21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 1 Class 14 - Review: Sorting & searching r What are sorting and searching? r Simple sorting algorithms m Selection sort m Insertion sort r “Asymptotic” efficiency of algorithms r Faster sorting algorithms m Quicksort m Mergesort

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 2 Sorting r Given array or list of values, and an ordering on the elements (e.g. “<“ on numbers, alphabetical ordering on string), put the elements of the array or list into increasing order. m In-place sorting: rearrange the elements without allocating a new array m Non-destructive sorting: create new array or list with the reordered elements.

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 3 Sorting (cont.)  In-place sort: void sort (double[] A) Suppose A = After calling sort(A), A =  Non-destructive sort: double[] sort (double[] A) If A is as above before call, then call sort(A) returns:

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 4 Searching r Given a set of elements (possibly with associated attributes, e.g. names with associated phone numbers), and given an element, find the element in the set. r E.g. we know how to find an element in a list: boolean find (int x, IntList L) { if (L == L.empty()) return false; else if (x == L.hd()) return true; else return find(x, L.tl()); }

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 5 Searching (cont.) r However, above method is not efficient enough for some applications (such as finding a phone number in an entire phone book). r Searching can be made more efficient by using different data structures to store the data. We will not talk about searching in today’s class.

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 6 Sorting algorithms r Many algorithms are known for sorting elements in a list or array. They include: m Selection sort m Insertion sort m Quicksort m Mergesort r Efficiency - that is, run time - can vary dramatically from one to another.

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 7 Selection sort  Idea is simple and intuitive, and easy to program. (We will assume we are sorting an array of length n, although the method could also be applied to sorting lists.)  Find smallest element among A[0]... A[n- 1 ]. Suppose it is A[i]. Swap A[0] and A[i].  Find smallest element among A[1]... A[n-1]. Suppose it is A[j]. Swap A[1] and A[j].  Continue up to A[n-2]... A[n-1].

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 8 Selection sort (cont.) Starting state of A: After step 1: After step 2: After step 3: After step 4: After step 5: After step 6: After step 7:

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 9 Selection sort (cont.) r As with any array-processing method, can use iteration or recursion. For the moment, we will use iteration. Basic algorithm: void selectionSort (double[] A) { for (i=0; i<n-1; i++) { // Invariant: A[0]..A[i-1] is sorted, and // elements in A[0]..A[i-1] are all less than // elements in A[i]..A[n-1] minloc =... location of smallest value in A[i]... A[n-1]... swap A[i] with A[minloc] }

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 10 Selection sort (cont.) r Fill in specific parts by making method calls; then define those methods. void selectionSort (double[] A) { int i, minloc; int n = A.length; for (i=0; i<n-1; i++) { // Invariant: A[0]..A[i-1] is sorted, and // elements in A[0]..A[i-1] are all less than // elements in A[i]..A[n-1] minloc = findMin(A, i); swap(A, i, minloc); }

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 11 Selection sort (cont.) int findMin (double[] A, int i) { int j, min = i; for (j=i+1; j<A.length; j++) // A[min] <= all elements in A[i]..A[j-1] if (A[j] < A[min]) min = j; return min; } void swap (double[] A, int i, int j) { double temp = A[i]; A[i] = A[j]; A[j] = temp; }

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 12 Insertion sort r Like selection sort, insertion sort is an easy algorithm to understand and program. ¬ Swap A[0] and A[1], if necessary, so that A[0]...A[1] is sorted. ­ Move A[2], if necessary, so that A[0]...A[2] is sorted. ® Move A[3], if necessary, so that A[0]...A[3] is sorted. ¯ Continue, finally moving A[n-1], if necessary, so that entire array is sorted.

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 13 Insertion sort (cont.) Starting state of A: After step 1: After step 2: After step 3: After step 4: After step 5: After step 6: After step 7:

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 14 Insertion sort (cont.) void insertionSort (double[] A) { for (i=1; i<n; i++) { // Invariant: A[0]..A[i-1] is sorted... insert A[i] into correct location in A[0]... A[i] } r Basic algorithm: r As before, fill in the informal part with a method call:

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 15 Insertion sort (cont.) void insertionSort (double[] A) { int i, n = A.length; for (i=1; i<n; i++) { // Invariant: A[0]..A[i-1] is sorted insertInOrder(A[i], A, i); }

21/3/00SEM107 - © Kamin & ReddyClass 14 - Sorting - 16 Insertion sort (cont.) void insertInOrder(double x, double[] A, int hi) { int j = hi; while (j > 0 && x < A[j-1]) { // Invariant: the elements originally in // A[0]..A[hi-1] are now in A[0]..A[j-1] and // A[j+1]..A[hi], and x < A[j+1] A[j] = A[j-1]; j--; } A[j] = x; }