M Pivot Sort – Faster Than Quick Sort! Presenter: James Edmondson Mentor: Dr. Judy Hankins Study funded by MTSU Ronald E McNair Program
M Pivot Sort – Faster than Quick Sort!
Table of Contents
Introduction ……………………………………………. Introduction to M Pivot Sort………………………….. Testing: System Specifications……………………… Testing: Ascending Lists……………………………… Testing: Descending Lists……………………………. Testing: Random Integer Lists………………………. Testing: Random String Lists………………………… Testing: Random Class Lists………………………… Conclusion……………………………………………... References…………………………………………….. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Introduction
Sorting Algorithms – Sorting algorithms constitute the core of most algorithm textbooks. – Most computer programmers still use sorting algorithms that were developed in the late 1950s and early 1960s. – O(n log n) is the theoretical lower bound for comparison based sorting algorithms. – New O(n log n) algorithms have been developed as late as Multikey Quick Sort in 1997 but are usually specialized sorts for certain types of data. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Introduction
Sorting Algorithms (continued) – To be accepted into common practice, new sorting algorithms have to compete with Quick Sort in speed and usability. – Two other algorithms, Merge Sort and Heap Sort, are also used widely and should be included in empirical comparisons.
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M Pivot Sort – Faster than Quick Sort!
Introduction: M Pivot Sort
M Pivot Sort – Based on partitioning principle in Quick Sort. – Applies the Strong Law of Large Numbers to produce better sampling than Quick Sort. – Handles duplicates in linear time (Quick Sort goes into polynomial time without three way partitioning.) – Performs on average in O(n log n) time. – Has a better worst case scenario (roughly half) than Quick Sort.
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M Pivot Sort – Faster than Quick Sort!
Introduction: M Pivot Sort
M Pivot Sort (continued) – Performs more targeted moves than Quick Sort (results in less moves) – Performs slightly more data comparisons than Quick Sort but far less than Heap Sort. – Performs less recursion than Quick Sort or Merge Sort. – May perform runtime changes to correct polynomial time [O(n2)] performance by changing the number of pivots (3-7). 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: System Specifications
System Specifications for Testing – – – – –
Quad Xeon, 2 GBs of RAM, Intel Architecture. Windows 2003 Server Standard Edition. Visual C++ command prompt based. Random Integer Generator designed by Andy Thomas. Random string generator produces 4-19 character random ANSI C++ strings. – Random class generator produces 4 7-17 character random ANSI C++ strings, one of which is the sorting key. – Ascending and descending lists are integer based. – Ten different sets of data ran for each algorithm for each type of test. Mean average taken for stats generated here. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Ascending Lists
* Tests on list of 100 million ascending integers
Asce nding List - Total Ope rations in N Units
Quick Sort (CM)
65.08
Quick Sort (MOT)
64.09
Pivot Sort
40.73
Merge Sort
66.83
Heap Sort 0.00
130.77 20.00
40.00
60.00
80.00
100.00
120.00
140.00
Units expressed in 100 million operations, ie Merge Sort performed 66.83 100 million operations, or 6,683,000,000 total operations. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Ascending Lists
* Tests on list of 100 million ascending integers
Ascending List - Sorting Time in Seconds Quick Sort (CM)
18
Quick Sort (MOT)
15
Pivot Sort
16
Merge Sort
34
Heap Sort
33 0
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M Pivot Sort – Faster than Quick Sort!
Testing: Descending Lists
* Tests on list of 100 million descending integers
Descending List - Total Operations in N Units Quick Sort (CM)
68.10
Quick Sort (MOT)
65.45
Pivot Sort
56.06
Merge Sort
66.46
Heap Sort 0.00
124.25 20.00
40.00
60.00
80.00
100.00
120.00
140.00
Units expressed in 100 million operations, ie Merge Sort performed 66.83 100 million operations, or 6,683,000,000 total operations. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Descending Lists
* Tests on list of 100 million descending integers
Descending List - Sorting Time in Seconds 20
Quick Sort (CM) 17
Quick Sort (MOT) Pivot Sort
18
Merge Sort
32
Heap Sort
35 0
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Integers
* Tests on list of 100 million random integers
Random Integer List - Total Operations in N Units Quick Sort (CM)
88.18
Quick Sort (MOT)
75.56
Pivot Sort
66.84 78.65
Merge Sort
127.28
Heap Sort 0.00
20.00
40.00
60.00
80.00
100.00 120.00 140.00
Units expressed in 100 million operations, ie Merge Sort performed 78.65 100 million operations, or 7,650,000,000 total operations. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Integers
* Tests on list of 100 million random integers
Random Integer List - Sorting Time in Seconds Quick Sort (CM)
33
Quick Sort (MOT)
27
Pivot Sort
27
Merge Sort
49 156
Heap Sort 0
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100
150
200
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Integers
* Test on list of 100 million random integers (200,000 possible values – duplicate records present) Time in Seconds - 100 Million Random Integers (0-200,000 possible values) 160
148
140 120
101
100 80 60
48
40 19
20 0 Heap Sort
Merge Sort
Pivot Sort
Quick Sort
Figure 8
Shows Quick Sort’s performance on duplicates vs. Pivot Sort. Source is M Pivot Sort – Replacing Quick Sort. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Strings
* Tests on list of 10 million random strings
Random String List - Total Operations in N Units Quick Sort (CM)
76.72
Quick Sort (MOT)
65.83
Pivot Sort
58.70
Merge Sort
68.65 110.61
Heap Sort 0.00
20.00
40.00
60.00
80.00
100.00
120.00
Units expressed in 10 million operations, ie Merge Sort performed 68.65 10 million operations, or 686,500,000 total operations. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Strings
* Tests on list of 10 million random strings
Random String List - Sorting Time in Seconds Quick Sort (CM)
61
Quick Sort (MOT)
54
Pivot Sort
43
Merge Sort
80 116
Heap Sort 0
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60
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100
120
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Classes
* Tests on list of one million random classes
Random Class List - Total Operations in N Units Quick Sort (CM)
65.60
Quick Sort (MOT)
56.39
Pivot Sort
50.15
Merge Sort
58.58 93.94
Heap Sort 0.00
20.00
40.00
60.00
80.00
100.00
Units expressed in 1 million operations, ie Merge Sort performed 58.58 one million operations, or 58,580,000 total operations. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
Testing: Random Classes
* Tests on list of one million random classes
Random Class List - Sorting Time in Seconds Quick Sort (CM)
18
Quick Sort (MOT)
17
Pivot Sort
12
Merge Sort
16 28
Heap Sort 0
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M Pivot Sort – Faster than Quick Sort!
Conclusion
Conclusion – M Pivot Sort meets or beats the time taken by Quick Sort for sorts on contiguous lists (arrays). – M Pivot Sort performs favorably on sorting presorted and descending lists (common database operations.) – M Pivot Sort performs less moves and less total operations than Quick Sort, Heap Sort, and Merge Sort. – M Pivot Sort is an ideal candidate for general purpose sorting libraries for varied or singular data types. 11/27/2011
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M Pivot Sort – Faster than Quick Sort!
References
References – [1] Bentley, Jon L. “The trouble with qsort.” UNIX Review. 10.2 (1992): 85-90. – [2] Bentley, Jon L. “Software Exploratorium: history of a Heapsort.” UNIX Review. 10.8 (1992): 71-77. – [3] Cormen, Thomas, et al. Introduction to Algorithms. 2nd ed. Massachusetts: MIT Press, 2001. – [4]Edmondson, James. “M Pivot Sort – Replacing Quick Sort.” Proceedings of the 2005 International Conference on Algorithmic Mathematics and Computer Science (2005): 47-53. – [5] Kruse, Robert, et al. Data Structures & Program Design in C –. 2nd ed. New Jersey: Prentice Hall, 1997. 335. – [6] McClave, James T. and Terry Sincich. A First Course in Statistics – Annotated Instructor’s Edition. 8th ed. New Jersey: Prentice Hall, 2003. 224, 277-308. – [7] Sedgewick, Robert and Jon Bentley. “Quicksort is Optimal.” Knuthfest, Stanford University, Stanford. January 2002. – [8] Sedgewick, Robert. Algorithms in Java. 3rd ed. Boston: Addison-Wesley, 2003. – [9] Thomas, Andy. “Randomal64 Pseudo Random Number Generator.” Generation5.org. 20 April 2001. 6 Feb. 2005 .
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