# Computer Algorithms: Minimum and Maximum

## Introduction

To find the minimum value into an array of items itsn’t difficult. There are not many options to do that. The most natural approach is to take the first item and to compare its value against the values of all other elements. Once we find a smaller element we continue the comparisons with its value. Finally we find the minimum.

First thing to note is that we pass through the array with n steps and we need exactly n-1 comparisons. It’s clear that this is the optimal solution, because we must check all the elements. For sure we can’t be sure that we’ve found the minimum (maximum) value without checking every single value.
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# Computer Algorithms: Insertion Sort

## Overview

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.

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# Computer Algorithms: Data Compression with Relative Encoding

## Overview

Relative encoding is another data compression algorithm. While run-length encoding, bitmap encoding and diagram and pattern substitution were trying to reduce repeating data, with relative encoding the goal is a bit different. Indeed run-length encoding was searching for long runs of repeating elements, while pattern substitution and bitmap encoding were trying to “map” where the repetitions happen to occur.

The only problem with these algorithms is that not always the input stream of data is constructed out of repeating elements. It is clear that if the input stream contains many repeating elements there must be some way of reducing them. However that doesn’t mean that we cannot compress data if there are no repetitions. It all depends on the data. Let’s say we have the following stream to compress.

`1, 2, 3, 4, 5, 6, 7`

We can hardly imagine how this stream of data can be compressed. The same problem may occur when trying to compress the alphabet. Indeed the alphabet letters the very base of the words so it is the minimal part for word construction and it’s hard to compress them.

Fortunately this isn’t true always. An algorithm that tryies to deal with non repeating data is relative encoding. Let’s see the following input stream – years from a given decade (the 90’s).

`1991,1991,1999,1998,1991,1993,1992,1992`

Here we have 39 characters and we can reduce them. A natural approach is to remove the leading “19” as we humans often do.

`91,91,99,98,91,93,92,92`

Now we have a shorter string, but we can go even further with keeping only the first year. All other years will as relative to this year.

`91,0,8,7,0,2,1,1`

Now the volume of transferred data is reduced a lot (from 39 to 16 – more than 50%). However there are some questions we need to answer first, because the stream wont be always formatted in such pretty way. How about the next character stream?

`91,94,95,95,98,100,101,102,105,110`

We see that the value 100 is somehow in the middle of the interval and it is handy to use it as a base value for the relative encoding. Thus the stream above will become:

`-9,-6,-5,-5,-2,100,1,2,5,10`

The problem is that we can’t decide which value will be the base value so easily. What if the data was dispersed in a different way.

`96,97,98,99,100,101,102,103,999,1000,1001,1002`

Now the value of “100” isn’t useful, because compressing the stream will get something like this:

`-4,-3,-2,-1,100,1,2,3,899,900,901,902`

To group the relative values around “some” base values will be far more handy.

`(-4,-3,-2,-1,100,1,2,3)(-1,1000,1,2)`

However to decide which value will be the base value isn’t that easy. Also the encoding format is not so trivial. In the other hand this type of encoding can be useful in som specific cases as we can see bellow.
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