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• Pure Java open-source libraries for portfolio selection (= constrained, non-linear optimization)?

  - by __roland__

  Does anyone know or has experience with a pure Java library to select portfolios or do some similar kinds of quadratic programming with constraints? There seems to be a lot of tools, as already discussed elsewhere - but what I would like to use is a pure Java implementation. Since I want to call the library from within another open-source software with a BSD-ish license I would prefer LGPL over GPL. Any help is appreciated. If you don't know such libraries, which is the most simple algorithm you would suggest to implement? It has to cope with an inequality constraint (all x_i = 0) and an equality constraint (sum of all x_i = 1).

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• Algorithm for tracking progress of controller method running in background

  - by SilentAssassin

  I am using Codeigniter framework for PHP on Windows platform. My problem is I am trying to track progress of a controller method running in background. The controller extracts data from the database(MySQL) then does some processing and then stores the results again in the database. The complete aforesaid process can be considered as a single task. A new task can be assigned while another task is running. The newly assigned task will be added in a queue. So if I can track progress of the controller, I can show status for each of these tasks. Like I can show "Pending" status for tasks in the queue, "In Progress" for tasks running and "Done" for tasks that are completed. Main Issue: Now first thing I need to find is an algorithm to track the progress of how much amount of execution the controller method has completed and that means tracking how much amount of method has completed execution. For instance, this PHP script tracks progress of array being counted. Here the current state and state after total execution are known so it is possible to track its progress. But I am not able to devise anything analogous to it in my case. Maybe what I am trying to achieve is programmtically not possible. If its not possible then suggest me a workaround or a completely new approach. If some details are pending you can mention them. Sorry for my ignorance this is my first post here. I welcome you to point out my mistakes. EDIT: Database outline: The URL(s) and keyword(s) are first entered by user which are stored in a database table called link_master and keyword_master respectively. Then keywords are extracted from all the links present in this table and compared with keywords entered by user and their frequency is calculated which is the final result. And the results are stored in another table called link_result. Now sub-links are extracted from the domain links and stored in a table called sub_link_master. Now again the keywords are extracted from these sub-links and the corresponding results are stored in a table called sub_link_result. The number of records cannot be defined beforehand as the number of links on any web page can be different. Only the cardinality of *link_result* table can be known which will be equal to multiplication of number of keyword(s) and URL(s) . I insert multiple records at a time using this resource. Controller outline: The controller extracts keywords from a web page and also extracts keywords from all the links present on that page. There is a method called crawlLink. I used Rolling Curl to extract keywords and web page content. It has callback function which I used for extracting keywords alongwith generating results and extracting valid sub-links. There is a insertResult method which stores results for links and sub-links in the respective tables. Yes, the processing depends on the number of records. The more the number of records, the more time it takes to execute: Consider this scenario: Number of Domain Links = 1 Number of Keywords = 3 Number of Domain Links Result generated = 3 (3 x 1 as described in the question) Number of Sub Links generated = 41 Number of Sub Links Result = 117 (41 x 3 = 123 but some links are not valid or searchable) Approximate time taken for above process to complete = 55 seconds. The above result is for a single link. I want to track the progress of the above results getting stored in database. When all results are stored, the task is complete. If results are getting stored, the task is In Progress. I am not clear how can I track this progress.

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• Auto-organized / smart inventory system?

  - by VeXe

  for the past week I've been working on an inventory system with Unity3D. At first I got help from the guys at Design3 but it wasn't too long till we split path, because I really didn't like the way they did their code, it didn't have any smell of OOP whatsoever. I took it further steps ahead - items take more than one slot, advanced placement system (items tries their best to find the best close fit), local mouse system (mouse gets trapped in active bag area), etc. Here's a demo of my work. What we would like to have in our game, is an auto-organizing feature - not auto-sort. We want this feature because our inventory's going to be in 'real-time' - not like in Resident Evil 1,2,3 etc where you would pause the game and do things in your inventory. Now imagine your self in a sticky situation surrounded by zombies, and you don't have bullets, you look around, you see that there are bullets nearby on the ground, so you go for them and try to pick them up, but they don't fit! you look at your inventory and find out that if you reorganize some of the items, it will fit! - now the player - in that situation doesn't have time to reorganize because he's surrounded with zombies and will die if he stops and organizes the inventory to make space (remember inventory in real-time, no pausing) - wouldn't it be nice for that to happen automatically? - Yes! (I believe this has been implemented in some games like Dungeon siege or something, so sure it's doable) take a look at this picture for example: Yes, so if you auto-sort the issue you will get your spaces but it's bad because: 1- Expensive: it doesn't need a whole sort operation to free those spaces, in the first picture, just slide the red item at the bottom to the very left, and you get the same spaces that you got from the auto-sort. 2- It's annoying to the player: "Who the F told you to re-order my stuff?" I'm not asking for "How to write the code" for this, I'm just asking for some guidance, where to look, what algorithms are involved? Is this something related to graphs and shortest path stuff? I hope not cuz I didn't manage to continue my college studies :/ But even if it is, just tell me and I will learn the stuff related. Notice there could be more than just one solution. So I guess the first thing I have to do is figure out if the situation is 'solvable' - if I know how to determine if a situation is solvable or not, then I can 'solve' it. I just need to know the conditions that makes it 'solvable'. And I believe there must be some algorithm/data structure for this. Here's a pic for more than one solution of trying to fit a 1x3 item: The arrows show just one of the solutions, but if you look you will find more than one. This is what I ultimately not auto-sorting but find a solution and applying it. Note that if I spend time on it I will come up with a way to solve it, but it wouldn't be the best way, it's like, holding a car wheel with your feet instead of your hands! XD Or just like trying to solve an issue that requires arrays, but you're not yet aware of their existence! So what is the right approach to this? Hope somebody helps, thanks a lot in advance :)

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• YouTube SEO: Video Optimization

  - by Mike Stiles

  SEO optimization is still regarded as one of the primary tools in the digital marketing kit. However and wherever a potential customer is conducting a search, brands want their content to surface in the top results. Makes sense. But without a regular flow of good, relevant content, your SEO opportunities run shallow. We know from several studies video is one of the most engaging forms of content, so why not make sure that in addition to being cool, your videos are helping you win the SEO game? Keywords:-Decide what search phrases make the most sense for your video. Don’t dare use phrases that have nothing to do with the content. You’ll make people mad.-Research those keywords to see how competitive they are. Adjust them so there are still lots of people searching for it, but there are not as many links showing up for it.-Search your potential keywords and phrases to see what comes up. It’s amazing how many people forget to do that. Video Title: -Try to start and/or end with your keyword.-When you search on YouTube, visual action words tend to come up as suggested searches. So try to use action words. Video Description: -Lead with a link to your site (include http://). -Don’t stuff this with your keyword. It leads to bad writing and it won’t work anyway. This is where you convince people to watch, so write for humans. Use some showmanship. -At the end, do a call to action (subscribe, see the whole playlist, visit our social channels, etc.) Video Tags:-Don’t over-tag. 5-10 tags per video is plenty. -If you’re compelled to have more than 10, that means you should probably make more videos specifically targeting all those keywords. Find Linking Pals:-45% of videos are discovered on video sites. But 44% are found through links on blogs and sites.-Write a blog about your video’s content, then link to the video in it. -A good site for finding places to guest blog is myblogguest.com-Once you find good linking partners, they’ll link to your future videos (as long as they’re good and you’re returning the favor). Tap the Power of Similar Videos:-Use Video Reply to associate your video with other topic-related videos. That’s when you make a video responding to or referencing a video made by someone else. Content:-Again, build up a portfolio of videos, not just one that goes after 30 keywords.-Create shorter, sequential videos that pull them deeper into the content and closer to a desired final action.-Organize your video topics separately using Playlists. Playlists show up as a whole in search results like individual videos, so optimize playlists the same as you would for a video. Meta Data:-Too much importance is placed on it. It accounts for only 15% of search success.-YouTube reads Captions or Transcripts to determine what a video is about. If you’re not using them, you’re missing out.-You get the SEO benefit of captions and transcripts whether the viewers has them toggled on or not. Promotion:-This accounts for 25% of search success.-Promote the daylights out of your videos using your social channels and digital assets. Don’t assume it’s going to magically get discovered. -You can pay to promote your video. This could surface it on the YouTube home page, YouTube search results, YouTube related videos, and across the Google content network. Community:-Accounts for 10% of search success.-Make sure your YouTube home page is a fun place to spend time. Carefully pick your featured video, and make sure your Playlists are featured. -Participate in discussions so users will see you’re present. The volume of ratings/comments is as important as the number of views when it comes to where you surface on search. Video Sitemaps:-As with a web site, a video sitemap helps Google quickly index your video.-Google wants to know title, description, play page URL, the URL of the thumbnail image you want, and raw video file location.-Sitemaps are xml files you host or dynamically generate on your site. Once you’ve made your sitemap, sign in and submit it using Google webmaster tools. Just as with the broadcast and cable TV channels, putting a video out there is only step one. You also have to make sure everybody knows it’s there so the largest audience possible can see it. Here’s hoping you get great ratings. @mikestiles

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• How would I instruct extconf.rb to use additional g++ optimization flags, and which are advisable?

  - by mohawkjohn

  I'm using Rice to write a C++ extension for a Ruby gem. The extension is in the form of a shared object (.so) file. This requires 'mkmf-rice' instead of 'mkmf', but the two (AFAIK) are pretty similar. By default, the compiler uses the flags -g -O2. Personally, I find this kind of silly, since it's hard to debug with any optimization enabled. I've resorted to editing the Makefile to take out the flags I don't like (e.g., removing -fPIC -shared when I need to debug using main() instead of Ruby's hooks). But I figure there's got to be a better way. I know I can just do $CPPFLAGS += " -DRICE" to add additional flags. But how do I remove things without editing the Makefile directly? A secondary question: what optimizations are safe for shared objects loaded by Ruby? Can I do things like -funroll-loops? What do you all recommend? It's a scientific computing project, so the faster the better. Memory is not much of an issue. Many thanks!

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• What's the steps for SQL optimization and changes without reflect live system ?

  - by Space Cracker

  we have a big portal that build using SharePoint 2007 , asp.net 3.5 , SQL Server 2005 .. many developers work in it since 01/2008 and we are now doing huge analysis for current SQL Databases [not share-point DB ] to optimize and enhance it. The main db have about 330 table and 1720 stored procedure (SP) created from 01/2008 till now Many table names / Columns is very long and we want to short it we found SP names is written in 25 format :( , some of them are very complex and also we want to rename many SP parameters need to be renamed one of the biggest table is Registered user table, that will be spitted in more than one table for some optimization, many columns name will be changed I searched for the way that i can rename table names ,columns and i found SQL refactor tool but i still trying it .. my questions : Is SQl Refactor is the best tool for renaming ? or is there any other one ? if i want to make it manually, is there any references or best practice for that ? How can i do such changes in fast and stable way .. i search for recommendations and case studies if exist ?

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• How is schoolbook long division an O(n^2) algorithm?

  - by eSKay

  Premise: This Wikipedia page suggests that the computational complexity of Schoolbook long division is O(n^2). Deduction: Instead of taking "Two n-digit numbers", if I take one n-digit number and one m-digit number, then the complexity would be O(n*m). Contradiction: Suppose you divide 100000000 (n digits) by 1000 (m digits), you get 100000, which takes six steps to arrive at. Now, if you divide 100000000 (n digits) by 10000 (m digits), you get 10000 . Now this takes only five steps. Conclusion: So, it seems that the order of computation should be something like O(n/m). Question: Who is wrong, me or Wikipedia, and where?

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• What is a convenient base for a bignum library & primality testing algorithm?

  - by nn

  Hi, I am to program the Solovay-Strassen primality test presented in the original paper on RSA. Additionally I will need to write a small bignum library, and so when searching for a convenient representation for bignum I came across this [specification][1]: struct { int sign; int size; int *tab; } bignum; I will also be writing a multiplication routine using the Karatsuba method. So, for my question: What base would be convenient to store integer data in the bignum struct? Note: I am not allowed to use third party or built-in implementations for bignum such as GMP. Thank you.

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• How to sort in-place using the merge sort algorithm?

  - by eSKay

  I know the question is too open. All I want is someone to tell me how to convert a normal merge sort into an in-place merge sort (or a merge sort with constant extra space overhead). All I can find (on the net) is pages saying "it is too complex" or "out of scope of this text". "The only known ways to merge in-place (without any extra space) are too complex to be reduced to practical program." (from here) Even if it is too complex, can somebody outline the basic concept of how to make the merge sort in-place?

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• The algorithm used to generate recommendations in Google News?

  - by Siddhant

  Hi everyone. I'm study recommendation engines, and I went through the paper that defines how Google News generates recommendations to users for news items which might be of their interest, based on collaborative filtering. One interesting technique that they mention is Minhashing. I went through what it does, but I'm pretty sure that what I have is a fuzzy idea and there is a strong chance that I'm wrong. The following is what I could make out of it :- Collect a set of all news items. Define a hash function for a user. This hash function returns the index of the first item from the news items which this user viewed, in the list of all news items. Collect, say "n" number of such values, and represent a user with this list of values. Based on the similarity count between these lists, we can calculate the similarity between users as the number of common items. This reduces the number of comparisons a lot. Based on these similarity measures, group users into different clusters. This is just what I think it might be. In Step 2, instead of defining a constant hash function, it might be possible that we vary the hash function in a way that it returns the index of a different element. So one hash function could return the index of the first element from the user's list, another hash function could return the index of the second element from the user's list, and so on. So the nature of the hash function satisfying the minwise independent permutations condition, this does sound like a possible approach. Could anyone please confirm if what I think is correct? Or the minhashing portion of Google News Recommendations, functions in some other way? I'm new to internal implementations of recommendations. Any help is appreciated a lot. Thanks!

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• Algorithm: how to check intersections of recurring events definitions?

  - by glaz666

  The question comes from MS Outlook calendar behavior. Imagine I have two recurring events (starting from today): "each second Monday" and "every odd date". Is there any way to check intersections and/or find the first intersecting date algorithmically without brute-forcing over each date? Definitions can be made in CRON's notations or ICal notation. I think it doesn't matter. Are there any solutions for this in Gregorian calendar?

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• Algorithm: How to tell if an array is a permutation in O(n)?

  - by Iulian Serbanoiu

  Hello, Input: A read-only array of N elements containing integer values from 1 to N. And a memory zone of a fixed size (10, 100, 1000 etc - not depending on N). How to tell in O(n) if the array represents a permutation? --What I achieved so far:-- I use the limited memory area to store the sum and the product of the array. I compare the sum with N*(N+1)/2 and the product with N! I know that if condition (2) is true I might have a permutation. I'm wondering if there's a way to prove that condition (2) is sufficient to tell if I have a permutation. So far I haven't figured this out ... Thanks, Iulian

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• How to find minimum weight with maximum cost in 0-1 Knapsack algorithm?

  - by Nitin9791

  I am trying to solve a spoj problem Party Schedule the problem statement is- You just received another bill which you cannot pay because you lack the money. Unfortunately, this is not the first time to happen, and now you decide to investigate the cause of your constant monetary shortness. The reason is quite obvious: the lion's share of your money routinely disappears at the entrance of party localities. You make up your mind to solve the problem where it arises, namely at the parties themselves. You introduce a limit for your party budget and try to have the most possible fun with regard to this limit. You inquire beforehand about the entrance fee to each party and estimate how much fun you might have there. The list is readily compiled, but how do you actually pick the parties that give you the most fun and do not exceed your budget? Write a program which finds this optimal set of parties that offer the most fun. Keep in mind that your budget need not necessarily be reached exactly. Achieve the highest possible fun level, and do not spend more money than is absolutely necessary. Input The first line of the input specifies your party budget and the number n of parties. The following n lines contain two numbers each. The first number indicates the entrance fee of each party. Parties cost between 5 and 25 francs. The second number indicates the amount of fun of each party, given as an integer number ranging from 0 to 10. The budget will not exceed 500 and there will be at most 100 parties. All numbers are separated by a single space. There are many test cases. Input ends with 0 0. Output For each test case your program must output the sum of the entrance fees and the sum of all fun values of an optimal solution. Both numbers must be separated by a single space. Example Sample input: 50 10 12 3 15 8 16 9 16 6 10 2 21 9 18 4 12 4 17 8 18 9 50 10 13 8 19 10 16 8 12 9 10 2 12 8 13 5 15 5 11 7 16 2 0 0 Sample output: 49 26 48 32 now I know that it is an advance version of 0/1 knapsack problem where along with maximum cost we also have to find minimum weight that is less than a a given weight and have maximum cost. so I have used dp to solve this problem but still get a wrong awnser on submission while it is perfectly fine with given test cases. My code is typedef vector<int> vi; #define pb push_back #define FOR(i,n) for(int i=0;i<n;i++) int main() { //freopen("input.txt","r",stdin); while(1) { int W,n; cin>>W>>n; if(W==0 && n==0) break; int K[n+1][W+1]; vi val,wt; FOR(i,n) { int x,y; cin>>x>>y; wt.pb(x); val.pb(y); } FOR(i,n+1) { FOR(w,W+1) { if(i==0 || w==0) { K[i][w]=0; } else if (wt[i-1] <= w) { if(val[i-1] + K[i-1][w-wt[i-1]]>=K[i-1][w]) { K[i][w]=val[i-1] + K[i-1][w-wt[i-1]]; } else { K[i][w]=K[i-1][w]; } } else { K[i][w] = K[i-1][w]; } } } int a1=K[n][W],a2; for(int j=0;j<W;j++) { if(K[n][j]==a1) { a2=j; break; } } cout<<a2<<" "<<a1<<"\n"; } return 0; } Could anyone suggest what am I missing??

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• How could I apply a genetic algorithm to a simple game that follows rollercoaster tracks?

  - by Chris

  I have free-rein over what I do on a final assignment for school, with respect to modifying a simple direct-x game that currently just has the camera follow some rollercoaster rails. I've developed an interest in genetic algorithms and would like to take this opportunity to apply one and learn something about them. However, I can't think of any way I could possibly apply one in this case. What are some options available to me?

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• Can SHA-1 algorithm be computed on a stream? With low memory footprint?

  - by raoulsson

  I am looking for a way to compute SHA-1 checksums of very large files without having to fully load them into memory at once. I don't know the details of the SHA-1 implementation and therefore would like to know if it is even possible to do that. If you know the SAX XML parser, then what I look for would be something similar: Computing the SHA-1 checksum by only always loading a small part into memory at a time. All the examples I found, at least in Java, always depend on fully loading the file/byte array/string into memory. If you even know implementations (any language), then please let me know!

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• Which is the best sorting algorithm: insertion,selection,bubble,shell,merge,heap,quick, or radix?

  - by flopex

  I wanted to know which of these algorithms is the best for sorting random,semi-sorted,reversed or unique data. I know that some of these are of O(n^2) and O(nlogn).

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• In Python, what is the fastest algorithm for removing duplicates from a list so that all elements ar

  - by Jeff Miller

  For example: >>> x = [1, 1, 2, 'a', 'a', 3] >>> unique(x) [1, 2, 'a', 3] Assume list elements are hashable. Clarification: The result should keep the first duplicate in the list. For example, [1, 2, 3, 2, 3, 1] becomes [1, 2, 3].

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• What is the best data structure and algorithm for comparing a list of strings?

  - by Chiraag E Sehar

  I want to find the longest possible sequence of words that match the following rules: Each word can be used at most once All words are Strings Two strings sa and sb can be concatenated if the LAST two characters of sa matches the first two characters of sb. In the case of concatenation, it is performed by overlapping those characters. For example: sa = "torino" sb = "novara" sa concat sb = "torinovara" For example, I have the following input file, "input.txt": novara torino vercelli ravenna napoli liverno messania noviligure roma And, the output of the above file according to the above rules should be: torino novara ravenna napoli livorno noviligure since the longest possible concatenation is: torinovaravennapolivornovilligure Can anyone please help me out with this? What would be the best data structure for this?

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• Is there any algorithm for determining 3d position in such case? (images below)

  - by Ole Jak

  So first of all I have such image (and ofcourse I have all points coordinates in 2d so I can regenerate lines and check where they cross each other) But hey, I have another Image of same lines (I know thay are same) and new coords of my points like on this image So... now Having points (coords) on first image, How can I determin plane rotation and Z depth on second image (asuming first one's center was in point (0,0,0) with no rotation)?

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• Algorithm for assigning a unique series of bits for each user?

  - by Mark

  The problem seems simple at first: just assign an id and represent that in binary. The issue arises because the user is capable of changing as many 0 bits to a 1 bit. To clarify, the hash could go from 0011 to 0111 or 1111 but never 1010. Each bit has an equal chance of being changed and is independent of other changes. What would you have to store in order to go from hash - user assuming a low percentage of bit tampering by the user? I also assume failure in some cases so the correct solution should have an acceptable error rate. I would an estimate the maximum number of bits tampered with would be about 30% of the total set. I guess the acceptable error rate would depend on the number of hashes needed and the number of bits being set per hash. I'm worried with enough manipulation the id can not be reconstructed from the hash. The question I am asking I guess is what safe guards or unique positioning systems can I use to ensure this happens.

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• How do you avoid an invalid search space in a genetic algorithm?

  - by Dave

  I am developing a GA for a school project and I've noticed that upon evaluating my functions for fitness, an individual is equivalent to its inverse. For example, the set (1, 1, -1, 1) is equivalent to (-1, -1, 1, -1). To shrink my search space and reach a solution more efficiently, how can I avoid my crossovers from searching in this second half of the search space?

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• Is this a bad version of the Merge Sort algorithm?

  - by SebKom

  merge1(int low, int high, int S[], U[]) { int k = (high - low + 1)/2 for q (from low to high) U[q] = S[q] int j = low int p = low int i = low + k while (j <= low + k - 1) and (i <= high) do { if ( U[j] <= U[i] ) { S[p] := U[j] j := j+1 } else { S[p] := U[i] i := i+1 } p := p+1 } if (j <= low + k - 1) { for q from p to high do { S[q] := U[j] j := j+1 } } } merge_sort1(int low, int high, int S[], U[]) { if low < high { int k := (high - low + 1)/2 merge_sort1(low, low+k-1, S, U) merge_sort1(low+k, high, S, U) merge1(low, high, S, U) } } I am really sorry for the terrible formating, as you can tell I am not a regular visitor here. So, basically, this is on my lecture notes. I find it quite confusing in general but I understand the biggest part of it. What I don't understand is the need of the "if (j <= low + k - 1)" part. It looks like it checks if there are any elements "left" in the left part. Is that even possible when mergesorting?

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• Is there a name for this type of algorithm?

  - by rehanift

  I have a 2 dimensional array forming a table: [color][number][shape ] ------------------------- [black][10 ][square ] [black][10 ][circle ] [red ][05 ][triangle] [red ][04 ][triangle] [green][11 ][oval ] and what I want to do is group largest common denominators, such that we get: 3 groups group #1: color=black, number=10, shapes = [square, circle] group #2: color=red, shape=triange, numbers = [05,04] group #3: color=green, number=11, shape = oval I wrote code that will handle a 2 "column" scenario, then I needed to adjusted it for 3 and I was figuring I might as well do it for n. I wanted to check first if there is some literature around this but I can't think of what to start looking for!

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• Algorithm design: can you provide a solution to the multiple knapsack problem?

  - by MalcomTucker

  I am looking for a pseudo-code solution to what is effectively the Multiple Knapsack Problem (optimisation statement is halfway down the page). I think this problem is NP Complete so the solution doesn't need to be optimal, rather if it is fairly efficient and easily implemented that would be good. The problem is this: I have many work items, with each taking a different (but fixed and known) amount of time to complete. I need to divide these work items into groups so as to have the smallest number of groups (ideally), with each group of work items taking no longer than a given total threshold - say 1 hour. I am flexible about the threshold - it doesnt need to be rigidly applied, though should be close. My idea was to allocate work items into bins where each bin represents 90% of the threshold, 80%, 70% and so on. I could then match items that take 90% to those that take 10%, and so on. Any better ideas?

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• algorithm to generate maximum number of character 'A' using keystrokes 'A', CTRL + 'A', CTRL + 'C' and CTRL + 'V'

  - by munda

  This is an interview question from google. I am not able to solve it by myself. Can somebody throw some light? The question goes like this. Write a program to print the sequence of keystrokes such that it generates the maximum number of character 'A's. You are allowed to use only 4 keys: 'A', CTRL + 'A', CTRL + 'C' and CTRL + 'V'. Only N keystrokes are allowed. All CTRL+ characters are considered as one keystroke, so CTRL+A is one keystroke. e.g.: A, ctrl+A, ctrl+C, ctrl+V generates two As in 4 keystrokes. Edit: CTRL + A is Select All CTRL + C is copy CTRL + V is paste I did some mathematics. For any N, using x numbers of A's , one CTRL+A, one CTRL+C and y CTRL+V, we can generate max ((N-1)/2)^2 numbers of A's. But for some N M, it is better to use as many ^A, ^C and ^V as it doubles the number of As. Edit 2: ^A, ^V and ^C will not overwrite on the existing selection but it will append the copied selection to selected one.

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