Dynamic Programming Problems经典问题集

Author posted @ 2012年3月05日 09:21 in Algorithm , 16917 阅读

1. The Integer Knapsack Problem (Duplicate Items Permitted):

You have $n$ types of items, where the $i^{th}$ item type has an integer size $s_i$ and a real value. $v_i$

You are trying to fill a knapsack of total capacity $C$ with a selection of items of maximum value.

You can add multiple items of the same type to the knapsack.

Solution: Let M(j) denote the maximum value you can pack into a size knapsack.

We can express M(j) recursively in terms of solutions to smaller problems as follows:

$M(j)= \left\{ \begin{aligned}\overset{}\max\{M(j-1) , \max_{i=1...n}M(j-s_i)+v_i \} \quad j\ge 1\\ \\ 0 \quad j \le 0 \end{aligned}\right.$

Computing each M(j) value will require $\Theta(n)$ time, and we need to sequentially compute $C$ such values.
Therefore, total running time is $\Theta(nC)$ . Total space is $\Theta(n)$ .

The value of $M(C)$ will contain the value of the optimal knapsack packing.

We can reconstruct the list of items in the optimal solution by maintaining and following “backpointers”

2. Maximum Value Contiguous Subsequence.

Given a sequence of $n$ real numbers $A_1,
... A_n$ ,

determine a contiguous subsequence $A_1, ... A_j$ for which the sum of elements in the subsequence is maximized.

Solution: Let $M(j)$ denote the max sum over all windows ending at $j$

$M(j) = max\{ M(j-1)+A[j], A[j]\}$

It only takes linear time $O(n)$ to run the program since it only need to solve $n$ sub-problems and each takes constnt time.

3. Making Change: You are given $n$ types of coin denominations of values $v_1 < v_2 ... < v_n$ (all integers).

Assume $v_1 =1$ , so you can always make change for any amount of money C.

Give an algorithm which makes change for an amount of money C with as few coins as possible.

Solution: Let $M(j)$ denote the min number of coins required to make changes for amount of money $j$

$M(j)=\min_{i=1...n}\{M(j-v_i)\}+1$

It takes $O(nC)$ time because we are solving $C$ subproblems,

each of which requires minimization $O(n)$ different terms. (Similar to integer knapsack problem)

4. Longest Increasing Subsequence. Given a sequence of n real numbers $A_1,
... A_n$ ,

determine a subsequence (not necessarily contiguous) of maximum length in which the values in the subsequence form a strictly increasing sequence.

Solution: Let $L(j) denote the longest strictly increasing subsequence ending at position $j$ , therefore,

$L(j) = \max\limits_{i<j}\limits_{A[i]<A[j]}\{ L(i)\}+1$

to find the solution overall, it has to find the maximum over all potential ending points $j$ of $L(j) :

$\max\limits_{j=1...n}\{L(j)\}$

since the longest increasing subsequence could end anywhere.

The algorithm takes $O(n^2)$ time to run because I have to sort $n$ subproblems and each of them takes $O(n)$ time.

5.Box Stacking:

You are given a set of $n$ types of rectangular 3-D boxes, where the $i^{th}$ box has height $h_i$ , width $w_i$ and depth $d_i$ (all real numbers).

You want to create a stack of boxes which is as tall as possible,

but you can only stack a box on top of another box if the dimensions of the 2-D base of the lower box are each strictly larger than those of the 2-D base of the higher box.

Of course, you can rotate a box so that any side functions as its base.

It is also allowable to use multiple instances of the same type of box.

Solution: constraint : can only stack box $i$ on box $j$ if $w_i<w_j$ and $d_i<d_j$

(without loss of generality, assume $w_i\le d_i$ ).

First sort the base area by decreasing order:

then let $H(j)$ denote the tallest box stack I could form with box $j$ on top, so

$H(j) = \max\limits_{i<j, w_i<w_j, d_i<d_j}\{ H(i)\}+h_j$

The algorithm takes $O(n^2)$ time to run because I have to sort $n$ subproblems and each of them takes $O(n)$ time.

At the end, the solution is found by taking the max of all potential top boxes $j$ of $H(j)$ .

(Since we are not sure in the optimal solution, which box is on top)

6. Building Bridges.

Consider a 2-D map with a horizontal river passing through its center.

There are $n$ cities on the southern bank with x-coordinates $a_1... a_n$ and $n$ cities on the northern bank with x-coordinates $b_1 ... b_n$ .

You want to connect as many north-south pairs of cities as possible with bridges such that no two bridges cross.

When connecting cities, you are only allowed to connect the the ith city on the northern ban to the ith city on the southern bank.

Solution: consider the south bank city (below the river) $i$ ,

let $x(i)$ denote the index of corresponding city on the north bank ( $x(i)=3$ in the figure).

So we could compute the $x(i)$ by sorting in $O(n\log n)$ time.

Now we want to find the longest increasing subsequence through $x(1),
... x(n)$ , so it is identical to problem 4.

7. Balanced Partition:You have a set of n integers each in the range 0 . . .K.

Partition these integers into two subsets such that you minimize $S_1 - S_2$ ,

where $S_1$ and $S_2$ denote the sums of the elements in each of the two subsets.

Solution: Use $P(i,j)= \left\{ \begin{aligned}\overset{}1 \quad \text{if some subset of } \{A_1, ... A_n\} \text{has sum of j}\\0 \quad \text{otherwise}\end{aligned}\right.$ where $i \in [0,n]$ and $j \in [0,nK]$

therefore, $P(i,j)=1$ if $P(i-1, j)=1$ or $P(i-1,
j-A_i)=1$

dynamic programming formula is:

$P(i,j)=\max\{P(i-1,j), P(i-1, j-A_i)\}$

this procedure takes $O(n^2K)$ time to compute.

To solve the original problem, let $S=\sum(A_i)/2$ , and the purpose is to make $S_1 - S_2$ close to 0 as possible.

what we want to find is:

$\min\limits_{i\le S}\{S-i: P(n, i)=1\}$

8. Edit Distance.

Given two text strings A of length n and B of length m,

you want to transform A into B with a minimum number of operations of the following types:

delete a character from A, insert a character into A,

or change some character in A into a new character.

The minimal number of such operations required to transform A into B is called the edit distance between A and B.

Solution: Use $C_i, C_d, C_r$ to denote the cost for inserting, deleting or replacing a letter.

The goal is to compute the minimum cost of translating A to B.

Let $T(i,j)$ denote the minimum cost of translating $A[1...i]$ into $B[1...j]$ , then

$T(i,j)= \left\{\begin{aligned}\overset{}C_d + T(i-1, j) \\ C_i+T(i, j-1) \\ \overset{} T(i-1, j-1) \quad if \quad A[i]=B[j] \\ C_d+T(i-1,j-1) \quad if \quad A[i] \neq B[j] \end{aligned}\right.$

It takes $O(nm)$ time to compute and final solution is stored in $T(n,m)$

[回复]

Biofit 说:
2021年7月13日 04:11

Hi, I find reading this article a joy. It is extremely helpful and interesting and very much looking forward to reading more of your work..

[回复]

seo 说:
2021年10月07日 04:59

Nice post! This is a very nice blog that I will definitively come back to more times this year! Thanks for informative post. join us

[回复]

seo 说:
2021年10月12日 07:25

오피를 모두 모아둔 곳에서 원하는 종류의 달콤월드,오피쓰,오피그램 등에서 전국에 있는 OP를 즐길 수 있는 완벽한 오피사이트를 모아 추천하고 쉽게 한눈에 선택이 가능합니다. 처음 오피를 이용하는 모든 분들에게 가장 편리한 구성으로 준비하여 안내해드립니다. 오피그램

[回复]

seo 说:
2021年10月15日 03:59

This blog is so nice to me. I will keep on coming here again and again. Visit my link as well.. 전국급전

[回复]

seo 说:
2021年10月19日 01:25

Your work is truly appreciated round the clock and the globe. It is incredibly a comprehensive and helpful blog. mobiledsng.com/product/levitra-20mg-price-in-pakistan

[回复]

seo 说:
2021年11月01日 07:03

Very informative post! There is a lot of information here that can help any business get started with a successful social networking campaign. 토토사이트

[回复]

seo 说:
2021年11月05日 08:41

I was surfing the Internet for information and came across your blog. I am impressed by the information you have on this blog. It shows how well you understand this subject. 먹튀

[回复]

seo 说:
2021年11月07日 04:31

Hello, this weekend is good for me, since this time i am reading this enormous informative article here at my home. 먹튀

[回复]

seo 说:
2021年11月08日 05:59 I recently came across your blog and have been reading along. I thought I would leave my first comment. I don't know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often. 먹튀검증

[回复]

seo 说:
2021年11月14日 11:06

Hello, I have browsed most of your posts. This post is probably where I got the most useful information for my research. Thanks for posting, maybe we can see more on this. Are you aware of any other websites on this subject. 먹튀검증

[回复]

seo 说:
2021年11月17日 07:59

You made such an interesting piece to read, giving every subject enlightenment for us to gain knowledge. Thanks for sharing the such information with us to read this... https://www.roadrunneremails.org

[回复]

seo 说:
2021年11月29日 22:40

Thanks for a wonderful share. Your article has proved your hard work and experience you have got in this field. Brilliant .i love it reading. public restroom

[回复]

seo 说:
2021年12月01日 08:30

I was reading some of your content on this website 먹튀검증

[回复]

918kiss 说:
2021年12月29日 03:04

Useful information shared. I am very happy to read this article..thanks for giving us nice info. Fantastic walk-through. I appreciate this post.I have Bookmark this webpage, I am sure I will visit this place again soon. Thanks for sharing.

[回复]

sagame66 说:
2021年12月29日 04:10

Thanks for this. I really like what you've posted here and wish you the best of luck with this blog!

[回复]

seo 说:
2021年12月29日 07:38

This is such a great resource that you are providing and you give it away for free. I love seeing blog that understand the value of providing a quality resource for free. 메이저사이트

[回复]

wild rift coaching 说:
2022年1月01日 23:19

I am hoping that you will continue writing this kind of blog. Thanks for sharing this information.

[回复]

seo 说:
2022年1月06日 23:27

I admit, I have not been on this web page in a long time... however it was another joy to see It is such an important topic and ignored by so many, even professionals. I thank you to help making people more aware of possible issues. account in london

[回复]

seoo 说:
2022年1月14日 07:14

I know your expertise on this. I must say we should have an online discussion on this. Writing only comments will close the discussion straight away! And will restrict the benefits from this information. is inbox dollars legit

[回复]

seoo 说:
2022年1月23日 12:02

I was reading some of your content on this website 안전놀이터

[回复]

satta 说:
2022年1月30日 22:30

I appreciate everything you have added to my knowledge base.Admiring the time and effort you put into your blog and detailed information you offer.Thanks.

[回复]

matka 说:
2022年1月31日 05:29

Website especially this blog page. Among the lots of comments on your articles. Thanks for sharing

[回复]

seo 说:
2022年2月02日 10:14

Positive site, where did u come up with the information on this posting?I have read a few of the articles on your website now, and I really like your style. Thanks a million and please keep up the effective work. 룰루바둑이

[回复]

satta matta 说:
2022年2月12日 23:26

I wanted to thank you for this in your liking ensnare!! I particularly enjoying all tiny little bit of it I have you ever bookmarked to check out delivered stuff you pronounce.

[回复]

matka guessing 说:
2022年2月13日 23:45

You have done a great job on this article. It’s very readable and highly intelligent. You have even managed to make it understandable and easy to read. You have some real writing talent. Thank you.

[回复]

Car Show Boards 说:
2022年2月21日 20:31

This is a good post. This post gives truly quality information. I’m definitely going to look into it. Really very useful tips are provided here. Thank you so much. Keep up the good works.

[回复]

Digital Global Times 说:
2022年2月22日 19:02

I just couldn't leave your website before telling you that I truly enjoyed the top quality info you present to your visitors? Will be back again frequently to check up on new posts.

[回复]

kroger feed my esche 说:
2022年2月23日 20:24

I definitely enjoying every little bit of it. It is a great website and nice share. I want to thank you. Good job! You guys do a great blog, and have some great contents. Keep up the good work

[回复]

car show displays 说:
2022年2月28日 00:54

This is a good post. This post gives truly quality information. I’m definitely going to look into it. Really very useful tips are provided here. Thank you so much. Keep up the good works.

[回复]

ali 说:
2022年3月02日 04:16

Microsoft account lets you flow freely from task to task and make the most of every moment. It not only gives you access to Microsoft services, but also makes them work together in a way that’s all about you. https://sites.google.com/view/outlookcom-login/

[回复]

ali 说:
2022年3月02日 20:33

It looks like the only spot that time steps on as it goes through, so it looks trodden and bleak. They look like ruthless places to me, and thus solitary. 9animelab

[回复]

ali 说:
2022年3月02日 22:55

I really appreciate the kind of topics you post here. Thanks for sharing us a great information that is actually helpful. Good day! telenovelahd

[回复]

ali 说:
2022年3月03日 21:41

Thanks for sharing this quality information with us. I really enjoyed reading. Will surely going to share this URL with my friends. children book online

[回复]

ali 说:
2022年3月04日 03:31

Great info! I recently came across your blog and have been reading along. I thought I would leave my first comment. I don’t know what to say except that I have. Picses p100

[回复]

Converter Tools 说:
2022年3月05日 02:17

This is a great inspiring article.I am pretty much pleased with your good work.You put really very helpful information

[回复]

satta matka 说:
2022年3月06日 19:57

i am until the call off of time greatly surprised by means of the amount of recommend understandable in financial credit score to this situation. What you presented became ably researched and adroitly worded in an effort to collect your stand within the location of this across to every and the entire single one one your readers.

[回复]

seoo 说:
2022年3月06日 20:31

Slot machine game information community in Korea 슬롯머신

[回复]

ali 说:
2022年3月06日 23:29

I think this is an informative post and it is very beneficial and knowledgeable. Therefore, I would like to thank you for the endeavors that you have. Online serije

[回复]

ali 说:
2022年3月08日 05:12

There couple of interesting points in time in this posting but I do not determine if I see them all center to heart. There is some validity but I most certainly will take hold opinion until I check into itfurther. 먹튀사이트

[回复]

SEO 说:
2022年3月08日 07:10

Very nice article, I enjoyed reading your post, very nice share, I want to twit this to my followers. Thanks!. 토토사이트

[回复]

seoo 说:
2022年3月09日 08:47

This is such a great resource that you are providing and you give it away for free. I love seeing blog that understand the value of providing a quality resource for free. 메이저사이트

[回复]

SEO 说:
2022年3月10日 08:49

You know your projects stand out of the herd. There is something special about them. It seems to me all of them are really brilliant! 토토사이트

[回复]

sattamatka 说:
2022年3月12日 05:06

hey, your websites are truly suitable. I appreciate your movement.

[回复]

SEO 说:
2022年3月12日 09:35

I like your post. It is good to see you verbalize from the heart and clarity on this important subject can be easily observed... เว็บค่าย JILI

[回复]

satta 说:
2022年3月13日 20:41

I read a article under the same title some time ago, but this articles quality is much, much better. How you do this..

[回复]

Riviere 说:
2022年3月18日 00:47

thanks due to the fact you have been precise-natured to percentage opinion subsequent to us. we are able to continually recognize all you have finished here because I understand you are selected worried thinking about our.

[回复]

seoo 说:
2022年3月19日 08:02

It was wondering if I could use this write-up on my other website, I will link it back to your website though.Great Thanks. 안전놀이터

[回复]

Biotechnolabs 说:
2022年3月22日 19:25

Very Nice Website, I really Appreciate your contents, this is very meaning for us.

[回复]

ali 说:
2022年3月22日 21:13

Thanks for taking the time to discuss this, I feel strongly about it and love learning more on this topic. If possible, as you gain expertise, would you mind updating your blog with extra information? It is extremely helpful for me. serials online

[回复]

ali 说:
2022年4月02日 23:55

I definitely enjoying every little bit of it. It is a great website and nice share. I want to thank you. Good job! You guys do a great blog, and have some great contents. Keep up the good work. Seriale Turcesti

[回复]

ali 说:
2022年4月03日 23:29

[回复]

Kingpure 说:
2022年4月04日 20:17

Your work is very good, and I appreciate you and hopping for some more informative posts.

[回复]

Lentor Modern Showfl 说:
2022年4月04日 22:55

I appreciate everything you have added to my knowledge base.Admiring the time and effort you put into your blog and detailed information you offer.Thanks.

[回复]

ali 说:
2022年4月04日 23:25

It looks like the only spot that time steps on as it goes through, so it looks trodden and bleak. They look like ruthless places to me, and thus solitary. Online serije

[回复]

Vending Services Dal 说:
2022年4月07日 03:46

I found so many interesting stuff in your blog especially its discussion. From the tons of comments on your articles, I guess I am not the only one having all the enjoyment here keep up the good work

[回复]

SEO 说:
2022年4月09日 11:46

I’m going to read this. I’ll be sure to come back. thanks for sharing. and also This article gives the light in which we can observe the reality. this is very nice one and gives indepth information. thanks for this nice article... 인천출장마사지

[回复]

prime advantage 说:
2022年4月10日 03:30

This is a good post. This post gives truly quality information. I’m definitely going to look into it. Really very useful tips are provided here. Thank you so much. Keep up the good works.

[回复]

Digital Global Times 说:
2022年4月14日 09:46

I read a lot of stuff and i found that the way of writing to clearifing that exactly want to say was very good so i am impressed and ilike to come again in future.

[回复]

SEO 说:
2022年4月15日 21:59

This is such a great resource that you are providing and you give it away for free. I love seeing blog that understand the value of providing a quality resource for free. companies that build prototypes

[回复]

ali 说:
2022年4月18日 05:48

I know your aptitude on this. I should say we ought to have an online discourse on this. Composing just remarks will close the talk straight away! What's more, will confine the advantages from this data. Seriale Turcesti

[回复]

Midtown Bay Balance 说:
2022年4月18日 06:19

This is a brilliant blog! I'm very happy with the comments!..

[回复]

yoyo 说:
2022年5月01日 10:28

Just admiring your work and wondering how you managed this blog so well. It’s so remarkable that I can't afford to not go through this valuable information whenever I surf the internet! Write for us

[回复]

yoyo 说:
2022年5月07日 05:31

If you set out to make me think today; mission accomplished! I really like your writing style and how you express your ideas. Thank you. Updated Ideas

[回复]

yoyo 说:
2022年5月11日 07:22

All the contents you mentioned in post is too good and can be very useful. I will keep it in mind, thanks for sharing the information keep updating, looking forward for more posts.Thanks 출장안마

[回复]

ali 说:
2022年5月13日 23:25

I woke up to be notified by this blog and I already knew my day started in right way! Your words of wisdom are overflowing and I truly appreciate it! Turske Online

[回复]

lokesh website desig 说:
2022年5月15日 19:08

Your website is very nice.
<a href="https://appfinz.com">Website Development company in delhi</a>

[回复]

yoyo 说:
2022年5月18日 04:28

This type of message always inspiring and I prefer to read quality content, so happy to find good place to many here in the post, the writing is just great, thanks for the post. https://helvetia-umzug.ch/

[回复]

yoyo 说:
2022年5月22日 06:57

[回复]

arsi 说:
2022年5月28日 00:35

Oh dear it seems as if your site Going Home Posted Stitches consumed my first remark it’s rather extensive we think We’ll simply sum it up the things i submitted and state, I really relishing your website. I as well am an ambitious blog writer but I’m still a new comer to the whole thing. Do you possess any kind of tips and hints regarding inexperienced bloggers! I truly really enjoy it… In addition did you hear Tunisia incredible announcement… Regards Flash Website Builder betting sites in south africa

[回复]

arsi 说:
2022年5月28日 09:49

Generally Are generally Diet ‘s an very affordable and flexible food regimen product suitable for induced the boycott . endeavouring to fast and as a result subsequently conserve a vibrant lifespan. lose weight betting sites in south africa

[回复]

arsi 说:
2022年5月28日 23:12

some genuinely choice content on this internet site , saved to my bookmarks . betting sites in switzerland

[回复]

yoyo 说:
2022年6月05日 07:55

Nice knowledge gaining article. This post is really the best on this valuable topic. 레깅스룸

[回复]

yoyo 说:
2022年6月17日 07:03

New Port Residences EC @ Tengah Garden Walk by CDL & MCL Land. Showflat Hotline 61009266. Get Discounts, Direct Developer Price, Discount, Brochure, Floor Plan, & More. New Port Residences Showroom

[回复]

Anonimouse 说:
2022年6月18日 01:44

Very <a href="https://mangotips.com/what-you-search-on-twitter/">efficiently</a> written <a href="https://mangotips.com/some-of-the-best-linus-tech-tips-meme/">information</a>. It will be <a href="https://mangotips.com/a-biography-of-john-henton-and-his-daughter-nicole-simone/">beneficial</a> to anybody who <a href="https://mangotips.com/how-to-fix-a-zipper/">utilizes </a>it, including me. Keep up the <a href="https://mangotips.com/who-is-michael-strahans-wife/">good work.</a> For <a href="https://mangotips.com/xosonam-net-worth-how-much-money-is-xosonam-worth/">sure </a>i will <a href="https://mangotips.com/varro-blair-mendez-and-jennifer-lopez/">check </a>out more posts. This <a href="https://mangotips.com/what-will-larry-king-net-worth-be-in-2020/">site </a>seems to get a <a href="https://mangotips.com/wizz-air/">good </a>amount of <a href="https://mangotips.com/gza-net-worth-2021/">visitors.</a><a href="https://mangotips.com/so-you-want-to-write-an-article/">So you want to write an article</a> than you need to know that <a href="https://mangotips.com/writing-is-thinking/">writing is thinking</a>, its time to <a <a href="https://mangotips.com/contribute/">contribute</a> to us

[回复]

yoyo 说:
2022年6月21日 00:57

Thanks for the blog loaded with so many information. Stopping by your blog helped me to get what I was looking for. Updated Ideas

[回复]

updatedverse 说:
2022年7月18日 21:45

<a href='https://updatedverse.com/10-best-android-cleaner-apps-to-clear-ram-and-cache-in-2021/' title='10 Best Android Cleaner Apps 2021' target='_blank'>10 Best Android Cleaner Apps 2021</a>
<a href='https://updatedverse.com/10-best-app-locks-for-android-to-secure-your-device-in-2022/' title='Best App Locks For Android in 2022' target='_blank'>Best App Locks For Android in 2022</a>
<a href='https://updatedverse.com/10-best-pdf-reader-apps-for-iphone-ipad-view-and-edit-pdfs-in-2022/' title='Best PDF Reader For iPhone' target='_blank'>Best PDF Reader For iPhone</a>
<a href='https://updatedverse.com/10-fail-proof-email-marketing-tips-for-small-businesses/' title='email marketing tips' target='_blank'>email marketing tips</a>
<a href='https://updatedverse.com/10-most-secure-and-secure-messaging-apps-for-android-and-ios-2022/' title='10 Most Secure and Secure messaging apps for (Android and iOS)' target='_blank'>10 Most Secure and Secure messaging apps for (Android and iOS)</a>
<a href='https://updatedverse.com/10-top-iphone-browsers-top-safari-alternatives-for-2022/' title='Ten Best iPhone Web Browser Apps For 2022' target='_blank'>Ten Best iPhone Web Browser Apps For 2022</a>
<a href='https://updatedverse.com/12-best-android-camera-apps-in-2022/' title='Best Android Camera Apps' target='_blank'>Best Android Camera Apps</a>
<a href='https://updatedverse.com/11-best-linux-distros-for-programming-and-development-2022-edition/' title='11 Best Linux Distros For Programming In 2022' target='_blank'>11 Best Linux Distros For Programming In 2022</a>

[回复]

uaecarpet 说:
2022年7月19日 00:04

<a href='https://uaecarpet.com/dubai-carpet-cleaning-services/' title=' carpet shop in deira dubai' target='_blank'> carpet shop in deira dubai</a>
<a href=' https://uaecarpet.com/carpet-shops-and-stores-in-dubai-and-uae/' title=' carpet shops and stores in dubai and uae' target='_blank'> carpet shops and stores in dubai and uae</a>
<a href='https://uaecarpet.com/modern-building-curtain-selection-in-dubai/' title='Modern building curtain selection in dubai' target='_blank'>Modern building curtain selection in dubai</a>
<a href='https://uaecarpet.com/roman-curtain-and-know-about-this-curtain/' title='Roman curtain and know about this curtain' target='_blank'>Roman curtain and know about this curtain</a>

[回复]

yoyo 说:
2022年7月20日 06:39

Impressive web site, Distinguished feedback that I can tackle. Im moving forward and may apply to my current job as a pet sitter, which is very enjoyable, but I need to additional expand. Regards. Updated Ideas

[回复]

yoyo 说:
2022年8月06日 23:49

[回复]

yoyo 说:
2022年8月09日 08:40

What is an outstanding post! “I’ll be back” (to read more of your content). Thanks for the nudge! Digital Business Time

[回复]

yoyo 说:
2022年8月10日 19:11

Took me time to read all the comments, but I really enjoyed the article. It proved to be Very helpful to me and I am sure to all the commenters here! It’s always nice when you can not only be informed, but also entertained! Wiki business now

[回复]

ayan 说:
2022年8月14日 01:55

Wow, excellent post. I'd like to draft like this too - taking time and real hard work to make a great article. This post has encouraged me to write some posts that I am going to write soon. Digital Global Times

[回复]

rajmandir 说:
2022年8月16日 23:13

Very Nice Website, I really Appreciate your contents, this is very meaning for us. Rajmandir Hyper Market

[回复]

ss water tanks 说:
2022年8月24日 00:37

Very Nice Website, I really Appreciate your contents, this is very meaning for us. Stainless Steel Water Tank

[回复]

ss water tanks 说:
2022年8月24日 00:38

Thank you for very useful information. Luxury Series Stainless Steel Water Tanks in India

[回复]

ss water tanks 说:
2022年8月24日 00:38

Nice post mate, keep up the great work. Platinum Series SS Water Tanks in India

[回复]

ss water tanks 说:
2022年8月24日 00:39

This is very nice one and gives indepth information. Solitaire Series Stainless Steel Water Tanks in India

[回复]

ss water tanks 说:
2022年8月24日 00:40

Nice Post Your content is very inspiring and appriciating I really like it. premium series SS Water Tanks in India

[回复]

yoyo 说:
2022年8月24日 17:59

Hi there! Nice stuff, do keep me posted when you post again something like this! 먹튀폴리스

[回复]

MBOSE Question Paper 说:
2022年8月24日 23:07

Meghalaya Board Model Paper 2023 Class 6 Pdf Download with Answers for Bengali Medium, English Medium, Hindi Medium, Urdu Medium & Students for Small Answers, Long Answer, Very Long Answer Questions, and Essay Type MBOSE Question Paper Class 6 Questions to Term1 & Term2 Exams at official website. New Exam Scheme or Question Pattern for Sammittive Assignment Exams (SA1 & SA2): Very Long Answer (VLA), Long Answer (LA), Small Answer (SA), Very Small Answer (VSA), Single Answer, Multiple Choice and etc.

[回复]

yoyo 说:
2022年8月27日 04:10

What a sensational blog! This blog is too much amazing in all aspects. Especially, it looks awesome and the content available on it is utmost qualitative. Digital Global Times

[回复]

yoyo 说:
2022年9月01日 03:29

You have a good point here!I totally agree with what you have said!!Thanks for sharing your views...hope more people will read this article!!! wiki business now

[回复]

yoyo 说:
2022年9月07日 04:30

Thanks for picking out the time to discuss this, I feel great about it and love studying more on this topic. It is extremely helpful for me. Thanks for such a valuable help again. https://www.simplybuzzes.com/angel-number-915/

[回复]

yoyo 说:
2022年9月08日 04:22

Thank you for the update, very nice site.. Digital Global Times

[回复]

yoyo 说:
2022年9月10日 08:01

It was any exhilaration discovering your website yesterday. I arrived here nowadays hunting new things. I was not necessarily frustrated. Your ideas after new approaches on this thing have been helpful plus an superb assistance to personally. We appreciate you leaving out time to write out these items and then for revealing your thoughts. Updated Ideas

[回复]

yoyo 说:
2022年9月11日 20:24

I think this is an informative post and it is very useful and knowledgeable. therefore, I would like to thank you for the efforts you have made in writing this article. where to buy crypto

[回复]

yoyo 说:
2022年9月22日 00:42

Hello there, just became alert to your blog through Google, and found that it’s truly informative. I’m gonna watch out for brussels. I will appreciate if you continue this in future. A lot of people will be benefited from your writing. Cheers! NagaPoker88 Asia

[回复]

yoyo 说:
2022年9月23日 23:43

You can still think about a number of advised organized tours with various limo professional services. A handful of offer medieval software programs a number of will administer you really to get automobile for any capital center, or perhaps checking out the upstate New York. ??????? Daftar Naga303

[回复]

yoyo 说:
2022年9月28日 05:18

I think this is an informative post and it is very useful and knowledgeable. therefore, I would like to thank you for the efforts you have made in writing this article. icon group

100

[回复]

ali 说:
2022年9月28日 05:42

I really loved reading your blog. It was very well authored and easy to understand. Unlike other blogs I have read which are really not that good.Thanks alot! online shop

101

[回复]

yoyo 说:
2022年9月29日 03:23

I think this is an informative post and it is very useful and knowledgeable. therefore, I would like to thank you for the efforts you have made in writing this article. 토토사이트

102

[回复]

yoyo 说:
2022年9月30日 00:21

Hello! Nice post! Please do keep us posted when we can see a follow up! Courier to USA

103

[回复]

yoyo 说:
2022年10月11日 05:44

This is such a great resource that you are providing and you give it away for free. I love seeing blog that understand the value of providing a quality resource for free. Temazepam 20mg

104

[回复]

yoyo 说:
2022年10月13日 07:32

I believe you have mentioned some very interesting points , regards for the post. Updated Ideas

105

[回复]

yoyo 说:
2022年10月26日 01:49

It has always been my belief that good writing like this takes research and talent. It’s very apparent you have done your homework. Great job! Digital Global Times

106

[回复]

ali 说:
2022年12月09日 09:57

Live broadcast of the game today Kora Live Online, you do not have to worry about the interruption of the live broadcast, Kora Live exclusively provides uninterrupted live broadcast of the game today, and it is one of the largest live broadcast sites in the Arab world. koora live

107

[回复]

yoyo 说:
2023年1月16日 04:42

I’m really impressed along with your writing abilities well with the structure in your weblog. Is that this a paid subject matter or did you modify it your self? Either way stay up the nice quality writing, it is uncommon to look a great weblog like this one nowadays. Digital Global Times

108

[回复]

yoyo 说:
2023年1月28日 02:12

I am glad to be one of the visitants on this outstanding internet site (:, regards for posting . Digital Global Times

109

[回复]

yoyo 说:
2023年2月10日 06:15

hi!,I like your writing so much! share we keep up a correspondence extra approximately your post on AOL? I require a specialist on this space to solve my problem. May be that is you! Looking ahead to peer you. UFABET ทางเข้ามือถือ

110

[回复]

yoyo 说:
2023年2月18日 02:28

I enjoy you because of all of your efforts on this website. Kim take interest in getting into investigations and it is easy to see why. We know all concerning the dynamic mode you produce priceless ideas by means of the web blog and as well as recommend participation from some other people on this area then our own daughter is certainly becoming educated a lot of things. Take pleasure in the remaining portion of the year. You’re carrying out a splendid job. General Questions about Nigeria

111

[回复]

gut vita buy 说:
2023年3月16日 02:33

If you are looking for more information about flat rate locksmith Las Vegas check that right away. gut vita buy

112

[回复]

yoyo 说:
2023年4月09日 00:11

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! Digital Global Times

113

[回复]

yoyo 说:
2023年4月22日 07:02

Loving the information on this web site , you have done great job on the posts . judi rolet

Something Useful

Dynamic Programming Problems经典问题集

我的头像

我的文章 · · · · · ·

文章分类 · · · · · ·

文章搜索

其它链接 · · · · · ·