A new project with TypeScript and Angular

July 2, 2018 Leave a comment

More than a year ago, I started a new adventure in a new startup company. New company, new adventure and a new project. New technology maybe?
Of course the risk of adopting a new technology in a new project is lower than migrating to a new technology in an existing one, but there still is a risk. Especially if the technology is young and almost no one in the team has experience with it.

I’m working in web projects for almost twenty years and with JavaScript for all this period. It is said that JavaScript is the least understood language. And even though you understand it you need a very high level of discipline in designing your application and writing you code if you want to keep away from spaghetti code. One of the biggest issues with JavaScript in my view is that it’s not a strong type language. In the past in my code I even tried to bring classes in JavaScript. But this solves the problem only partially.

You can understand my enthusiasm when I saw TypeScript. A strong typed language for the web. Yoohoo! And an entire framework built on top – Angular. Angular, not AngularJS. I worked with both frameworks, but basically what they have in common is the name. Angular is also known as the next version of AngularJS, or Angular 2, 4, 5, 6 …

Now coming back to the project. I proposed for it as the development language/framework the new TypeScript/Angular. At that moment it seemed like a big risk: no one in the team was used it before and even myself I have used in only couple of projects, none of which made it into production. But now, in retrospective, I believe it was one of the best decisions when it comes to technology selection for a new project.

I would not insist too much about TypeScript and Angular, but I still would like to point out a few advantages that I really like to make my case.


TypeScript it’s a strong typed language for the web with a lot of similarities with JavaScript. It’s not an interpreted language, but a hybrid one that compiles to JavaScript. This way you’ll catch a lot of errors right in the development phase, even better they’ll be flagged by your favorite IDE/editor.

Looking into the future, I think new projects and libraries should be written in TypeScript, even the ones targeting JavaScript. TypeScript is interoperable with JavaScript, the code compiles to JavaScript and the library is augmented also with type information for TypeScript users. The compiled script is optimized, obfuscated and easy to integrate. JavaScript acts like some kind of assembler code in this case.

A lot of the TypeScript improvements came to JavaScript through the latest ECMAScript standards, but not all are widely supported. There are also initiatives for native TypeScript support directly in the browser. But I would still see quite a few advantages from the ones outlined above still standing in a hybrid approach (compiled + interpreted).

In conclusion, I believe TypeScript is the modern and the best choice when it comes to programming languages for the web. It’s so cool, that sometimes I cannot believe it was made by Microsoft. Of course, it was a joint effort and maybe this approach will make them think about their future in a more and more open community.


Angular is the perfect companion as a framework for TypeScript. Its component-ized approach could seem an overkill in the beginning, but in an enterprise project you’ll quickly see its value. Components can be easily isolated and reused. It’s so easy to develop such a component, that sometimes could be easier to develop your own from scratch than customize an existing 3rd party one. Of course, this should be the exception, rather than the rule :).

As I said earlier AngularJS and Angular have basically only the name in common. Because of that it’s pretty hard to upgrade from the former to the latter. Quite the opposite is to upgrade between different versions of Angular as they maintain a high level of backward compatibility and features are deprecated progressibely. Usually it took me just a few hours to upgrade from Angular 2 to 4, from 4 to 5, from 5 too 6. The fact that TypeScript is strong typed, the compiler, or even better the IDE, points out the errors, making it extremely easy and straightforward.

Of course, a homogeneous product is the ideal case, but those are so rare … We had to integrate our project with an existing one built on AngularJS. It was like a case study – how to upgrade and interoperate between Angular’s. Angular came with a nice rescue solution here and with a decent effort we came up with a clean way of doing this. I will not enter into details here, but the nicest part in that, which definitely gained my vote, was that you could actually upgrade module by module, or even component by component. And the effort finally paid off when we started to reuse parts of the new project into the old one.

If you want to start a new project, Angular is a very well equipped framework that comes out of the box with TypeScript linter and compiler, webpack, SCSS support, unit and automation testing, polyfills etc. AngularJS did not have an official scaffolding tool, but Angular has Angular CLI which does a nice job.

TypeScript and Angular offered us a development landscape with emphasize on ease of development, less errors and lots of reusing opportunities. I think it was the best foundation on top of which we could build a modular toolkit, based on atomic design principles. We also managed to create a continuous build system, where the code was lint-checked and compiled for different environments, catching a lot of issues, right from that phase, a much harder or even impossible endeavor with JavaScript and other frameworks. We also integrated unit and automation tests and we’re working on extending the coverage of these. This will give us the confidence to build new features at higher speeds and shorten release cycles.

So, every time when you start a new project, especially if you’re unhappy with your development ecosystem, try investigating new ones – technologies are evolving nowadays at much higher speeds. For the past decade or so, the biggest issue in software and web development is maintainability, even before performance. And more importantly, don’t be afraid of change, embrace it.

Categories: Software, Web

Atomic design

January 23, 2018 3 comments

I recently read Atomic Design by Brad Frost. It was like a breath of fresh air – look! someone else is thinking the same, phew, I’m not alone. And not only that, someone else took the time to write a book and formalize everything. I sincerely believe that the book should be a mandatory read (it’s easy and takes only a few hours) for anyone involved in web projects – UX designers, visual designers, copywriters, front-end developers, back-end developers, testers, project managers, CTOs … anyone! Are you in the web business? Then read it.

Why do I liked it so much? Not only because it lays down some very good design principles and offers a common language to it, but mainly because it preaches a mindset change.

And now I come to the point where I want to tell you about my experience in this field. About six years ago, about the same time Brad started to apply the principles in his book, I was working for a major IT company. Mobile web was on the rise, but the company had no presence there. The desktop website was rendering on mobile just as a zoomed out version, making it unreadable and, of course, unusable. So I pitched the idea of building such a mobile web presence to my manager and I was in luck. I was tasked to create a proof-of-concept and the idea caught. One other manager was on board, so this way a new team was born. The proof-of-concept went live so that we can actually get some usage metrics.

Few months later it was another lucky development. Periodically the company was going through brand redesign. And this included the web presence too. Frankly speaking, my case was not exactly like the ones in Brad’s book. The company had structure and clear brand, including web, guidelines. They even had a governance team. But the problems were from another nature. First, to give you an idea of the magnitude, the company website consisted of hundreds of thousands of web pages and tens/hundreds of web applications. A lot of teams, even external agencies were working on these.

These were raising some very interesting challenges:

  • Redesign was tedious, long and expensive. Many people for many months were working on this updating all these. Many times entire sections or applications did not benefit of the facelift, so the same website had different living designs.
  • Because the web guidelines were quite extensive, the learning curve for creating web pages and applications was very steep. Thus high costs.
  • The governance team was overwhelmed and most of the time busy with checking websites if they meet the standards they put in place. Due to this check, going live with a website was delayed, sometimes inexplicably for the developers and stakeholders
  • Special needs of some development teams were almost never addressed so this was ending up in frustration and either going rogue or doing an ok-ish work with the sole purpose of just delivering something

Then I said to one of the UX designers:
– Cool! We have now the chance to do things the right way. We build not only a mobile presence, but a responsive web presence for all the devices and we will build it like a toolkit that can be used by everyone.
– Nah! Building such a toolkit will be very tedious. Not technically, but politically, to get the buy-in of all the high levels involved.
– Then let’s do it at a smaller scale to show them the advantages.
– Yup, here you may have something, let’s do this for mobile.

Then I connected directly with the UX team and asked them to create a list of semantic components. I explained them the concept of semantic design(CSS), its advantages and the fact that it doesn’t add up any work – it’s just a paradigm shift. They were very open (maybe I infected them with my enthusiasm, maybe they wanted to try something new) and they agreed. After just a few iterations we had an entire box full of semantic components. At that time I wasn’t aware of atoms – molecules – organisms, even though we organized them incrementally too. But I think this naming convention is much clearer.

I just want to make a small parenthesis here – the biggest challenge here was finding good names and most of the iterations were related to it. We even applied this when naming colors. I believe if you cannot find a good, SEMANTIC name to stand the test of style, then it would not stand the test of time. Test of style means that the name will make sense even if you completely change its style.

Initially, the components were presented in desktop style, but this was no issue. Creating a mobile style was fairly easy and fast. Only then we jumped to development – and it was a breeze this time. We even used the same names in CSS and the fact that we could reuse styles and components made all that planning effort and mindset change worthwhile.

We ended up with a toolkit in just a few weeks. And this was the output of a team of just 4-5 developers, not fully dedicated to this project. Now, other front-end developers and external agencies were able now to develop mobile pages. But how was this better or faster? First of all they got rid of all the web guidelines, a huge book to read and, if you wanted to be proficient, memorize. Now they had a few templates, a handful of components and just a few pages of documentation. If they stick to those templates and those components, their pages will be compliant. And here is how you get a quite fast approval from the standards team. Also this toolkit had embedded all the quirks of mobile development (back then were much more than nowadays 🙂 ). They were able to test their mobile pages on desktop browsers and have the confidence that they will work on mobile devices too. We, as the development team, took care of all the inconsistencies between different devices and browsers. This also gave them the opportunity to focus on their task at hand – develop mini websites fast and not care about a plethora of devices and associated quirks.

We also had toolkit guidelines, but ours were much simpler: do not introduce any new CSS classes or any new custom tags – just stick to our templates and components. Would you think it’s too restrictive? Not a chance. We also advertised to all our users that we will create ourselves any new components that they might need, if the current ones are not sufficient. And we got a lot of requests. But most of the time, those requests were practically a misunderstanding of the naming convention – they were looking for synonym component. We ended up adding synonyms to the documentation. And to make it official, we always responded with a link to that documentation. Sometimes, creating a new component wasn’t actually necessary, but just tweaking and customizing an existing one. This way the UI toolkit became more powerful and the documentation more comprehensive with each request coming from our users. And the number of requests was decreasing, freeing us the time to extend and improve the toolkit.

For static pages we had templates and components developed in …, actually the name of the tool is not important, but the fact that the users were not starting with a blank page. And the learning curve was not steep anymore. We even had JSP templates and custom tags available for web application development. We also created a transcoder from desktop pages to mobile optimized pages using the same toolkit.

And now I will tell you about two cases demonstrating the power of this approach, cases that stuck into my mind.

Less than a year later, by the time our toolkit became the one stop shop for mobile pages development, the time for a new company-wide redesign came. Most of the components stood the test of time and they just needed a facelift. Probably the most eye-catching facelift was to move from a black background to a white one. I’ll tell you later why I revealed this detail.

After UX and visual designers hand us over the new specs, for us was mostly a matter of rewriting a new CSS. Which we did in less than 3 weeks. When we were ready to go live, the desktop team was simply amazed:
– What? We haven’t finished yet the homepage. But I guess you did it only for a few pages, so it’s not actually ready to go live yet.
– No, we did for ALL the pages.
– Including the transcoded pages?
– Yup!
– For all the countries and languages?
– Of course.
– But you could not go live, we’re not ready.
Then the frustration was on our side and of another kind. But they agreed a compromise – to change the style just on the homepage and all the subsequent pages to remain on the old style. Most probably hoping that this would buy them another 2-3 weeks at least. Next day we came up with this version – in the end was just a matter of conditionally including one CSS or the other.
– But we also want to do some A/B testing and release the new design gradually.
– Ok, no problem, we already have support for this. We just need to know the target users for A and B.
Finally they let us go live with the new version in full. Few weeks later, they managed to release the homepage, many months later approximately 80% of the pages got the new design. A/B testing never happened.

Next day, after we went live with the new design, one of the front-end developers came to my desk and told me:
– I was developing yesterday a mobile page. In the evening I saved it and shutdown my computer. This morning I came, opened it and it turned from black to white. But I swear that I didn’t do anything.
I started laughing and I assured him that it’s fine, that this is the new redesign communicated by the standards team just few weeks back. I have to admit that I was happy about this level of upgrade. But we also sent a communication stating that front-end developers don’t have to do anything to get the new version. They just have to use the same tools, guidelines and toolkit. The replies were almost instantaneously: “We want this for desktop too!”

This is how we got the buy-in for creating a new RESPONSIVE toolkit. But that’s another story.

Update Jan 24th, 2018 (thanks to Meghan)

There are just a few, but very valuable things that I took from the “Atomic Design” book. First of all this book formalizes the entire process and gives pretty good names to everything.

I was using the term of semantic design, but atomic suggests the idea of modularity. On the other hand, semantic clearly states the separation between content and style. We were using the term of components, and even though we were describing them incrementally, I think atoms-molecules-organism makes a much clearer separation and gives a better idea of magnitude.

Another nice idea is the clear separation between UX comps and visual design. If you do this, it will be an additional check that your design system is both modular and semantic.

Categories: Web

Paperless user manuals

August 12, 2016 Leave a comment

In the last years an idea is bugging me. You probably saw those user manuals that come with almost any product. Yeah, those thick ones that nobody reads them. Even better some are accompanied by a CD. Really? CD? Has a time machine been invented and are those manufacturers producing goods to ship them in the past?

Who’s using a CD anymore? Or should I ask, who’s owning a CD or DVD anymore? From thousands of laptops on the market I think you can find DVD drives in only few tens or hundreds of them. And when it comes to tablets, and many laptop owners switched to tablets, the ratio is clear x:0.

The funny part is when you get such a CD for setting up a laptop or even a DVD drive. Come on! Everyone now has an Internet connection and optic fiber is even de-facto standard in many countries. Yeah, I agree that there are some exceptions, but I would say that there are only exceptions. And if they’re not exceptions how would you call owning a CD/DVD drive?

So, why the manufacturers are so stubborn in printing and packing those? Is there any kind of law to force them? I think it should be one to force them not to print them. As an ecologist and a parent, I care about what happens to our trees and environment in general.

I would think that the best thing should be a small standard sticker with a QR code and a tiny URL printed on it. From that point on you’ll get a web page with a much better interaction. Starting guide, usage sections in a mobile optimized experience. Nowadays many people are owning a smartphone and this will be an improved experience. Not to mention that few months after you bought a product, most probably you lost the user manual. But not if you bookmarked the page or if you read/scan the sticker again.

Also imagine the costs of creating such user manuals and printing them. Costs that you pay as the end user.

Unfortunately, until a law will enforce high taxes on this kind of manuals and the total waste of resources, I don’t think the situation will change.

Categories: Ideas

PhoneGap setup

March 22, 2016 1 comment

It’s not the first time that I played with PhoneGap, but I haven’t done in quite some time. But I always liked the idea of creating a platform independent application. And if that application can be tested directly in the web browser, even better.

Creating a user interface in a descriptive language like HTML is easier as opposed to a programmatic approach where you have to write code to create your visual components. Nowadays, most frameworks also offer the descriptive approach, usually through XML, but learning a new language when you already know another one more powerful is not that appealing. HTML is also augmented by CSS that easily offers a high degree of customization and JavaScript that comes along with functionality. And all together create a platform-independent framework with a high degree of customization and a clear separation of layers.

So it’s clear why I like the idea of PhoneGap right from the start. Now, let’s set it up.

To develop a Phonegap application you don’t need to many things. The best thing will be to install nodejs and then phonegap: npm install -g phonegap.

Then you can create a sample application with phonegap create my-app, command which will create all the necessary files and subfolders under my-app folder.

Now it comes the testing part and for this you need to install PhoneGap Desktop. As I said, it’s nice that you can test your app directly in your browser by visiting the link displayed at the bottom of Phonegap Desktop window, e.g. (hint: it doesn’t work with localhost or And if you install PhoneGap Developer App you can easily test on your mobile too without the hassle of installing the application itself every time you make a change – changes will be automatically deployed (reloaded).

When you’re done it comes the fun part – actually building the application. Let’s do this for Android.

First you need to install JDK (I tested with version 8) and Android Studio.

And then you need to setup some enviroment variables

  • JAVA_HOME – this must be set to the folder where your JDK, not JRE, is installed.
  • ANDROID_HOME – this must be set to the folder where your Android environment is installed.
  • add to PATH the following %ANDROID_HOME%\tools;%ANDROID_HOME%\platform-tools;%JAVA_HOME%\bin in Windows or ${ANDROID_HOME}\tools;${ANDROID_HOME}\platform-tools;${JAVA_HOME}\bin in Linux

If the above are not correctly set or the PATH is invalid (like it has an extra quote(“) or semicolon(;)) you can run into errors like

  • Error: Failed to run "java -version", make sure that you have a JDK installed. You can get it from: http://www.oracle.com/technetwork/java/javase/downloads. Your JAVA_HOME is invalid: /usr/lib64/jvm/java-1.8.0-openjdk-1.8.0
  • Error: Android SDK not found. Make sure that it is installed. If it is not at the default location, set the ANDROID_HOME environment variable.

I also had to run

phonegap platforms remove android
phonegap platforms add android@4.1.1

By default I had installed Android 5.1.1, but I was getting the error Error: Android SDK not found. Make sure that it is installed. If it is not at the default location, set the ANDROID_HOME environment variable. You can check what platforms you have installed by running the command phonegap platforms list.

Make sure that you have all the Android tools and SDKs installed by running android on the command line and select all the ones not installed and install them.

Finally, you can build the application by running the following command in your project folder:

phonegap build android

and if everything goes well you’ll find your apk at <your-project-dir>/platforms/android/build/outputs/apk.

Categories: Software, Web

Banking apps

October 8, 2015 1 comment

I was thinking few days ago what I want from a banking application. So I decided to write an article from the users point of view and their expectations when it comes to banking applications. So this one is for everyone, not only technical people :).

We have to admit that banks are huge dinosaurs, especially when it comes to their web site and the web application offered to users. And it shouldn’t be the case. They’re making huge piles of money out of thin air, I’m trusting them with my money, at least they should give me some good tools.

At first, there are some features that every banking app should incorporate. You should be able to access any of your accounts, current, savings or loans, have an AGGREGATED status of all these, create new ones in any currency or delete existing, be able to easily transfer between them and any other external accounts. And if I want to transfer money, I don’t care if it’s internal, national or international, I’ll just give the recipient, the IBAN (or any account identification string), the amount and that’s it. Show me the transaction fee (and the exchange rate if that’s the case) and if I accept it, just do it. And the same for scheduled transfers in the future or recurrent (weekly, monthly, yearly) ones. If the transaction fee will change on any of these in the future, deactivate them and just let me know so I can reactivate them.

Direct debits are a must. I like a bank where I have to spend less of my time in the bank. Offline or online. So I should be able to easily setup direct debits like any other transfer – specify a recipient, IBAN and a limited amount per week, month or year. Also any company should be able to easily request debits from my account, through some kind of API. And then I could do this not only for my gas and electric bill, but also for my internet provider or gym subscription. And I can cancel them at anytime or just set an expiration date.

Another must is to be able to associate my card with any of my accounts. Imagine that I go abroad and then I would like to spend money from a foreign currency account. Be able to switch and switch back instantly, when traveling is not uncommon, shouldn’t be impossible or a hassle.

A mobile app with which I can pay, without any credit/debit card is also something that should stay in the ordinary area, not cutting edge tech.

Categories: Web

IMW 2015 or How to make an (un)interesting presentation

October 8, 2015 Leave a comment

I’ve just attended for the fourth (I guess) year in a row the biggest Romanian conference on Internet and mobile. Bigger and bigger each year, I always had a love-hate relationship with Internet and Mobile World. At the beginning I say that it’s not interesting and it’s probably the last year I will attend, but then a few presentations and exhibitors change my mind. This year was no different.

I read lately a Dale Carnegie book and he was saying that if you want to sell something you shouldn’t talk about you and your product, but about the customer need. Nothing but true. In this idea, I saw some really boring presentations. First of all, some of them were given by managers or CEOs. I don’t have something against CEOs, but let’s get one thing straight. They got there, not because they know how to captivate the audience, but because they know how to run a business. Unfortunately, some managers, as soon as they become managers they also instantly gain access to the entire world knowledge and become proficient in every skill known to mankind. No! Again, they became managers because they recognized talent and they knew how to acquire it into their business. They should do the same here – get someone else to give a really interesting and captivating presentation.

I understand that you want to brag about what cool things are you creating and how good your company is doing. But, me in the audience, I’m also doing cool things. And if I’m not, I probably hate you. Or I simply don’t care about it. You don’t care either about my needs or interests, why shouldn’t this be a fair relationship?

I can easily get a financial report, projection or a product/service portfolio from any company website. So don’t come and present long and boring slides about any of these topics. But if you talk to me about any of my needs and interests and then slowly you introduce me on how a new technology can help me you can arouse my curiosity. And now that you got me, you can also happen to mention about your cool and innovative product that is the incarnation of that technology. And don’t make any false claims, either on the product or your expertise on the field. If I realized it and I will surely do, if I’m interested on the subject, taking into account the multiple channels of gathering information nowadays, then I will drop your presentation like a rotten apple. As a shadow of doubt, at the very best, felt on its entirety.

High level managers of large corporations, which they haven’t evolve from startups, usually tend to have a results oriented presentation. Whoa! I did not pay a ticket to came and make your business here. I got it – your company is the greatest and I think we established that already. Developers on the other hand tend to lose themselves in small technical details – I even saw slides of code in front of a general audience. Whoa again! I did not came here to work. I came to get fresh ideas, new contacts, to see where the market (not a company or product) sits as a whole. So you better arouse me, show me something cool, interesting, spiced with clear use cases. Walk in my shoes and show me a road to your product, but an interesting road.

Few presentations were like this, but those that were, they felt in the interesting area, even though I knew some of their content. And for God’s sake, speak loudly and in no way in a monotone voice. And if you don’t master English better ask your audience if you can do your presentation in your mother tongue.

I also need to highlight an interesting idea materialized in a cool product by a Romanian company. Altom built a robot that by using two motors and a camera was able to automate testing on real(!) devices like cameras and tablets. Using the camera it was recognizing images and then it was able to move on XY axis and tap on the device. And you could write your testing scenarios directly from your IDE, just like a Selenium test case! I would vote them as the most innovative product on this fair.

In general, the market trends seemed to be streaming and video on mobile and Internet of Things. Robotics seemed to be catching on, but this was always the case with this field for years and years, having spikes, but never really becoming mainstream. Indeed, in different forms and shapes they already entered our life, but not at the expectancy that SF fans always hoped. Cloud moved more to a mainstream level, which is kind of the case. I would definitely not think today of creating my own infrastructure, no matter the size of the project or company.

So, all in all, IMW 2015 wasn’t a waste of time with some, not majority, interesting presentations and exhibitors. Will I go next year? Will see :).

Categories: Technology

Shift an array in O(n) in place

April 7, 2015 Leave a comment

Below I copied the code for shifting an array in place.

void shift(Object[] array, int startIndexInclusive, int endIndexExclusive, int offset) {
if (array == null) {
if (startIndexInclusive >= array.length - 1 || endIndexExclusive <= 0) {
if (startIndexInclusive = array.length) {
    endIndexExclusive = array.length;
int n = endIndexExclusive - startIndexInclusive;
if (n  0) {
    int n_offset = n - offset;
    if (offset > n_offset) {
        swap(array, startIndexInclusive, startIndexInclusive + n - n_offset,  n_offset);
        n = offset;
        offset -= n_offset;
    } else if (offset < n_offset) {
        swap(array, startIndexInclusive, startIndexInclusive + n_offset,  offset);
        startIndexInclusive += offset;
        n = n_offset;
    } else {
        swap(array, startIndexInclusive, startIndexInclusive + n_offset, offset);

The swap(array, index1, index2, len) method swaps in the given array the elements from [index1, index1 + len) with the ones [index2, index2 + len).

Even though it may seem complicated at first, the idea is pretty simple. If the offset is half of array length, or in other words if offset == (n – offset), where n is the total number of elements to be shifted, then the shift is equivalent of swapping the two halves of the array.
For the first two cases we swapped the portions at the ends and one of them will come in place and we will continue the iteration for the rest as shown in the below figure.

shift algorithm

Space complexity is clearly O(1), but what about time complexity. I’m gonna prove that it is O(n).

Let Sh(n, k) be the problem of shifting k positions in an array of size n and Sw(k) be the problem of swapping k elements in an array. For the sake of simplicity I left out the start and end indices.

It is obvious that O(Sw(k)) = O(k).

It is also obvious that O(Sh(1, k)) = O(1), with k < 1. Also O(Sh(x, 0)) = O(1).

Now let's assume that O(Sh(n, k)) = O(n), whatever n, with k < n. I'll try to prove that O(Sh(n + 1, k')) = O(n), with k' < n + 1.

Analyzing the algorithm we have

O (Sh(n + 1, k’)) = max(
O(Sw(n + 1 – k’)) + O(Sh(k’, 2k’ – n – 1)), if 2k’ > n + 1
O(Sw(k’)) + O(Sh(n + 1 – k’, k’)), if 2k’ < n + 1
O(Sw(k')), if 2k' == n + 1

  • First case 2k’ > n + 1

    O(Sh(n + 1, k’)) = O(Sw(n + 1 – k’)) + O(Sh(k’, 2k’ – n – 1)) = O(n) + O(Sh(k’, 2k’ – n – 1))

    Because (k’ < n +1) ⇒ (k' ≤ n) and (2k' ≤ 2n) ⇒ (2k' – n – 1 ≤ n – 1) ⇒ (2k' – n – 1 < n), then O (Sh(n + 1, k')) = O(n) + O(n) = O(n).

  • Second case 2k’ < n + 1

    O(Sh(n + 1, k')) = O(Sw(k')) + O(Sh(n + 1 – k', k')) = O(k') + O(Sh(n + 1 – k', k')) = O(n) + O(Sh(n + 1 – k', k'))

    Because (0 < k') ⇒ (n + 1 – k' < n + 1) ⇒ (n + 1 – k' ≤ n) and (k' < (n + 1)/2) ⇒ (k' ≤ n/2) ⇒ (k' < n), then O(Sh(n + 1, k')) = O(n) + O(n) = O(n).

  • Third case 2k’ == n + 1

    O(Sh(n + 1, k’)) = O(Sw(k’)) = O(n)

So O(Sh(n + 1, k’)) = O(n). As a consequence O(Sh(n, k)) = O(n), whatever n, with k < n.

The code will become part of commons-lang, ArrayUtils class, as of version 3.5.

Categories: Web