Tag Archives: binary search

Computer Algorithms: Balancing a Binary Search Tree


The binary search tree is a very useful data structure, where searching can be significantly faster than searching into a linked list. However in some cases searching into a binary tree can be as slow as searching into a linked list and this mainly depends on the input sequence. Indeed in case the input is sorted the binary tree will seem much like a linked list and the search will be slow.

Inserting into a binary search tree
A binary search tree may seem much like a linked lists if the input is nearly sorted!

To overcome this we must change a bit the data structure in order to stay well balanced. It’s intuitively clear that the searching process will be better if the tree is well branched. This is when finding an item will become faster with minimal effort.

Balanced tree
Searching into a balanced tree is significantly faster than searching into a non-balanced tree!

Since we know how to construct a binary search tree the only thing left is to keep it balanced. Obviously we will need to re-balance the tree on each insert and delete, which will make this data structure more difficult to maintain compared to non-balanced search trees, but searching into it will be significantly faster. Continue reading Computer Algorithms: Balancing a Binary Search Tree

Computer Algorithms: Binary Search Tree


Constructing a linked list is a fairly simple task. Linked lists are a linear structure and the items are located one after another, each pointing to its predecessor and its successor. Almost every operation is easy to code in few lines and doesn’t require advanced skills. Operations like insert, delete, etc. over linked lists are performed in a linear time. Of course on small data sets this works fine, but as the data grows these operations, especially the search operation becomes too slow.

Indeed searching in a linked list has a linear complexity and in the worst case we must go through the entire list in order to find the desired element. The worst case is when the item doesn’t belong to the list and we must check every single item of the list even the last one without success. This approach seems much like the sequential search over arrays. Of course this is bad when we talk about large data sets.

Search over Linked Lists and Arrays
Sequential search over arrays seems much like searching in linked lists and it is a basically ineffective opration!
Continue reading Computer Algorithms: Binary Search Tree

Computer Algorithms: Insertion Sort


Sorted data can dramatically change the speed of our program, therefore sorting algorithms are something quite special in computer science. For instance searching in a sorted list is faster than searching in an unordered list.

There are two main approaches in sorting – by comparing the elements and without comparing them. A typical algorithm from the first group is insertion sort. This algorithm is very simple and very intuitive to implement, but unfortunately it is not so effective compared to other sorting algorithms as quicksort and merge sort. Indeed insertion sort is useful for small sets of data with no more than about 20 items.

Insertion sort it is very intuitive method of sorting items and we often use it when we play card games. In this case the player often gets an unordered set of playing cards and intuitively starts to sort it. First by taking a card, making some comparisons and then putting the card on the right position.

So let’s say we have an array of data. In the first step the array is unordered, but we can say that it consists of two sub-sets: sorted and unordered, where on the first step the only item in the sorted sub-set is its first item. If the length of the array is n the algorithm is considered completed in n-1 steps. On each step our sorted subset is growing with one item. The thing is that we take the first item from the unordered sub-set and with some comparisons we put it into its place in the sorted sub-set, like on the diagram bellow.

Main principle of insertion sort
Main principle of insertion sort.

Continue reading Computer Algorithms: Insertion Sort

Computer Algorithms: Data Compression with Run-length Encoding


No matter how fast today’s computers and networks are, the users will constantly need faster and faster services. To reduce the volume of the transferred data we usually use some sort of compression. That is why this computer sciences area will be always interesting to research and develop.

There are many data compression algorithms, some of them lossless, others lossy, but their main goal aways will be to spare storage space and traffic. These algorithms are very useful when talking about data transfer between two distant places. Perhaps the best example is the transfer between a web server and a browser.

In the last few years a lot of research has been done on compressing files, executed on the client side. Such files are javascript, css, htmls and images. In fact servers and clients already have some techniques to compress data, like using GZIP for instance, that can dramatically decrease the transfer. In the other hand there are lots of tools and tricks in order to decrease the size of the data.

Actually when a file is executed by the client’s virtual machine, it doesn’t matter how “beautifully” it is formatted from a programmer’s point of view. Thus the spaces, tabs and the new lines don’t bring any significant information for the environment. That is why such compressing tools like YUI Compressor, Google Closure Compiler, etc. remove those symbols. Well, they can achieve even more in order to improve the compression rate. In this post I won’t cover this, but this shows how important data compression algorithms are.

It would be great if we could just compress data with some tool. Unfortunately this is not the case and usually the compression rate depends on the data itself. It is obvious that the choice of data compression algorithm depends mainly on the data and first of all we must explore the data.

Here I’ll cover one very simple lossless data compression algorithm called “run-length encoding” that can be very useful in some cases.

Run-length Encoding


This algorithm consists of replacing large sequences of repeating data with only one item of this data followed by a counter showing how many times this item is repeated. To become clearer let’s see a string example.


This string’s length is 24 and as we can see there are lots of repetitions. Using the run-length algorithm, we replace any run with shorter string followed by a counter.


The length of this string is 17, which is approximately 70% of the initial length. Continue reading Computer Algorithms: Data Compression with Run-length Encoding

Computer Algorithms: Interpolation Search


I wrote about binary search in my previous post, which is indeed one very fast searching algorithm, but in some cases we can achieve even faster results. Such an algorithm is the “interpolation search” – perhaps the most interesting of all searching algorithms. However we shouldn’t forget that the data must follow some limitations. In first place the array must be sorted. Also we must know the bounds of the interval.

Why is that? Well, this algorithm tries to follow the way we search a name in a phone book, or a word in the dictionary. We, humans, know in advance that in case the name we’re searching starts with a “B”, like “Bond” for instance, we should start searching near the beginning of the phone book. Thus if we’re searching the word “algorithm” in the dictionary, you know that it should be placed somewhere at the beginning. This is because we know the order of the letters, we know the interval (a-z), and somehow we intuitively know that the words are dispersed equally. These facts are enough to realize that the binary search can be a bad choice. Indeed the binary search algorithm divides the list in two equal sub-lists, which is useless if we know in advance that the searched item is somewhere in the beginning or the end of the list. Yes, we can use also jump search if the item is at the beginning, but not if it is at the end, in that case this algorithm is not so effective.

So the interpolation search is based on some simple facts. The binary search divides the interval on two equal sub-lists, as shown on the image bellow.

Binary search basic approach
The binary search algorithm divides the list in two equal sub-lists!

What will happen if we don’t use the constant ½, but another more accurate constant “C”, that can lead us closer to the searched item.

Interpolation search
The interpolation search algorithm tries to improve the binary search!

Continue reading Computer Algorithms: Interpolation Search