read more. 여우알바
Feb 22, 2023 08:52:38 PM
Also, whatever happened to those vaunted shields that in the television show always protected the ship from harm? In this movie the shields are about as effective as paper-mache as the Enterprise is strafed, bombed, rocketed, smashed, tossed, toppled, and shaken like a baby’s toy. 여우알바
Oct 05, 2023 11:21:21 PM
I was recommended this web site by my cousin. I am not sure whether this post is written by him as nobody else know such detailed about my difficulty. You are amazing! Thanks! 花藝設計
Dec 09, 2023 03:37:43 PM
There are a couple of interesting points in time on this page but I don’t determine if I see them all center to heart. There exists some validity but I’ll take hold opinion until I consider it further. Great post , thanks and now we want a lot more! Included in FeedBurner at the same time 玫瑰花
===================
Thanks for the good critique. Me & my friend were just preparing to do a little research about this. We got a book from our area library but I think I’ve learned more from this post. I’m very glad to see such magnificent info being shared freely out there… 開張花籃花牌
====================
There may be jacksonville florida health insurance a means in making cash not having doing work yourself tired in addition to without having restricting your moment. You need to use the ability on the Online to earn more income and now have any time to truly enjoy it. No matter the way tiny you realize about the Net as well as how much internet business encounter you will have, you possibly can take advantage of the countless incredible opportunities to generate income right on your computer. 訂花推介
====================
I am curious to find out what blog system you’re working with? I’m experiencing some small security issues with my latest blog and I would like to find something more safeguarded. Do you have any suggestions? 訂花
Dec 09, 2023 11:22:42 PM
This will be the correct weblog for really wants to learn about this topic. You are aware of much its nearly challenging to argue with you (not that I actually would want…HaHa). You actually put a new spin on the topic thats been discussing for some time. Fantastic stuff, just great! amazon
Dec 27, 2023 02:11:46 AM
Very interesting subject , thanks for putting up. universitas medan
Jan 11, 2024 04:41:50 PM
Pretty nice post, thanks for the awesome article. I’m having troubles subscribing to your blogs feed. Thought I’d let you know Agencja marketingowa Gdańsk
===================
Hi there this is kinda of off topic but I was wondering if bloger use editors or if you have to manually code with HTML. I’m starting a blog soon but have no coding know-how so I wanted to get guidance from someone with experience. Any help would be greatly appreciated! 訂花
===================
I like the precious information you offer for your articles. I will be able to bookmark your weblog and feature my youngsters check up here generally. I’m somewhat certain they are going to learn numerous new stuff here than anyone else! 訂花推介
===================
Hey you. I do not know whether it’s acceptable, but this blog is really well designed. 開張花籃花牌
===================
I’m impressed, I have to admit. Truly rarely will i encounter a blog that’s both educative and entertaining, and without a doubt, you have hit the nail on the head. Your concept is outstanding; the problem is something that too few folks are speaking intelligently about. I’m delighted that we came across this within my seek out some thing relating to this. 玫瑰花
===================
This is the type of movie that the less you know, the more enjoyable it will be. Seiko King Samurai
===================
This post something certainly worth reading. There are tons of sites that really make no sense at all. Please keep up the fresh blogging and many more people will keep coming. seiko watch bands
===================
Youre so cool! I dont suppose Ive read anything such as this just before. So nice to discover somebody with some original ideas on this subject. realy thanks for starting this up. this fabulous website is a thing that is needed on the net, someone after a little originality. beneficial problem for bringing a new challenge on the internet! watch straps
===================
Hiya. Very cool web site!! Man .. Beautiful .. Superb .. I’ll bookmark your web site and take the feeds also…I’m glad to find a lot of useful info here in the article. Thank you for sharing.. seiko 5 sports gmt
===================
Hello, you used to write fantastic, but the last few posts have been kinda boring… I miss your tremendous writings. Past few posts are just a bit out of track! come on! 二手電腦
Jan 21, 2024 10:29:07 PM
i would love to get some free calendars on the internet, are there are sites or company that gives one?,. AI Workflow Automation Guide
Jan 26, 2024 06:33:40 AM
You made some good points there. I did a search on the subject and found most individuals will go along with with your website. FM카지노먹튀
Feb 16, 2024 05:35:25 AM
I feel that is among the so much important info for me. And i’m satisfied studying your article. However wanna commentary on some normal things, The site taste is ideal, the articles is actually nice . Good task, cheers. black dollar cleaning machine
==============
Hi, your blog is full of comments and it is very active, 母親節
==============
Yes, really. I join told all above. We can communicate on this theme. Here or in PM. 玫瑰花
==============
Very good written blog. It will be useful to everyone who usess it, including myself. Keep up the good work. 生日訂花
==============
being a blogger myself , i can see someone with great potential, 法式花藝
==============
I visited a lot of website but I believe this one has got something extra in it. 手機回收
==============
There are a few interesting points in time in this posting but I do not determine if I see these people center to heart. There may be some validity but Let me take hold opinion until I take a look at it further. Very good write-up , thanks and we want much more! Included in FeedBurner likewise Seiko King Samurai
==============
I was just imagine about it and you provided me the correct information I really bookmark it seiko watch bands
==============
Awsome site! I am loving it!! Will be back later to read some more. I am taking your feeds also watch straps
==============
An fascinating discussion will be worth comment. I do think that you should write on this topic, it might not often be a taboo subject but generally persons are insufficient to talk on such topics. To the next. Cheers seiko 5 sports gmt
Feb 17, 2024 07:01:21 AM
Good info, good to be knowledge and distributed to the public ZLD SYSTEM MANUFACTURER (ZERO LIQUID DISCHARGE SYSTEM)
=======================
You ought to be a part of a tournament personally of the most useful blogs over the internet. I’m going to suggest this site! ARSENIC REMOVAL PLANT MANUFACTURER
Mar 20, 2024 03:24:10 PM
Spot i’ll carry on with this write-up, I must say i believe this excellent website requirements far more consideration. I’ll probably be once more to see additional, thank you that information. Download WordPress
Mar 21, 2024 11:26:56 PM
hey there and thank you for your info – I have definitely picked up something new from right here. I did however expertise some technical issues using this website, as I experienced to reload the web site a lot of times previous to I could get it to load correctly. I had been wondering if your web host is OK? Not that I’m complaining, but sluggish loading instances times will very frequently affect your placement in google and could damage your high-quality score if ads and marketing with Adwords. Anyway I’m adding this RSS to my email and can look out for a lot more of your respective interesting content. Ensure that you update this again soon.. 母親節
===================
puppies and dogs are very cute, i always love to play with them during my spare time’ 玫瑰花
===================
This is Great. So much good, useful and well written information and entertaining too! I appreciate the time you have spent on this. It is a big help for me. – Thank you! 生日訂花
===================
Awesome! I thank you your input to this matter. It has been useful. my blog: maple syrup 法式花藝
===================
I believe there’s a issue with your blog working with Safari internet browser. 回收iphone
===================
Just as before, you need to alteration your tax bill obligations for anybody who is expected an important substantial return. Seiko King Samurai
===================
Awesome site! You’ve some quite interesting posts.. Nice background as well haha. Keep up the nice work, Ill make sure to come across to see really your page! seiko watch bands
===================
I am glad to read this post, it’s an impressive piece. I am always searching for quality posts and articles and this is what I found here, I hope you will be adding more in future. watch straps
====================
Only wanna say that this is extremely helpful, Thanks for taking your time to write this. seiko 5 sports gmt
Mar 28, 2024 06:46:49 PM
I would like to point out my gratitude for your kind-heartedness in support of women who actually need help with that situation. Your real commitment to passing the solution all through became pretty insightful and has consistently encouraged professionals like me to realize their objectives. Your new useful tips and hints can mean a lot to me and still more to my peers. Thanks a lot; from all of us. Tech
====================
evening dresses should always be classy, simple but elegant. you don’t need to invest several hundred bucks on a classy evening dress“ Tech
Mar 31, 2024 03:51:07 AM
Thanks, I’ve recently been searching for info about this subject matter for ages and yours is the best I’ve discovered so far. klub piłkarski
========================
Appreciate bothering to talk about this unique, Personally i think eagerly measurements and thus really enjoy figuring out regarding this unique theme. Any time prospects, any time you generate expertise, on earth would you ideas writing your personal web site utilizing even further important information? This is of great help for anyone. Clean black money in Africa
Apr 12, 2024 03:40:09 PM
You actually make it seem so easy along with your presentation however I to find this topic to be actually one thing which I feel I’d by no means understand. It kind of feels too complicated and extremely vast for me. I’m looking forward on your subsequent put up, I will try to get the cling of it! Yep Marketing Agency
Apr 27, 2024 04:30:53 AM
There is obviously a lot to realize about this. I believe you made various good points in features also. 法式花藝
===============
Good day! I simply wish to give an enormous thumbs up for the great info you’ve got right here on this post. I might be coming back to your blog for more soon. 生日訂花
===============
Can I simply say what a aid to find someone who actually knows what theyre speaking about on the internet. You definitely know how one can bring an issue to gentle and make it important. More individuals need to learn this and perceive this aspect of the story. I cant believe youre not more well-liked since you definitely have the gift. 玫瑰花
===============
How do I start to learn how to Hip Hop dance? 母親節
May 04, 2024 02:21:34 AM
I must say, as a lot as I enjoyed reading what you had to say, I couldnt help but lose interest after a while. Its as if you had a wonderful grasp on the subject matter, but you forgot to include your readers. Perhaps you should think about this from far more than one angle. Or maybe you shouldnt generalise so considerably. Its better if you think about what others may have to say instead of just going for a gut reaction to the subject. Think about adjusting your own believed process and giving others who may read this the benefit of the doubt. AVデータベース
May 06, 2024 05:15:05 PM
I want to show my thanks to you just for rescuing me from such a predicament. Because of looking throughout the world wide web and coming across thoughts that were not helpful, I thought my entire life was well over. Living without the approaches to the issues you’ve solved as a result of your main guide is a crucial case, as well as the kind which may have adversely damaged my career if I had not noticed your web page. Your actual skills and kindness in dealing with all the stuff was crucial. I don’t know what I would’ve done if I had not come upon such a step like this. I can at this time look ahead to my future. Thanks so much for the impressive and effective help. I will not be reluctant to recommend your web sites to any individual who should receive guidelines on this problem. Ośrodek Terapii
====================
you got a very excellent website, Sword lily I found it through yahoo. Kredyty dla Polaków w Norwegii
====================
A powerful share, I just given this onto a colleague who was doing a bit evaluation on this. And he actually bought me breakfast as a result of I discovered it for him.. smile. So let me reword that: Thnx for the deal with! However yeah Thnkx for spending the time to debate this, I really feel strongly about it and love reading more on this topic. If potential, as you turn out to be expertise, would you thoughts updating your blog with more particulars? It’s highly helpful for me. Huge thumb up for this weblog put up! Lechia Gdańsk Klub Piłkarski
May 15, 2024 08:10:13 PM
Thanks for the one of the nicest blogs i have read in a while. 온라인슬롯
May 17, 2024 03:31:39 PM
I just added this blog site to my feed reader, excellent stuff. Cannot get enough! 62mas
================
Just where maybe you’ve discovered the supply for the purpose of this kind of post? Wonderful studying I have subscribed to your site feed. seiko skx
================
I have been in the same situation before. It’s not as easy an answer as you think it is, it is something that you will need to think through for yourself over a period of time. rubber strap
================
of fighters to grace the Manny Pacquiao vs Shane Mosley Pay Per View event scheduled today Seiko King Samurai
================
You could definitely see your expertise in the paintings you write. The arena hopes for more passionate writers such as you who are not afraid to mention how they believe. All the time go after your heart. rentacarkosovo 回收手提電腦
================
I will invite all my friends to your blog, you really got a great blog . iphone回收價
================
You completed certain good points there. I did searching on the subject matter and found most persons will go together with your blog 手機回收價格表
================
Excellent post, thank you a lot for sharing. Do you have an RSS feed I can subscribe to? 魔術表演
================
Thank you a lot for sharing this with all people you actually recognize what you’re speaking about! Bookmarked. Please additionally talk over with my web site =). We could have a hyperlink alternate arrangement among us! 兒童魔術班
================
I recognize there’s lots of spam on this website. Do you need help cleansing them up? I may help among courses! 魔術
================
Some really great info , Gladiola I found this. magic class
May 21, 2024 01:17:44 AM
I’d need to talk to you here. Which is not some thing I do! I love reading an article that can get people to believe. Also, many thanks allowing me to comment! AVメーカー一覧ナビ
=================
An impressive share, I with all this onto a colleague who was simply performing a small analysis on this. And hubby in reality bought me breakfast due to the fact I discovered it for him.. smile. So well then, i’ll reword that: Thnx for your treat! But yeah Thnkx for spending the time to discuss this, Personally i think strongly over it and love reading regarding this topic. If it is possible, as you grow expertise, might you mind updating your blog site to comprehend details? It’s highly useful for me. Massive thumb up for this writing! AV監督一覧
==================
Hello there, I found your site by the use of Google even as searching for a similar subject, your web site came up, it appears good. I’ve bookmarked it in my google bookmarks. エロ動画保存ランキング
May 25, 2024 12:32:16 AM
Your current blogs usually have got a lot of really up to date info. Where do you come up with this? Just saying you are very inspiring. Thanks again 온라인 바카라
May 26, 2024 09:25:11 AM
After study a number of the blog posts in your site now, i really such as your technique of blogging. I bookmarked it to my bookmark web site list and will be checking back soon. Pls take a look at my web site likewise and make me aware what you believe. Flower shop
=============
This website is often a walk-through it really is the information it suited you relating to this and didn’t know who need to. Glimpse here, and you’ll definitely discover it. 線上花店
=============
Usually I do not read post on blogs, but I wish to say that this write-up very pressured me to take a look at and do it! Your writing style has been amazed me. Thanks, very nice post. 玫瑰花
=============
Thank you for this specific facts I was basically seeking all Bing to find it! 情人節
Jun 03, 2024 12:26:54 AM
I think this website holds very great pent written content blog posts. Tip de redactie
Jun 13, 2024 03:02:48 PM
I was very lucky to find this internet site on bing, just what I was searching for likewise saved to favorites . Brabant Vandaag
Jun 17, 2024 07:51:26 PM
This might be among the best mentions of this subject I’ve seen in a long time. It is obvious that the knowledge of the topic is deep and this made for a very fascinating read. Tilburg
Jun 19, 2024 12:49:45 PM
I discovered your blog website on google and check a number of your early posts. Always keep the really good operate. I recently additional the Rss to my MSN News Reader. Seeking forward to reading far more of your stuff later on!… 情人節
==============
Nice post. I discover something harder on different blogs everyday. It will always be stimulating to learn to read content from other writers and rehearse a little at their store. I’d would rather apply certain with all the content in my small blog regardless of whether you don’t mind. Natually I’ll supply you with a link on the web blog. Many thanks sharing. 玫瑰花
==============
very good post, i surely enjoy this amazing site, persist in it 線上花店
==============
I have to show some thanks to the writer for bailing me out of such a difficulty. After looking out through the world-wide-web and finding techniques which were not beneficial, I figured my entire life was over. Existing without the answers to the difficulties you’ve fixed all through the article is a crucial case, as well as the kind that might have badly affected my career if I had not come across your blog post. Your own talents and kindness in maneuvering the whole thing was precious. I’m not sure what I would’ve done if I hadn’t come upon such a point like this. I am able to at this time look forward to my future. Thanks a lot so much for this professional and sensible help. I will not be reluctant to recommend the blog to any person who ought to have recommendations about this area. Flower shop
Jun 24, 2024 11:48:27 PM
Such a well written post.. Thnkx for sharing this post !! Also you can check my ?? av4 crypto
Jun 26, 2024 02:51:29 AM
Very interesting details you have mentioned , thankyou for posting . 魔術表演
==============
Retinoid is always one of the best medication that you can use on your skin~ 兒童魔術班
==============
You really should take part in a contest for one of the best blogs on the web. Let me suggest this web site! 魔術
==============
Judging by the way you write, you seem like a professional writer.,;~*- magic class
==============
Oh my goodness! an amazing post dude. Many thanks Nonetheless We are experiencing problem with ur rss . Do not know why Unable to join it. Could there be any person getting identical rss dilemma? Anybody who knows kindly respond. Thnkx 熟女動画一覧データベース
==============
Good day,your writing style is great and i love it, AV単体作品一覧データベース
==============
Hey there! This is kind of off topic but I need some help from an established blog. Is it hard to set up your own blog? I’m not very techincal but I can figure things out pretty fast. I’m thinking about creating my own but I’m not sure where to begin. Do you have any tips or suggestions? Appreciate it AV女優一覧データベース
Jun 30, 2024 04:12:05 PM
I recently added this website to my bookmarks. I like reading you. It had been quite helpful and helped me tons! Weer Vandaag
===============
Can I recently say what relief to seek out someone that truly knows what theyre referring to on the web. You certainly understand how to bring a concern to light and earn it important. The diet need to read this and appreciate this side from the story. I cant think youre no more well-liked when you certainly contain the gift. 62mas
===============
fantastic post. Ne’er knew this, appreciate it for letting me know. seiko skx
===============
Youre so cool! I dont suppose Ive read anything similar to this just before. So nice to uncover somebody with some original thoughts on this subject. realy appreciate beginning this up. this fabulous website are some things that is needed online, a person with a bit of originality. beneficial project for bringing new things to your net! rubber strap
===============
Write more, thats all I have to say. Literally, it seems as though you relied on the video to make your point. You obviously know what youre talking about, why throw away your intelligence on just posting videos to your site when you could be giving us something enlightening to read? Seiko King Samurai
===============
Its like you read my mind! You seem to know a lot about this, like you wrote the book in it or something. I think that you could do with a few pics to drive the message home a bit, but other than that, this is excellent blog. A fantastic read. I’ll definitely be back. ipad回收
===============
This web site may be a walk-through rather than the internet you wished about it and didn’t know who to ask. Glimpse here, and you’ll certainly discover it. iphone trade in
===============
Hi” your blog is full of comments and it is very active” iphone回收
Jul 14, 2024 11:46:37 PM
An fascinating discussion is worth comment. I believe that you need to write read more about this topic, may possibly not be described as a taboo subject but generally everyone is too few to chat on such topics. An additional. Cheers boots
Jul 18, 2024 01:09:01 AM
You ought to take part in a tournament personally of the greatest blogs on the web. I’ll recommend this website! john spencer ellis new photos
=======================
It’s really a great and useful piece of info. I am glad that you shared this helpful information with us. Please keep us informed like this. Thank you for sharing. john spencer ellis news
Jul 23, 2024 09:35:32 PM
Oh my goodness! an amazing write-up dude. Thanks Nevertheless We are experiencing issue with ur rss . Do not know why Struggling to register for it. Possibly there is anyone obtaining identical rss difficulty? Anybody who knows kindly respond. Thnkx エロ同人漫画一覧
===================
Helpful info. Lucky me I found your website accidentally, and I’m stunned why this accident didn’t came about in advance! I bookmarked it. 中出し動画一覧
===================
Wonderful blog. I learn a little something more challenging on different blogs everyday. It’s stimulating to read through articles from other writers. I’d like to use some of the content on my website if you don’t mind. For sure I’ll give you a backlink to your blog. Nice one for sharing. 巨乳動画一覧
===================
What are you saying, man? I recognize everyones acquired their own thoughts and opinions, but genuinely? Listen, your web site is cool. I like the hard work you put into it, specially with the vids and the pics. But, appear on. Theres gotta be a better way to say this, a way that doesnt make it appear like everybody here is stupid! 素人動画一覧
===================
Would love to always get updated great blog ! . 人妻動画一覧
===================
Only a few blogger would discuss this topic the way you do.”.~;~ マニアエロ動画
Jul 25, 2024 04:10:13 PM
I am not sure where you are getting your info, but great topic. I needs to spend some time learning more or understanding more. Thanks for great information I was looking for this info for my mission. 插花技巧
===============
I like this post, enjoyed this one thanks for posting. 花藝課程
===============
your writing style really enjoying me. 花藝證書
===============
It’s difficult to get knowledgeable individuals about this topic, however, you could be seen as you know what you’re preaching about! Thanks 插花
Jul 29, 2024 03:17:51 AM
Youre so cool! I dont suppose Ive read anything such as this just before. So nice to seek out somebody with some original thoughts on this subject. realy we appreciate you starting this up. this amazing site is one thing that is needed on the internet, someone with a little originality. beneficial problem for bringing interesting things to your web! drzwi na wymiar
Aug 02, 2024 01:41:01 AM
You can definitely see your skills in the work you write. The world hopes for even more passionate writers like you who aren’t afraid to say how they believe. Always go after your heart. African FollTales
Aug 26, 2024 02:21:19 AM
I am not sure where you are getting your info, but great topic. I needs to spend some time learning more or understanding more. Thanks for great info I was looking for this info for my mission. trading in mobile
Aug 29, 2024 11:19:05 PM
I gotta bookmark this web-site it seems invaluable extremely helpful 메이저놀이터
Sep 07, 2024 02:28:24 PM
I like this website it's a master piece! Glad I found this on google. mobile trading platform
Sep 24, 2024 11:54:24 PM
you might have a great blog right here! would you wish to make some invite posts on my blog? trading signal online
Oct 07, 2024 03:27:58 AM
very good submit, i actually love this website, keep on it binary options trading
Oct 25, 2024 09:55:13 PM
I definitely wanted to compose a quick message to be able to appreciate you for those amazing advice you are giving out on this website. My time intensive internet investigation has finally been honored with good quality facts and strategies to go over with my great friends. I would state that that most of us site visitors actually are quite fortunate to be in a notable website with so many outstanding people with insightful strategies. I feel really privileged to have discovered your entire web page and look forward to plenty of more amazing times reading here. Thanks again for everything. Clean black money in Japan
Oct 25, 2024 09:55:44 PM
Okay Eighteen Old Country Boy Living On A Farm Looking For A Great Way To Make Money Online 日本でブラック マネーを洗浄
Oct 25, 2024 09:56:05 PM
This is my first time i visit here. I found so many informative stuffs on your blog, especially its discussion. Form the tons of comments and posts, I guess I am not the only one having all enjoyment here. Keep up the good work. Limpieza de dinero negro en Argentina
Oct 30, 2024 03:31:29 AM
I like the helpful info you provide in your articles. I will bookmark your weblog and check again here frequently. I am quite certain I’ll learn many new stuff right here! Good luck for the next! trading binary options
Nov 19, 2024 08:04:20 PM
Nice post. I understand some thing harder on diverse blogs everyday. Most commonly it is stimulating to study content using their company writers and rehearse something from their store. I’d would rather use some with the content on my weblog whether you do not mind. Natually I’ll supply you with a link on your web blog. Thanks for sharing. 池袋 痩身
Nov 28, 2024 08:40:38 PM
Youre so cool! I dont suppose Ive learn something like this before. So nice to search out any person with some unique thoughts on this subject. realy thank you for starting this up. this web site is one thing that’s needed on the net, someone with a bit of originality. useful job for bringing something new to the internet! Headphones for a PC with a Microphone
Nov 28, 2024 08:40:53 PM
Thank you, I’ve just been searching for info about this subject for a long time and yours is the best I have discovered till now. But, what in regards to the conclusion? Are you positive about the source? binary options signals
Nov 29, 2024 12:41:47 AM
You have noted very interesting points ! ps nice website . デジポリス 東京だけ
Nov 29, 2024 12:42:02 AM
Hi, Neat post. There’s a problem with your website in internet explorer, would test this… IE still is the market leader and a big portion of people will miss your fantastic writing because of this problem. カチコチtvの無料動画
Dec 15, 2024 10:06:17 PM
When do you think this Real Estate market will go back up? Or is it still too early to tell? We are seeing a lot of housing foreclosures in Casselberry Florida. What about you? Would love to get your feedback on this. best binary options brokers
Dec 16, 2024 07:46:46 PM
Great content material and great layout. Your website deserves all of the positive feedback it’s been getting. affiliate program
Dec 18, 2024 03:20:20 PM
Great post, you have pointed out some great points , I also think this s a very excellent website. trading bots