FASTER SORTING using RECURSION : QUICKSORT 2014-T2 Lecture 16 School of Engineering and Computer...
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Transcript of FASTER SORTING using RECURSION : QUICKSORT 2014-T2 Lecture 16 School of Engineering and Computer...
FASTER SORTING using RECURSION :
QUICKSORT
2014-T2 Lecture 16
School of Engineering and Computer Science, Victoria University of Wellington
CO
MP 1
03
Marcus Frean
2
RECAP-TODAYRECAP MergeSort cost O(n log n)
TODAY QuickSort
ANNOUNCEMENTS Test on Thursday – 4pm
Rooms FROM TO MCLT103 Abubakr Geuze HMLT002 Gibb Jones HMLT205 Julian Pappafloratos KKLT303 Patel Zhu
sorting algorithms put to music: (quicksort is at 38 sec) http://www.youtube.com/watch?v=kPRA0W1kECg
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MergeSort Cost (analysis order)
LEVEL 1 (two mergesorts)
LEVEL 1 (merge)
LEVEL 2 (two mergesorts)
LEVEL 2 (merge)
LEVEL 3 (two mergesorts)
LEVEL 3 (merge)
LEVEL 4 (two mergesorts)
LEVEL 4 (merge)
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Divide and Conquer SortsTo Sort:
Split Sort each part
(recursive) Combine
Where does thework happen?
MergeSort: split is trivial combine does all the work
QuickSort: split does all the work combine is trivial
Array
Sorted Array
Split
Combine
SubArray SubArray
SortedSubArray SortedSubArray
Sort Sort
Split
Combine
SubArray SubArray
Sort Sort
SortedSubArraySortedSubArray
Split
Combine
SubArray SubArray
Sort Sort
SortedSubArraySortedSubArray
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QuickSort uses Divide and Conquer, but does its work in the
“split” step
split the array into parts, by choosing a “pivot” item, and making sure that:
all items < pivot are in the left part all items > pivot are in the right part
Then (recursively) sort each part
Here's how we start it off:
public static <E> void quickSort( E[] data, int size, Comparator<E> comp) { quickSort (data, 0, size, comp);
}
“wrapper” version starts it off
recursive version carries it on (and on...)
note: it won’t usually be an
equal split
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QuickSort – the recursion
public static <E> void quickSort(E[ ] data, int low, int high, Comparator<E> comp){
if (high-low < 2) // only one item to sort.
return;if (high - low < 4) // only two or three items to sort.
sort3(data, low, high, comp);else { // split into two parts, mid = index of boundary
int mid = partition(data, low, high, comp);
quickSort(data, low, mid, comp);
quickSort(data, mid, high, comp);}
}
6 7 8 9 10 110 1 2 3 4 5
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QuickSort: simplest version1. Choose a pivot:
6 7 8 9 10 110 1 2 3 4 5
pivot
6 7 8 9 10 110 1 2 3 4 5
2. Use pivot to partition the array:
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not yet sorted not yet sorted
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QuickSort: in-place version1. Choose a pivot:
6 7 8 9 10 110 1 2 3 4 5
pivot
6 7 8 9 10 110 1 2 3 4 5
2. Use pivot to partition the array:
low highleft(gets returned)
not yet sorted not yet sorted
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QuickSort: coding in-place partition/** Partition into small items (low..mid-1) and large
items (mid..high-1)private static <E> int partition(E[] data, int low, int high,
Comparator<E> comp){
E pivot =int left = low-1;int right = high;
while( left <= right ){do { left++; // just skip over items on the left < pivot} while (left<high &&comp.compare(data[left], pivot)< 0);do { right--; // just skip over items on the right > pivot} while (right>=low && comp.compare(data[right], pivot)> 0);if (left < right) swap(data, left, right);
}return left;
}
6 7 8 9 10 110 1 2 3 4 5
data[low]; // simple but poor choice!median(data[low], data[high-1], data[(low+high)/2], comp);
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QuickSortdata array : [p a1 r f e q2 w q1 t z2 x c v b z1 a2 ] indexes : [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ]
do 0..16 : [p a1 r f e q2 w q1 t z2 x c v b z1 a2 ]part@p ->6 : [a2 a1 b f e c w q1 t z2 x q2 v r z1 p ]do 0..6 : [a2 a1 b f e c ]part@c ->4 : [a2 a1 b c e f ]do 0..4 : [a2 a1 b c ]part@b ->3 : [a2 a1 b c ]do 0..3 : [a2 a1 b ]done 0..3 : [a2 a1 b ]do 3..4 : [ c ]done 0..4 : [a2 a1 b c ]do 4..6 : [ e f ]done 4..6 : [ e f ]done 0..6 : [a2 a1 b c e f ]do 6..16 : [ w q1 t z2 x q2 v r z1 p ]part@q2->8 : [ p q2 t z2 x q1 v r z1 w ]do 6..8 : [ p q2 ]done 6..8 : [ p q2 ]
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Quick Sortdo 6..16 : [ w q1 t z2 x q2 v r z1 p ]part@q2->8 : [ p q2 t z2 x q1 v r z1 w ]do 6..8 : [ p q2 ]done 6..8 : [ p q2 ]do 8..16 : [ t z2 x q1 v r z1 w ]part@v ->12: [ t r v q1 x z2 z1 w ]do 8..12 : [ t r v q1 ]part@t ->10: [ q1 r v t ]do 8..10 : [ q1 r ]done 8..10 : [ q1 r ]do 10..12: [ v t ]done 10..12: [ t v ]done 8..12 : [ q1 r t v ]do 12..16: [ x z2 z1 w ]part@x ->13: [ w z2 z1 x ]do 12..13: [ w ]do 13..16: [ z2 z1 x ]done 13..16: [ x z2 z1 ]done 12..16: [ w x z2 z1 ]done 8..16 : [ q1 r t v w x z2 z1 ]done 6..16 : [ p q2 q1 r t v w x z2 z1 ]done 0..16 : [a2 a1 b c e f p q2 q1 r t v w x z2 z1 ]
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QuickSortpublic static <E> void quickSort (E[ ] data, int low, int high,
Comparator<E> comp) { if (high > low +2) { int mid = partition(data, low, high, comp); quickSort(data, low, mid, comp); quickSort(data, mid, high, comp); }
}
Cost of Quick Sort: three steps:
partition: has to compare (high-low) pairs first recursive call second recursive call
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QuickSort Cost: If Quicksort divides the array exactly in half, then:
C(n) = n + 2 C(n/2) n log(n) comparisons
= O(n log(n)) (best case)
If Quicksort divides the array into 1 and n-1: C(n) = n + (n-1) + (n-2) + (n-3) + … + 2 + 1 = n(n-1)/2 comparisons
= O(n2) (worst case)
Average case? very hard to analyse. still O(n log(n)), and very good.
Quicksort is “in place”, MergeSort is not
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Stable or Unstable? Faster if almost-sorted?
MergeSort: Stable: doesn’t jump any item over an unsorted region
⇒ two equal items preserve their order
Same cost on all input “natural merge” variant doesn’t sort already sorted
regions⇒ will be very fast: O(n) on almost sorted lists
QuickSort: Unstable: Partition “jumps” items to the other end
⇒ two equal items likely to reverse their order
Cost depends on choice of pivot. simplest choice is very slow: O(n2) even on almost sorted
lists better choice (median of three) ⇒ O(n log(n)) on almost
sorted lists
15
Some Big-O costs revisitedImplementing Collections:
ArrayList: O(n) to add/remove, except at end
Stack: O(1)
ArraySet: O(n) (cost of searching)
SortedArraySet O( log(n) ) to search (with binary
search)
O(n) to add/remove (cost of moving
up/down)
O( n2 ) to add n itemsO( n log(n) ) to initialise with n items. (with fast sorting)
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the test this Thursday 20% of final grade, and takes 45 mins
Rooms FROM TO MCLT103 Abubakr Geuze HMLT002 Gibb Jones HMLT205 Julian Pappafloratos KKLT303 Patel Zhu
Please be on time, and bring your ID card. “Exam conditions”:
sit in alternate rows, so someone can walk right along every 2nd row
sit in alternate seats put your ID card where it can be seen (but no need to leave bags at the front as happens in final
exams) What’s in?
everything up to now no complex coding questions on recursion or the fast sorts old tests are a guide for what to expect