Basic Techniques. Recursion. Today: Later on: Recursive methods who call themselves directly or indirectly.

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1 La 2 Recursion Basic Techniques SMD135 Programs and Data Structures Lecture 7 Sorting: Divide-and- Conquer Brute force Selection sort Insertion sort Linear proing [seaching through an unordered array] Divide-and-Conquer Binary Search [search in ordered arrays] Mergesort Quicksort Recursion Mergesort Quicksort Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 2 Today: Recursion Recursive methods who call themselves directly or indirectly. Later on: Recursive data structures, where one oject of type A has references to ojects of type A. Known as linked or dynamic structures. Example Consider the following class specification for a class that stores a unch of characters. - int loc Bunch «constructor» + Bunch(char c) «query» + int countofremaining() + char nextchar() «update» + void startnextchar() + void insert(char c) Bunch Class Specifications /* class invariant this unch contains one or more char values stored in the order they are inserted */ /* post: this unch consists of the char c */ pulic Bunch(char c) /* post: result == numer of chars in this unch */ pulic int countofremaining() /* pre: loc+1 <= numer of chars in this post: result == (loc+1)st char in this and loc == loc@pre + 1 */ pulic char nextchar() /* post: loc == 0 */ pulic void startnextchar() Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 4 The Oject of Data Astraction Addison Wesley pu.

2 What happens when the following code executes? pulic void printem( Bunch ) {.startnextchar(); while (.countofremaining() > 0) { System.out.print(.nextChar() ); Q: How would you rewrite this method, using a loop to print the unch in reverse without encountering Bunch chars multiple times? A: You can t! pulic void printemreversed( Bunch ) {?????? A recursive method is one that calls itself either directly or indirectly. Example /* post: all chars following loc have een output in reverse order */ pulic void printemreversed(bunch ) { if (.countofremaining() =.nextchar(); printemreversed();... without recusion (or a second data structure) Föreläsning 7 SMD135 - Håkan Jonsson 5 The Oject of Data Astraction Addison Wesley pu Föreläsning 7 SMD135 - Håkan Jonsson 6 The Oject of Data Astraction Addison Wesley pu. Example /* assert: somebunch is a unch consisting of P, I, G */ SomeBunch.startNextChar(); printemreversed( somebunch ); /* post: all chars following loc have een output in reverse order */ pulic void printemreversed(bunch ) { if (.countofremaining() =.nextchar(); printemreversed(); Each call results in a separate activation, and each activation has its own copy of local variales and parameter indings. ==? pulic void printemreversed(bunch ) { if (.countofremaining() =.nextchar(); printemreversed(); loc == Föreläsning 7 SMD135 - Håkan Jonsson 7 The Oject of Data Astraction Addison Wesley pu Föreläsning 7 SMD135 - Håkan Jonsson 8 The Oject of Data Astraction Addison Wesley pu.

3 pulic void printemreversed(bunch ) { if (.countofremaining() =.nextchar(); printemreversed(); loc == 1 pulic void ACTIVATION printemreversed(bunch 2 ) { ==? if (.countofremaining() pulic =.nexchar(); void printemreversed(bunch ) { printemreversed(); if (.countofremaining() =.nextchar(); printemreversed(); loc == Föreläsning 7 SMD135 - Håkan Jonsson 9 The Oject of Data Astraction Addison Wesley pu Föreläsning 7 SMD135 - Håkan Jonsson 10 The Oject of Data Astraction Addison Wesley pu. pulic void ACTIVATION printemreversed(bunch 2 ) { == I if (.countofremaining() pulic =.nexchar(); void printemreversed(bunch ) { printemreversed(); if (.countofremaining() =.nextchar(); printemreversed(); loc == 2 pulic void ACTIVATION printemreversed(bunch 2 ) { == I if (.countofremaining() pulic =.nexchar(); void printemreversed(); ACTIVATION printemreversed(bunch 3 ) { System.out.print( if (.countofremaining() ); == G =.nexchar(); pulic void printemreversed(bunch ) { printemreversed(); if (.countofremaining() =.nextchar(); printemreversed(); loc == Föreläsning 7 SMD135 - Håkan Jonsson 11 The Oject of Data Astraction Addison Wesley pu Föreläsning 7 SMD135 - Håkan Jonsson 12 The Oject of Data Astraction Addison Wesley pu.

4 pulic void ACTIVATION printemreversed(bunch 2 ) { == I if (.countofremaining() pulic =.nexchar(); void printemreversed(); ACTIVATION printemreversed(bunch 3 ) { System.out.print( if (.countofremaining() ); == G =.nexchar(); pulic void printemreversed(); ACTIVATION printemreversed(bunch 4 ) { System.out.print( if (.countofremaining() ); ==? =.nexchar(); pulic void printemreversed(bunch ) { printemreversed(); loc == 3 if (.countofremaining() =.nextchar(); printemreversed(); Föreläsning 7 SMD135 - Håkan Jonsson 13 The Oject of Data Astraction Addison Wesley pu. pulic void ACTIVATION printemreversed(bunch 2 ) { == I if (.countofremaining() pulic =.nexchar(); void printemreversed(); ACTIVATION printemreversed(bunch 3 ) { System.out.print( if (.countofremaining() ); == G =.nexchar(); pulic void printemreversed(bunch ) { printemreversed(); if (.countofremaining() =.nextchar(); printemreversed(); loc == Föreläsning 7 SMD135 - Håkan Jonsson 14 The Oject of Data Astraction Addison Wesley pu. How does the ehavior of the following variation compare to the printemreversed method from the preceding slides? /* post: all chars following loc have een output in reverse order */ pulic void printemsomehow(bunch ) { if (.countofremaining() =.nextchar(); printemsomehow(); pulic void ACTIVATION printemreversed(bunch 2 ) { == I if (.countofremaining() pulic =.nexchar(); void printemreversed(bunch ) { printemreversed(); if (.countofremaining() =.nextchar(); printemreversed(); loc == Föreläsning 7 SMD135 - Håkan Jonsson 15 The Oject of Data Astraction Addison Wesley pu Föreläsning 7 SMD135 - Håkan Jonsson 16 The Oject of Data Astraction Addison Wesley pu.

nextchar(); printemreversed(); 2004-02-05 Föreläsning

Methods loc == 3 (Handed out separately.

2004-02-05 Föreläsning 7 SMD135 - Håkan Jonsson 18

5 pulic void printemreversed(bunch ) { if (.countofremaining() =.nextchar(); printemreversed(); Föreläsning 7 SMD135 - Håkan Jonsson Examples of Recursive Methods loc == 3 (Handed out separately.) The Oject of Data17 Astraction Addison Wesley pu Föreläsning 7 SMD135 - Håkan Jonsson 18 (pyramid) (sumupto) Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 20

6 Recursion vs Iteration/Repetition One can e expressed using the other! Easy to go from iteration to recursion. Hard[er] to go the other way. Tail-recursive methods are easy. In principle, what any recursive method may compute can also e computed y an iterative method [y simulation] ut the translation can e very complicated. Anything that can e accomplished with a loop, can also e written recursively. General form of a while loop. while ( condition ) { loopbody; Equivalent recursive method pulic void recwhile() { if ( condition ) { loopbody; recwhile(); Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 22 The Oject of Data Astraction Addison Wesley pu. Sorting methods Divide-and-Conquer Transposition sorting BuleSort Insert and keep sorted Comparison-ased sorting Priority queue sorting Main sorting themes Divide and conquer Proxmap Sort Address- -ased sorting Diminishing increment sorting RadixSort Divide-and conquer is a general algorithm design paradigm: Divide: divide the input data S in two disjoint susets S 1 and S 2 Recur: solve the suprolems associated with S 1 and S 2 Conquer: comine the solutions for S 1 and S 2 into a solution for S The ase case for the recursion are suprolems of size 0 or 1 Insertion sort Tree sort Selection sort Heap sort QuickSort MergeSort ShellSort Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 24

7 Mergesort Divide... Mergesort on an input sequence S with n elements consists of three steps: Divide: partition S into two sequences S 1 and S 2 of aout n/2 elements each Recur: recursively sort S 1 and S 2 (using Mergesort) Conquer: merge S 1 and S 2 into a unique sorted sequence Algorithm mergesort(s, C) Input sequence S with n elements, comparator C Output sequence S sorted according to C if S.size() > 1 (S 1, S 2 )! partition(s, n/2) mergesort(s 1, C) mergesort(s 2, C) S! merge(s 1, S 2 ) Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 26 and conquer Mergesort For Mergesort an initial array is repeatedly divided into halves (usually each is a separate array), until arrays of just one element remain At each level of recomination, two sorted arrays are merged into one This is done y copying the smaller of the two elements from the sorted arrays into the new array, and then moving along the arrays Föreläsning 7 SMD135 - Håkan Jonsson 27 Aptr Bptr Cptr Föreläsning 7 SMD135 - Håkan Jonsson 28

Merging Analysis of Mergesort 1 13 24 26 2 15 27 38 1 Aptr Bptr Cptr 1 13 24 26 2 15 27 38 1 2 The height h of the mergesort tree is O(log n) at each recursive call we divide in half the sequence,

8 Merging Analysis of Mergesort Aptr Bptr Cptr The height h of the mergesort tree is O(log n) at each recursive call we divide in half the sequence, The overall amount or work done at the nodes of depth i is O(n) we partition and merge 2 i sequences of size n/2 i we make 2 i+1 recursive calls Thus, the total running time of mergesort is O(n log n) Aptr Bptr Cptr depth #seqs size n etc. Aptr Bptr Cptr Föreläsning 7 SMD135 - Håkan Jonsson 29 1 i 2 2 i n/2 n/2 i Föreläsning 7 SMD135 - Håkan Jonsson 30 Quicksort C. A. R. Hoare. "Quicksort." Computer Journal, 5(1):10--15, C. A. R. Hoare. Algorithm 64: Quicksort. Commun. ACM 4(7): 321 (1961) Sir C. A. R. ( Tony ) Hoare Föreläsning 7 SMD135 - Håkan Jonsson 31 Quicksort To understand quick-sort, let s look at a high-level description of the algorithm 1) Divide : If the sequence S has 2 or more elements, select an element x from S to e your pivot. Any aritrary element, like the last, will do. Remove all the elements of S and divide them into 3 sequences: L, holds S s elements less than x E, holds S s elements equal to x G, holds S s elements greater than x 2) Recurse: Recursively sort L and G 3) Conquer: Finally, to put elements ack into S in order, first inserts the elements of L, then those of E, and those of G Föreläsning 7 SMD135 - Håkan Jonsson 32

Idea of Quicksort 1) Select: pick an element 2) Divide: rearrange elements so that x goes to its final position E 3) Recurse and Conquer: recursively sort QuickSort The idea is as follows: 1.

9 Idea of Quicksort 1) Select: pick an element 2) Divide: rearrange elements so that x goes to its final position E 3) Recurse and Conquer: recursively sort QuickSort The idea is as follows: 1.If the numer of elements to e sorted is 0 or 1, then return 2.Pick any element, v (this is called the pivot) 3.Partition the other elements into two disjoint sets, S 1 of elements " v, and S 2 of elements > v 4.Return QuickSort (S 1 ) followed y v followed y QuickSort (S 2 ) Föreläsning 7 SMD135 - Håkan Jonsson Föreläsning 7 SMD135 - Håkan Jonsson 34 Quicksort example Pick any element (here the middle element) as the pivot (i.e., 10) Partition into the two susets elow Sort the susets (using Quicksort!) Recomine with the pivot Föreläsning 7 SMD135 - Håkan Jonsson 35 Some oservations aout Quicksort A poor choice of pivot can lead to O(n 2 ) time performance Use first element as pivot when sorting an already sorted sequence A good strategy is to pick the middle value of the left, centre, and right elements For small arrays, with n less than (say) 20, Quicksort does not perform as well as simpler sorts such as Selection sort Because Quicksort is recursive, these small cases will occur frequently A common solution is to stop the recursion at n = 10, say, and use a different, non-recursive sort This also avoids nasty special cases, e.g., trying to take the middle of three elements when n is one or two Föreläsning 7 SMD135 - Håkan Jonsson 36

10 Comparison Mergesort O(n log n) time Quicksort O(n log n) in the average case O(n 2 ) in the worst case But in practice, it s quick And the worst case doesn t happen often (input already sorted) Advice: Choose pivot median of three at random Föreläsning 7 SMD135 - Håkan Jonsson 37

CS 137 Part 8. Merge Sort, Quick Sort, Binary Search. November 20th, 2017

CS 137 Part 8. Merge Sort, Quick Sort, Binary Search. November 20th, 2017 CS 137 Part 8 Merge Sort, Quick Sort, Binary Search November 20th, 2017 This Week We re going to see two more complicated sorting algorithms that will be our first introduction to O(n log n) sorting algorithms.