Lecture 4 Sort(2) Merge sort Quick sort ACKNOWLEDGEMENTS: Some contents in this lecture source from COS226 of Princeton University by Kevin Wayne and Bob Sedgewick

Roadmap Merge sort Quick sort

Merge sort Transylvanian-Saxon folk dance

Merge sort Divide array into two halves. Recursively sort each half. Merge two halves. 2 13 6 24 43 1 51 9 10 2 13 6 24 1 43 9 10 2 6 13 24 9 10 51 1 9 10 43 51 1 2 6 9 10 13 24 43 51

Merging 2, 13, 43, 45, 89 6, 24, 51, 90, 93 2, 6, 13, 24, 43, 45, 51, 89, 90, 93 Assume we need to merge two sorted arrays, with M, N elements respectively. Then How many comparisons in best case? How many comparisons in worst case? A. M+N B. max{M,N} C. min{M,N} D. M+N-1

MERGE(A,p,q,r) ..., 2, 13, 43, 45, 89, 6, 24, 51, 90, 93, ... A p q r

Complexity D (N) = 2 D (N/2) + N Compute by picture. D (N) D (N / 2) D (N / 2k) 2 (N/2) 2k (N/2k) N/2 (2) ... lg N N lg N = N 4 (N/4)

Complexity D (N) = 2 D (N/2) + N Compute by expansion. D (N) / N = 2 D (N/2) / N + 1 = D (N/2) / (N/2) + 1 = D (N/4) / (N/4) + 1 + 1 = D (N/8) / (N/8) + 1 + 1 + 1 . . . = D (N/N) / (N/N) + 1 + 1 + ... + 1 = lg N

Complexity Good algorithms are better than supercomputers. Laptop executes 108 compares/second. Supercomputer executes 1012 compares/second. insertion sort (N2) mergesort (N log N) computer thousand million billion home instant 2.8 hours 317 years 1 second 18 min super 1 week Good algorithms are better than supercomputers.

Discussion Mergesort is stable? in place? Recursions cost spaces.

Bottom-up merge sort 1 2 6 9 10 13 24 43 51 1 2 6 9 13 24 43 51 10 2 6 13 24 1 9 43 51 10 2 13 6 24 1 43 9 51 10 2 13 6 24 43 1 51 9 10

Bottom-up merge sort

Quiz Give the array that results immediately after the 7th call to merge() by using: a) top-down mergesort b) bottom-up merge sort M B X V Z Y H U N S I K A. B M V X Y Z H N U S I K B. B M V X Y Z H N U S K I C. B M V X Y Z H U N S I K D. B M V X Y Z H N U S K I

Quiz Mergesort, BottomupMergesort, which one is more efficient? which one is easier for understanding and debugging? which one you prefer? A. Mergesort B. BottomupMergesort

O(nlogn) sorting algorithms n=2k Best case Worst case Bottom-up-merge Merge Space Θ(n) Quicksort: worst O(n2), average O(nlogn), space O(logn) Heapsort: worst O(nlogn), space O(1)

Sorting

Where are we? Merge sort Quick sort

Quick sort Quicksort honored as one of top 10 algorithms of 20th century in science and engineering. Mergesort. Java sort for objects. Perl, C++ stable sort, Python stable sort, Firefox JavaScript, ... Quicksort. Java sort for primitive types. C qsort, Unix, Visual C++, Python, Matlab, Chrome JavaScript, ...

Quick sort Sir Charles Antony Richard Hoare 1980 Turing Award

Quick sort Shuffle the array. Partition so that, for some j entry a[j] is in place no larger entry to the left of j no smaller entry to the right of j Sort each piece recursively.

Partition#1

Partition#1 10 13 6 24 43 1 51 9 i j

Partition #2 - by Sedgewick

Complexity Best case: array always split into two equal-sized subarrays. similar to mergesort, O(NlogN) Worst case: array always split into a 0-sized and an N-1-sized subarrays similar to selection sort, O(N2) Average case:

Complexity Good algorithms are better than supercomputers. Laptop executes 108 compares/second. Supercomputer executes 1012 compares/second. insertion sort (N2) mergesort (N log N) quicksort (N log N) computer thousand million billion home instant 2.8 hours 317 years 1 second 18 min 0.6 sec 12 min super 1 week Good algorithms are better than supercomputers. Great algorithms are better than good algorithms.

Duplicate keys Dijkstra’s 3-way partition B A A B A B B B C C C A A A A A A A A A A A Dijkstra’s 3-way partition

Quiz Give the array that results after applying quicksort partitioning to the following array by using: a) partition #1 b) partition #2 c) Dijkstra’s 3-way partition H J B R H R H B C V S A. C B H H B H R J R V S B. H C B B H H R R J V S C. C B B H H H R R V S J D. none of above

Sort summary inplace? stable? worst average best remarks selection x N exchanges insertion N 2 / 4 N use for small N or partially ordered shell ? tight code, subquadratic quick 2 N ln N N lg N N log N probabilistic guarantee fastest in practice 3-way quick improves quicksort in presence of duplicate keys merge N log N guarantee, stable

Exercise Download hw4.pdf from our course homepage Due on Oct. 11 Watch the week#4 video of Algorithm(Part I) Coursera. Optional: Programming Assignment 4 4/14/2017 Xiaojuan Cai