quicksort (normally) faster than mergesort and heapsort runtime heapsort is another O(n log n)

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1 , etc 3/25

2 (normally) faster than mergesort and heapsort heapsort is another O(n log n) runtime like shell sort, depends somewhat on part of the algorithm n log n on most data

3 the idea: pick some element p (the pivot) split the array into two halves: < p > p move p into place sort the left and right sub-parts

4 partitioning (splitting array into small and big numbers) iterate: find leftmost element greater than pivot find rightmost element less than pivot swap them stop when the indices cross

5 red = pivot green = left partition blue = right partition Step 1: pick pivot

6 red = pivot green = left partition blue = right partition example: pick last element Step 2: find leftmost element greater than (or equal to) pivot

7 red = pivot green = left partition blue = right partition Step 2: find leftmost element greater than (or equal to) pivot

8 red = pivot green = left partition blue = right partition Step 2: find leftmost element greater than (or equal to) pivot

9 red = pivot green = left partition blue = right partition Step 2: find leftmost element greater than (or equal to) pivot found it!

10 red = pivot green = left partition blue = right partition Step 3: find rightmost element less than (or equal to) pivot

11 red = pivot green = left partition blue = right partition Step 3: find rightmost element less than (or equal to) pivot

12 red = pivot green = left partition blue = right partition Step 3: find rightmost element less than (or equal to) pivot found it!

13 red = pivot green = left partition blue = right partition Step 4: swap the elements we found Keep iterating with steps 2/3/4

14 red = pivot green = left partition blue = right partition partitioning stops when left/ right index variable cross or equal each other

15 red = pivot green = left partition blue = right partition swap pivot with end of left partition or start of right partition

16 after partitioning, the pivot is in the place it should be (for the sorted array) this property is important

17 recursively sort the left and right partitions why does this work? our partitioning creates two piles each of those piles becomes two finergrained piles until it s completely sorted!

18 if we split into two evenly-sized partitions: complexity? O(n log n) what s the most unbalanced the partitions can be? does this affect the complexity? recurrence relation?

19 speed depends on how we pick the pivot the actual data

20 picking the pivot first (or last) element (last shown in example) what happens on pre-sorted or reversesorted data? we re picking the min or max one partition is empty every time O(n 2 )

21 picking the pivot middle element swap it to the end after picking it swap back later O(1) extra work, but makes the code easier how well will this work on presorted or reversed?

22 is quicksort O(n2 ) when picking middle element? yes, though it s rare how about a random element? still O(n 2 ), but can provide a probability for how unlikely it is

23 what s the ideal pivot? the median can we find the median in O(n)? why do we need to find it in O(n) or better? yes, but it s complicated and slow normal methods involve sorting

24 can we do better than middle element? median of some fixed number of elements median of three (first, mid, last) how to compute median of three? manual insertion sort or tree of if/then

25 median of three only O(1) extra work (because 3 is constant) if N is large, it s negligible, but provides better partitioning if N is small, the extra work may not be worth it what happens when we do mo3 for N=3?

26 median of three pros closer to true median than others don t need to check first/last for partitioning first serves as a sentinel (can remove loop condition) cons need special case for small arrays

27 picking pivots can we do better? depends... whatever we try must be O(n) though or else no chance of O(n log n) for the sort

28 picking pivots Tukey s ninther do three different median-of-threes take median of the medians setting it up to get a sentinel value in the first element can be challenging

29 picking pivots could insertion sort sqrt(n) data or shell sort, or merge sort, etc etc common idea: running O(n 2 ) algorithm on sqrt(n) data is O(n) overall

30 iterative version manually maintain a stack (can use vector<int>) push the left/right on, pop them off like a stack of work orders this is akin to implementing the function calls in assembly

31 small arrays can use insertion sort for small arrays just like with mergesort main difference: quicksort processes, then recurses can do a single insertion sort call after the entire tree of quicksort recursion ends usually faster than millions of insertion sort calls

32 small arrays can use insertion sort for small arrays we may want to modify the first-pass findmin if we re doing a single insertion sort call after also, beware that quicksort can be broken and it ll still sort but it ll be O(n 2 )

33 random data N = 10m 20m Shell (SW) 6-7s 13-14s Merge (alt, small cutoff) Quicksort (basic) Quicksort (mo3, small cut) 4s 8-9s 4s 8-9s 3s 7s

34 pre-sorted data N = 10m 20m Shell (SW) 1s 3s Merge (alt, small cutoff) Quicksort (basic) Quicksort (mo3, small cut) 1s 2-3s 1-2s 2s 1s 2s

35 notes quicksort optimizations interact with -O3 a little can have a different pivot selection for large vs medium vs small arrays med-of-3 is relatively less costly for larger quicksort performs especially well for reversed arrays - why?

36 future directions midterm Q&A sorting specialty sorts: bucket sort, radix sort searching - linear, binary, interpolated trees and graphs and hashtables!

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