A thesis on virtual network embedding with genetic algorithms

Recently I finished my master thesis which is a study on the problem of virtual network embedding (VNE). It concerns the virtualization of network resources and topologies so as to create a fully functional virtual network embedded in a detacenter, instead of using a physical stand alone network. There are many advantages in this approach as it is much easier to update and maintain virtual devices and links instead of physical ones.

There are many ways to map virtual on physical resources, a relatively small virtual topology may be embedded in many ways in the physical network of a datacenter. The problem of VNE concerns the finding of the ideal mapping so that virtualized network will have the optimal performance.

It is a hard problem to solve so traditional analytical methods are inefficient for its solution. Artificial intelligence methods provide in such cases good and efficient solutions. In the thesis I developed a genetic algorithm that approaches the problem as an optimization problem and provides good practical solutions. 

My thesis is written in Greek and is accessible in the repository of the Hellenic Open University.  A direct download of the paper is possible in this link.

A Fibonacci complexity class

While working on a project that will come out soon, I developed an algorithm with strange asymptotic behavior. In one of the functions of the algorithm there was an arithmetic progression that had to be computed. It was not a big deal as it concerned small sets of numbers each time, at least that's what I thought.

So instead of using the formula of arithmetic progression, I just put a recursive function for it. Something like

int d = 1000;
public void progression(a){
    while(d>0){
         progression(a+1);
    }
    d--;

}
progression(0);

Where d was proportional to the input.

It worked fine, but it was slower than it should be. The reason was that the progression was computed quite a lot of times and eventually such a simple function was a huge computation burden. I was really surprised to discover that the asymptotic behavior of the algorithm could be described by the Fibonacci sequence.
So that, given n input the complexity of the algorithm was O(f(n)∙c), where f(n) is the n-th member of the Fibonacci sequence and c is constant.
After replacing the function with the formula of arithmetic progression the complexity became polynomial, thankfully. 

There are two remarks about all these.

An innocent, simple and naive function may become a problem. We have to be careful on how many times this function will be executed.

Although the first edition of the algorithm was not optimal, there comes the question. Could we define a Fibonacci complexity class? Perhaps even find problems in this class.

A function for the Fibonacci sequence

A very fast and easy function to compute Fibonacci sequence

import java.math.BigInteger;

public class main {
    int cnt=0; 
    public static void main(String[] args) {
        main m=new main();
        m.f();
    }
   
    public void f(){
        BigInteger a = new BigInteger("0");
        BigInteger b = new BigInteger("1");
        System.out.println("F("+cnt+"): "+0);
        cnt++;
        System.out.println("F("+cnt+"): "+1);
        cnt++;
        while(cnt<22001){
            BigInteger i = new BigInteger("0");
            i=i.add(a);
            i=i.add(b);
            a=b;
            b=i;
            cnt++;
        }
        System.out.println("F("+(cnt-1)+"): "+b);
    }
}

The output, the 22000th member of the sequence

F(22000):

23911151843088049549559913282124240747696450497332393096044892530723732731293925768111253903890731043326256888603201506944633327289997819833277130504901675470982030552507094162859800238507116975505854341394434948447565033026221210674687237739231690676803475798182981709459374260692436901977628581806475354563630219526845715243350120708966860434133018586240550140499321343707423004223534607204678776111145268537233116243802761082543246510402792429498707827152647989073216179648475243965598252066378230654130470829925986857896675242722600485435164105264916456225839778968745829788512608356363913804761755901467120731200188413083136579975059928902776220812920982189971242379916708698973621766751597334216368620508938142664716772647688988834270374687235249785796824594603884916806990714534340114784869357840251209972361365742785864148242067483656445079559019489390577428621648443326010160477386933121033620560251402766516093547505148407529090149475348773752540043271767234242084292100628738483524181221019416392799181393231992347653438406600914228647401007351836588160487756479497407968315031381589638689295174823322470087642165598672121365266344137564858444622711576081771078496641844937390545682240273855295137238355228099317669135050155578932164209299731907594076460356882408869295135953607469229651099120705693263857034687843626991261121860614751345764505435552449681904020194972948652249854216623551013979700998791070339585263264828474927294673678682855378403136525049646034651097985005523792023353505230733318315806249870620106255127740873138847984020791508370779261660785053242257951531507791170889410085082494377608208359506865872052269194573000651768246582561397589379054459666059728582161757153393761840155076715427874951737024712380325595601597280455609229448744861922777558470731225112774660744411373183007786591379470772659610931545615527361828527091916649882203801119543947473205919119948687005839582793430636346335172367632567614224230754519059566437786383250603562336173795450845023555052163775749567488680209929301733953771659440841851896501176310938478061013508743746794886211545059385083016686715569969754105054345618188437480456355103779247725818147048349484128145728116464124348995944145983490657741814168978702539394407772782423978914262812859165776960650497186523875998775018517623119052276070479290941716138770758463104282213404679330007239145717905126309090972993970782474750784981648644639346866248972640808995649185451943534239803644658697695716458521133485188727182407257352988202971646888599939358651008064484936913916373530359894095131542041397727523148718163746575229775017491009940588025249473123594753441211634435730001790835648327090911035577452758024952482629199328590011397832889757271508213625527057331090954110825073969299898958013994599894944208687850453500173980744223000760078898697128052192071515006870831715957325971845783409157683811011917636814704595836802955582938763983561376453999740428687093957605525631048148288024553370500665522100169024309239057322581866840037497559911574765827496593193700042249031389395129435568195231113075207830333318500469629193676005498811061550348234102052774122312886541766341471582469457347117636205703661078984515116258554803095676657019285090837775746036193164235341465548098904615127611062939206161346093617265405906367027661546845621356149559040798776562241563525511725505792648393371898115939776591096309370497408492873254883402409802564872039524422858633347863626333558155009162515951780107533626468012894434520530192845273457409548744656177387603307356149287599938730196582636274410931646393247886112418444661056857161146184933011078210524893794117348746892516723498126540948178477524603638775070638523850390891025039264400618873393052111880108096369408947789274041790558597294722651729924404836342862376149286822275330813331363925609055924677777163416615796764181780714265864396474084320765015608284585854217131086151555790325994153510435163207497453674171928909161205383175664268065518581620152617418382937870516278875447542815346729709387258529667056688675843466545834345387745629279011899890664127165078236901460104874038389111495950291594316779654469302636244158996341777359509518944948859542336178551804878882015290324970263731030303062802591853791022670735831866555863543199284768114955047509691099289344617944901044767695479022055197983956300649586454906832181127379710025435852700124842320460296470665284849646627146368239854763930002211169617338509922746447728861929093704279396632724922533039876469981523255796252507976652376398665656696184253857275835860915242223337528463584609575885682566329602411550875

Killing time while in quarantine

There are many posts in blogs and social networks these days about what some people do or what suggest to others to do while they are isolated in quarantine. Well, this is my suggestion.
Find a difficult and interesting open problem and try to solve it. Some problem that you may understand its definition but you and the rest of the world have no idea about its resolution. In computer science and logic good candidates are the famous NP vs P problem or the Continuum Hypothesis; of course there are numerous others.
Reading about the problem and working on any idea that you may have about it will not only be an interesting task but will probably make you learn something interesting and it will surely help you in killing time. Who knows, eventually you might actually solve the problem and gain fame and fortune.
In case you follow this advice here is another one on how you should approach it. If you want to solve an open long standing problem you have to think innovative and differently than everybody else. If thinking like everybody else and following the mainstream would be enough to solve the problem then it wouldn’t have been open as someone would have already solved it.

CONVID-19 coronavirus quarantine. A chance to promote e-conferences.

The quarantine that has been imposed to many parts of the world due to the spread of the CONVID-19 virus creates a lot of problems in transportation and in events like conferences. This may be a chance to reconsider the way that scientific conferences are organized. Although modern technologies of teleconferencing are used in a wide scale in education they are not used in conferences. So in parts of the world were the organization of gatherings is nowadays impossible, conferences have to be canceled.
A good alternative in all these trouble is simply teleconferencing, probably including an archive of video presentations and lectures. But if the scientific community will ever move to this direction there will be more benefits than just dealing with a temporary uncomfortable situation. Promoting teleconferencing and establishing a new type of e-conferences will make conferences open to a wider audience, not just to those people that make it to attend the event. Even without the quarantine, traveling to other countries and continents to attend a conference is not always an easy case and for many people is a difficult issue.
The demand for open science and data should not be restricted to the free dissemination of papers and preprints. In an era were multimedia flood the web and MOOCs are the latest trend in education, conferences should follow the same direction and become open platforms for exchanging ideas and interesting results in global scale.