Tag Archives: Shell sort

Computer Algorithms: Radix Sort

March 19, 2012algorithmsBest worst and average case, Bubble sort, Bucket sort, faster algorithm, faster linear complexity algorithms, Heapsort, input algorithms, Insertion sort, Introduction Algorithms, linear complexity algorithms, Merge sort, PHP, purpose sorting algorithms, Quicksort, Radix sort, Shell sort, Sort, Sorting algorithmsStoimen

Introduction

Algorithms always depend on the input. We saw that general purpose sorting algorithms as insertion sort, bubble sort and quicksort can be very efficient in some cases and inefficient in other. Indeed insertion and bubble sort are considered slow, with best-case complexity of O(n²), but they are quite effective when the input is fairly sorted. Thus when you have a sorted array and you add some “new” values to the array you can sort it quite effectively with insertion sort. On the other hand quicksort is considered one of the best general purpose sorting algorithms, but while it’s a great algorithm when the data is randomized it’s practically as slow as bubble sort when the input is almost or fully sorted.

Now we see that depending on the input algorithms may be effective or not. For almost sorted input insertion sort may be preferred instead of quicksort, which in general is a faster algorithm.

Just because the input is so important for an algorithm efficiency we may ask are there any sorting algorithms that are faster than O(n.log(n)), which is the average-case complexity for merge sort and quicksort. And the answer is yes there are faster, linear complexity algorithms, that can sort data faster than quicksort, merge sort and heapsort. But there are some constraints!

Everything sounds great but the thing is that we can’t sort any particular data with linear complexity, so the question is what rules the input must follow in order to be sorted in linear time.

Such an algorithm that is capable of sorting data in linear O(n) time is radix sort and the domain of the input is restricted – it must consist only of integers.

Overview

Let’s say we have an array of integers which is not sorted. Just because it consists only of integers and because array keys are integers in programming languages we can implement radix sort.

First for each value of the input array we put the value of “1” on the key-th place of the temporary array as explained on the following diagram.

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Computer Algorithms: Merge Sort

March 5, 2012algorithmsAdaptive sort, Best worst and average case, Bubble sort, comparison model sorting algorithm, Divide and conquer algorithm, Insertion sort, interative solution, interative solutions, Merge sort, PHP, Quicksort, recursive solution, Shell sort, Sorting algorithms, Strand sort, three algorithms, USDStoimen

Introduction

Basically sorting algorithms can be divided into two main groups. Such based on comparisons and such that are not. I already posted about some of the algorithms of the first group. Insertion sort, bubble sort and Shell sort are based on the comparison model. The problem with these three algorithms is that their complexity is O(n²) so they are very slow.

So is it possible to sort a list of items by comparing their items faster than O(n²)? The answer is yes and here’s how we can do it.

The nature of those three algorithms mentioned above is that we almost compared each two items from initial list.

Insertion sort and bubble sort make too many comparisons, exactly what merge sort tries to overcome!

This, of course, is not the best approach and we don’t need to do that. Instead we can try to divide the list into smaller lists and then sort them. After sorting the smaller lists, which is supposed to be easier than sorting the entire initial list, we can try to merge the result into one sorted list. This technique is typically known as “divide and conquer”.

Normally if a problem is too difficult to solve, we can try to break it apart into smaller sub-sets of this problem and try to solve them. Then somehow we can merge the results of the solved problems.

If it's too difficult to sort a large list of items, we can break it apart into smaller sub-lists and try to sort them!

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Computer Algorithms: Shell Sort

February 27, 2012algorithmsAdaptive sort, Bubble sort, Comb sort, Combinatorics, Donald Shell, Insertion sort, Knuth, Mathematics, Merge sort, Order theory, PHP, Pratt, Quicksort, Shell sort, Sorting algorithmsStoimen

Overview

Insertion sort is a great algorithm, because it’s very intuitive and it is easy to implement, but the problem is that it makes many exchanges for each “light” element in order to put it on the right place. Thus “light” elements at the end of the list may slow down the performance of insertion sort a lot. That is why in 1959 Donald Shell proposed an algorithm that tries to overcome this problem by comparing items of the list that lie far apart.

Insertion Sort vs. Shell Sort — Insertion sort compares every single item with all the rest elements of the list in order to find its place, while Shell sort compares items that lie far apart. This makes light elements to move faster to the front of the list.

In the other hand it is obvious that by comparing items that lie apart the list can’t be sorted in one pass as insertion sort. That is why on each pass we should use a fixed gap between the items, then decrease the value on every consecutive iteration. Continue reading Computer Algorithms: Shell Sort →

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Tag Archives: Shell sort

Computer Algorithms: Radix Sort

Introduction

Overview

Computer Algorithms: Merge Sort

Introduction

Computer Algorithms: Shell Sort

Overview