SORTING 9/26/18. Prelim 1. Prelim 1. Why Sorting? InsertionSort. Some Sorting Algorithms. Tonight!!!! Two Sessions:

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1 Prelim 1 2 "Organizing is wat you do efore you do someting, so tat wen you do it, it is not all mixed up." ~ A. A. Milne SORTING Tonigt!!!! Two Sessions: You sould now y now wat room to tae te final. Jenna ed you. Bring your Cornell ID!!! We will grade tis evening, and if everyting wors out well, you will receive an in early morning from Gradescope telling you to loo at your grade. Lecture 11 CS2110 Fall 2018 Prelim 1 Wy Sorting? 3 4 Recitation 5. next wee: Enums and Java Collections classes. Noting to prepare for it! Sorting is useful Dataase indexing Operations researc Compression But get A3 done. Tere are lots of ways to sort Tere isn't one rigt answer You need to e ale to figure out te options and decide wic one is rigt for your application. Today, we'll learn several different algoritms (and ow to develop tem) Some Sorting Algoritms InsertionSort 5 Insertion sort Selection sort 6 pre: 0.lengt? post: 0.lengt Quic sort Merge sort 0 i.lengt? or: or: [0..i-1] is 0 i.lengt processed? [0..i-1] is processed A loop tat processes elements of an array in increasing order as tis invariant --- ust replace y processed. 1

2 Eac iteration, i= i+1; How to eep inv true? Wat to do in eac iteration? 7 0 i.lengt? 8 0 i.lengt? e.g. 0 i.lengt ? e.g. 0 i.lengt ? 0 i.lengt ? Loop ody (inv true efore and after) ? ? ? Pus [i] to its position in [0..i], ten increase i ? Tis will 0 tae time proportional i to te numer of.lengt swaps needed ? Insertion Sort Insertion Sort 9 // sort [], an array of int // [0..i-1] is for (int i= 0; i <.lengt; i= i+1) { // Pus [i] down to its // position in [0..i] Present algoritm lie tis Note Englis statement in ody. Astraction. Says wat to do, not ow. Tis is te est way to present it. We expect you to present it tis way wen ased. Later, can sow ow to implement tat wit an inner loop 10 // sort [], an array of int // [0..i-1] is for (int i= 0; i <.lengt; i= i+1) { // Pus [i] down to its // position in [0..i] int = i; wile ( > 0 && [] < [-1]) { Swap [] and [-1]; = 1; invariant P: [0..i] is except tat [] may e < [-1] i ? example start? stop? progress? maintain invariant? Insertion Sort // sort [], an array of int // [0..i-1] is for (int i= 0; i <.lengt; i= i+1) { // Pus [i] down to its // position in [0..i] Pusing [i] down can tae i swaps. Worst case taes n-1 = (n-1)*n/2 swaps. Let n =.lengt Worst-case: O(n 2 ) (reverse- input) Best-case: O(n) ( input) Expected case: O(n 2 ) Algoritm Ave time. Worst-case time Space Stale? Insertion Sort! " #.!(" # )!(1) Yes Merge Sort Quic Sort We ll tal aout staility later 2

3 SelectionSort SelectionSort 13 pre: 0.lengt? post: 0.lengt 0 i.lengt, <= [i..] >= [0..i-1] Additional term in invariant Keep invariant true wile maing progress? 0 i.lengt e.g.: Increasing i y 1 eeps inv true only if [i] is min of [i..] 14 //sort [], an array of int Anoter common way for // [0..i-1] AND people to sort cards // [0..i-1] <= [i..] for (int i= 0; i <.lengt; i= i+1) { int m= index of min of [i..]; Swap [i] and [m]; Runtime wit n =.lengt Worst-case O(n 2 ) Best-case O(n 2 ) Expected-case O(n 2 ) 0 i lengt, smaller values larger values Eac iteration, swap min value of tis section into [i] QuicSort Algoritm Ave time. Worst-case time Space Stale? Insertion sort!(# $ ).!(# $ )!(1) Yes Selection sort! # $.!(# $ )!(1) No Quic sort Merge sort Quicsort developed y Sir Tony Hoare (e was nigted y te Queen of England for is contriutions to education and CS). 84 years old. Developed Quicsort in But e could not explain it to is colleague, so e gave up on it. Later, e saw a draft of te new language Algol 58 (wic ecame Algol 60). It ad recursive procedures. First time in a procedural programming language. A!, e said. I now ow to write it etter now. 15 minutes later, is colleague also understood it. Distras, Hoares, Grieses, 1980s Partition algoritm of quicsort pre: +1 x? x is called te pivot Swap array values around until [..] loos lie tis: post: 3

4 Partition algoritm of quicsort Partition algoritm 19 pivot Not yet partition tese can e in any order tese can e in any order Not yet Te 20 could e in te oter partition 20 pre: post: +1 x? Comine pre and post to get an invariant t <= x x? >= x invariant needs at least 4 sections Partition algoritm QuicSort procedure 21 t <= x x? >= x = ; t= ; wile ( < t) { if ([+1] <= []) { Swap [+1] and []; = +1; else { Swap [+1] and [t]; t= t-1; Taes linear time: O(+1-) Initially, wit = and t =, tis diagram loos lie te start diagram Terminate wen = t, so te? segment is empty, so diagram loos lie result diagram 22 /** Sort [..]. */ pulic static void QS(int[], int, int ) { if ([..] as < 2 elements) return; Base case int = partition(,, ); // We now [.. 1] <= [] <= [+1..] // Sort [..-1] and [+1..] QS(,, -1); QS(, +1, ); Function does te partition algoritm and returns position of pivot Worst case quicsort: pivot always smallest value Best case quicsort: pivot always middle value 23 x0 x0 x1 x0 x1 x2 >= x0 >= x1 >= x2 /** Sort [..]. */ pulic static void QS(int[], int, int ) { if ([..] as < 2 elements) return; int = partition(,, ); QS(,, -1); QS(, +1, ); n partioning at dept 0 partioning at dept 1 partioning at dept 2 Dept of recursion: O(n) Processing at dept i: O(n-i) O(n*n) 24 0 n <= x0 x0 >= x0 <=x1 x1 >= x1 x0 <=x2 x2 >=x2 dept 0. 1 segment of size ~n to partition. Dept 2. 2 segments of size ~n/2 to partition. Dept 3. 4 segments of size ~n/4 to partition. Max dept: O(log n). Time to partition on eac level: O(n) Total time: O(n log n). Average time for Quicsort: n log n. Difficult calculation 4

5 25 QuicSort complexity to sort array of lengt n Time complexity Worst-case: O(n*n) /** Sort [..]. */ Average-case: O(n log n) pulic static void QS(int[], int, int ) { if ([..] as < 2 elements) return; int = partition(,, ); // We now [.. 1] <= [] <= [+1..] // Sort [..-1] and [+1..] QS(,, -1); QS(, +1, ); Worst-case space:? Wat s dept of recursion? Worst-case space: O(n)! --dept of recursion can e n Can rewrite it to ave space O(log n) Sow tis at end of lecture if we ave time 26 Partition. Key issue. How to coose pivot pre: post: x? Popular euristics: Use w first array value (not so good) w middle array value (not so good) w Coose a random element (not so good) w median of first, middle, last, values (often used)! Coosing pivot Ideal pivot: te median, since it splits array in alf But computing te median is O(n), quite complicated Merge two adacent segments Algoritm Ave time. Worst-case time Space Stale? Insertion sort!(# $ ).!(# $ )!(1) Yes Selection sort! # $.!(# $ )!(1) No Quic sort!(# log #).!(# $ ) O(log n)* No Merge sort /* Sort [..]. Precondition: [..t] and [t+1..] are. */ pulic static merge(int[], int, int t, int ) { Copy [..t] into a new array c; Merge c and [t+1..] into [..]; t * Te first algoritm we developed taes space O(n) in te worst case, ut it can e reduced to O(log n) merged, Merge two adacent segments Merge two adacent segments 29 /* Sort [..]. Precondition: [..t] and [t+1..] are. */ pulic static merge(int[], int, int t, int ) { t t merged, 30 // Merge c and [t+1..] into [..] 0 t- pre: c x t? y invariant: post: x and y, c 0 i c.lengt ead of x tail of x u v? tail of y ead of x and ead of y, x, y are 5

6 Merge Mergesort int i = 0; int u = ; int v = t+1; wile( i <= t-){ if(v <= && [v] < c[i]) { [u] = [v]; u++; v++; else { [u] = c[i]; u++; i++; 0 t- pre: c post: t? c 0 i c.lengt u v? /** Sort [..] */ pulic static void mergesort(int[], int, int ]) { if (size [..] < 2) return; int t= (+)/2; mergesort(,, t); mergesort(, t+1, ); merge(,, t, ); t merged, QuicSort versus MergeSort 33 /** Sort [..] */ pulic static void QS (int[], int, int ) { if ( < 1) return; int = partition(,, ); QS(,, -1); QS(, +1, ); /** Sort [..] */ pulic static void MS (int[], int, int ) { if ( < 1) return; MS(,, (+)/2); MS(, (+)/2 + 1, ); merge(,, (+)/2, ); One processes te array ten recurses. One recurses ten processes te array. 34 Algoritm Ave time. Worst-case time Space Stale? Insertion sort!(# $ ).!(# $ )!(1) Yes Selection sort! # $.!(# $ )!(1) No Quic sort!(# log #).!(# $ ) O(log n)* No Merge sort!(# log #).!(# log #) O(n) Yes * Te first algoritm we developed taes space O(n) in te worst case, ut it can e reduced to O(log n) Sorting in Java 35 Java.util.Arrays as a metod sort(array) implemented as a collection of overloaded metods for primitives, sort is implemented wit a version of quicsort for Oects tat implement Comparale, sort is implemented wit timsort, a modified mergesort developed in 1993 y Tim Peters Tradeoff etween speed/space and staility/performance guarantees 6