sorting algorithms and analysis robert duncan. refresher on big-o o(2^n)exponential ...
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Sorting
Algorithms and Analysis
Robert Duncan
Refresher on Big-ORefresher on Big-O
O(2^N) ExponentialO(N^2) QuadraticO(N log N) Linear/LogO(N) LinearO(log N) LogO(1) Constant
Hierarchy of Big-O functions from slowest to fastest
Generic running times Generic running times
N O(log N) O(N) O(N log N) O(N^2) O(2^N)
1 0 1 0 1 2
8 3 8 24 64 256
128 7 128 896 16384 3.4x10^38
1024 10 1024 10240 1048576 1.8x10^308
O(N log N) vs. O(N^2)O(N log N) vs. O(N^2)
0
5000
10000
15000
20000
0 1000
O(N log N) O(N^2)
Two Common CategoriesTwo Common Categories
Sorting Algorithms of O(N^2)
Bubble SortSelection SortInsertion Sort
Sorting Algorithms of
O(N log N)Heap SortMerge SortQuick Sort
For small values of NFor 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) SortsO(N^2) Sorts
Easy to program
Simple to understand
Very slow, especially for large values of N
Almost never used in professional software
Bubble SortBubble 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 1
2
3
6
4
1
5
2
3
6
4
1
5
2
3
4
6
1
5
2
3
4
1
6
5
2
3
4
1
5
6
Bubble Sort – Pass 2
2
3
4
1
5
6
2
3
4
1
5
6
2
3
1
4
5
6
2
3
1
4
5
6
2
3
1
4
5
6
Bubble Sort – Pass 3
2
3
1
4
5
6
2
1
3
4
5
6
2
1
3
4
5
6
2
1
3
4
5
6
2
1
3
4
5
6
Bubble Sort – Pass 4
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
Selection SortSelection 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 1-3
6 8 10 2 4
6 8 4 2 10
6 2 4 8 10
Selection Sort – Pass 4-5
4 2 6 8 10
2 4 6 8 10
2 4 6 8 10
Insertion SortInsertion 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
22
22 35
22 35 48
Insertion Sort
13 22 35 48
8 13 22 35 48
8 13 22 27 35 48
O(N log N) SortsO(N log N) SortsFastEfficientComplicated, not easy to understandMost make extensive use of recursion and
complex data structures
Heap SortHeap 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 SortHeap 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
Forming the heap from an unsorted array
11 26 14 2 19 32 7
32
214711
1926
Populating the new arrayPopulating the new array
32
214711
1926
Reforming the heapReforming the heap
32
19
2711
1426
Reforming the heapReforming the heap
32
26
2711
1419
Repeat the processRepeat the process
32 26
2711
1419
Repeat the processRepeat the process
32 26
14
711
219
Repeat the processRepeat the process
32 26
19
711
214
Repeat the processRepeat the process
32 26 19
711
214
Merge SortMerge SortUses recursion. Slightly faster than heap, but
uses twice as much memory from the 2nd 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.
11 26 14 2 32 18 7 21
32 18 7 2111 26 14 2
11 26 14 2 32 18 7 21
11 26 2 14 18 32 7 21
18 32 7 2111 26 2 14
11 26 2 14 18 32 7 21
7 18 21 322 11 14 26
2 11 14 26 7 18 21 32
7 18 21 322 11 14 26
2 11 14 26 7 18 21 32
2 7 11 14 18 21 26 32
Quick SortQuick 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 SortQuick 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 SortQuick 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.
11 26 14 2 32 18 7 21 5
Pivot
11 26 14 2 32 18 7 21 5
11 2 7 5
Pivot
11 2 7 5
Pivot
11 26 14 2 32 18 7 21 5
11 26 14 2 32 18 7 21 5
11 2 7 5
2 5
Pivot
11 26 14 2 32 18 7 21 5
2 5 711 2 7 5
2 5 2 5
Pivot
11 26 14 2 32 18 7 21 5
2 5 7 1111 2 7 5
2 5 2 5
What’s the pointWhat’s the point
Binary Searching Binary Searching Binary searches achieve O(log N) efficiency on
sorted dataSimilar to High-Low gameEach execution eliminates half of the elements
to search forAlthough hashing offers a quicker search of
O(1), binary searches are simpler, and use much less memory.
Binary SearchingBinary Searching
2 7 11 14 16 23 26 29 32 37 43
14?
Binary SearchingBinary Searching
2 7 11 14 16 23 26 29 32 37 43
14?
Binary SearchingBinary Searching
2 7 11 14 16 23 26 29 32 37 43
14
Conclusion…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.
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