Sorting Algorithms and Analysis Robert Duncan

Refresher on Big-O  O(2^N)Exponential  O(N^2)Quadratic  O(N log N)Linear/Log  O(N)Linear  O(log N)Log  O(1)Constant Hierarchy of Big-O functions from slowest to fastest

Generic running times NO(log N)O(N)O(N log N)O(N^2)O(2^N) x10^ x10^308

O(N log N) vs. O(N^2)

Two Common Categories Sorting Algorithms of O(N^2)  Bubble Sort  Selection Sort  Insertion Sort Sorting Algorithms of O(N log N)  Heap Sort  Merge Sort  Quick Sort

For small values of N  It is important to note that all algorithms appear to run equally as fast for small values of N.  For values of N from the thousands to the millions, The differences between O(N^2) and O(N log N) become dramatically apparent

O(N^2) Sorts  Easy to program  Simple to understand  Very slow, especially for large values of N  Almost never used in professional software

Bubble Sort  The most inefficient of the O(n^2) algorithms  Simplest sorting algorithm available  Works by comparing sequential items, and swapping them if the first one is larger than the second. It makes as many passes through an array as are needed to complete the sort

Bubble Sort – Pass

Bubble Sort – Pass

Bubble Sort – Pass

Bubble Sort – Pass

Selection Sort  More efficient than Bubble Sort, but not as efficient as Insertion Sort  Works by finding the largest element in the list and swapping it with the last element, effectively reducing the size of the list by 1.

Selection Sort – Pass

Selection Sort – Pass

Insertion Sort  One of the most efficient of the O(n^2) algorithms  Roughly twice as fast as bubble sort  Works by taking items from unsorted list and inserting them into the proper place.

Insertion Sort

Insertion Sort

O(N log N) Sorts  Fast  Efficient  Complicated, not easy to understand  Most make extensive use of recursion and complex data structures

Heap Sort  Slowest O(N log N) algorithm.  Although the slowest of the O(N log N) algorithms, it has less memory demands than Merge and Quick sort.

Heap Sort  Works by transferring items to a heap, which is basically a binary tree in which all parent nodes have greater values than their child nodes. The root of the tree, which is the largest item, is transferred to a new array and then the heap is reformed. The process is repeated until the sort is complete.

Forming the heap from an unsorted array

Populating the new array

Reforming the heap

Reforming the heap

Repeat the process

Repeat the process

Repeat the process

Repeat the process

Merge Sort  Uses recursion. Slightly faster than heap, but uses twice as much memory from the 2 nd array.  Sometimes called “divide and conquer” sort.  Works by recursively splitting an array into two equal halves, sorting the items, then re- merging them back into a new array.

Quick Sort  The most efficient O(N log N) algorithm available  Works by first randomly choosing a pivot point which is hopefully a median element in the array. All elements less than the pivot are transferred to a new array, and all elements greater are transferred to a second array.

Quick Sort  These new arrays are recursively quick sorted until they have been split down to a single element. The sorted elements smaller than the pivot are placed in a new array, then the pivot is placed after, and finally the elements greater than the pivot. These elements are now sorted.

Quick Sort  Note: Although it is the quickest sorting algorithm, a badly chosen pivot may cause Quick Sort to run at O(N^2).  Different versions of quick sort address this problem in different ways.  For example, one way is to randomly choose 3 elements, then use the median element as the pivot.

Pivot

Pivot

11275 Pivot

Pivot

Pivot

What’s the point

Binary Searching  Binary searches achieve O(log N) efficiency on sorted data  Similar to High-Low game  Each execution eliminates half of the elements to search for  Although hashing offers a quicker search of O(1), binary searches are simpler, and use much less memory.

Binary Searching ?

Binary Searching ?

Binary Searching

Conclusion…  O(N^2) algorithms are almost never used in professional software  Quick sort is generally considered to be the best overall sorting algorithm currently available.