1. SORTING 2014-T2 Lecture 13 School of Engineering and Computer Science, Victoria University of Wellington COMP 103 Marcus Frean

2. RECAP  ArraySet algorithms and costs  Binary Search is O(log n) TODAY  Two algorithms for sorting a list:  Selection Sort  Insertion Sort Announcements:  Test is getting closer – keep up to date with your work so it’s not stressful next week. 2 RECAP-TODAY

3. Why Sort ?  we can find items in an array much faster if they are sorted finding1-in-a-billion takes as long as the ArraySet takes to find1-in-30  But... constructing a sorted array one item at a time is very slow : O(n 2 )  for 10 million items you need  50,000,000,000,000 steps at 10nS per step ⇒ 500,000 seconds = 58 days.  There are sorting algorithms that are much faster if you can sort whole array at once: O(n log (n) )  for 10 million items you need  250,000,000 steps ⇒ 2.5 seconds 3

4. Ways of sorting:  Selecting sorts:  Find the next largest/smallest item and put in place  Builds the correct list in order  Inserting sorts:  For each item, insert it into an ordered sublist  Builds a sorted list, but keeps changing it  Compare-and-Swap sorts:  Find two items that are out of order, and swap them  Keeps “improving” the list 4

5. Ways to rate sorting algorithms  Efficiency  What is the (worst-case) order of the algorithm ?  Is the average case much faster than worst-case ?  Requirements on Data  Does the algorithm need random-access data?  Does it need anything more than “compare” and “swap” ?  Space Usage  Can the algorithm sort in-place, or does it need extra space ?  Stability  Is the algorithm “stable” (will it ever reverse the order of equivalent items?)  Performance on Nearly Sorted Data  Is the algorithm faster when given sorted (or nearly sorted) data ? (All items close to where they should be, or only a few out of order.) 5

6. 01234567891011 Selecting Sorts  Selection Sort (slow)  HeapSort (fast) 01234567891011 search for minimum here 6

7.  Insertion Sort (slow)  Shell Sort (pretty fast)  Merge Sort (fast) (Divide and Conquer) 01234567891011 Inserting Sorts 01356781011 249 7

8. Compare and Swap Sorts  Bubble Sort (easy but terrible)  QuickSort (the fastest) (Divide and Conquer) things bubble up quickly, but bubble down slowly things bubble up quickly, but bubble down slowly 01234567891011...and so on... 8

9. Other Sorts  Radix Sort (only works on some data)  Permutation Sort(very slow)  Random Sort(Generate and Test) 01234567891011 9

10. Coding it: some design decisions...  We could sort Lists ⇒ general and flexible but efficiency depends on how the List is implemented or just sort Arrays ⇒ less general efficiency is well defined NB: method toArray() converts any Collection to an array  We could require items to be Comparable i.e. any item can call compareTo(otherObj)...on another ("natural ordering") OR provide a Comparator i.e. the sorter can call compare(obj1, obj2)...on any two items. We will sort an Array, using a Comparator: public void …Sort(E[] data, int size, Comparator comp) number of items 10 comparator array

11. Selection Sort public void selectionSort(E[ ] data, int size, Comparator comp){ // for each position, from 0 up, find the next smallest item // and swap it into place for (int i=0; i<size-1; i++){ int minIndex = i; for (int j=i+1; j<size; j++) if (comp.compare(data[j], data[minIndex]) < 0) minIndex=j; swap(data, i, minIndex); } 0123456789101101234567891011

12. Efficiency of Selection Sort  Cost:  step ?  best case: what is it ? cost:  worst case: what is it ? cost:  average case: what is it ? cost: 12

13. Selection Sort Analysis  Efficiency  worst-case: O(n 2 )  average case exactly the same.  Requirements on Data  Needs random-access data, but easy to modify for files  Needs compare and swap  Space Usage  in-place  Stability  ??  Performance on Nearly Sorted Data  No faster when given nearly sorted data 13

14. Insertion Sort public void insertionSort(E[] data, int size, Comparator comp){ // for each item, from 0, insert into place in the sorted region (0..i-1) for (int i=1; i<size; i++){ E item = data[i]; int place = i; while (place > 0 && comp.compare(item, data[place-1]) < 0){ data[place] = data[place-1]; place--; } data[place]= item; } 01234567891011 14

15. Efficiency of Insertion Sort  first: what is the step ?  best case: what is it ? cost:  worst case: what is it ? cost:  average case: what is it ? cost:  almost sorted cost: 15

16. Insertion Sort Analysis  Efficiency  worst-case: O(n 2 )  average case: half the cost of worst case.  Requirements on Data  Needs random-access data  Needs compare and swap  Space Usage  in-place  Stability  ??  Performance on Nearly Sorted Data  Very fast (O(n)) when given nearly sorted data 16

17.  InsertionSort, SelectionSort and BubbleSort are all slow (apart from Insertion on nearly-sorted lists)  Problem:  Selection compares every pair of items, ie. it ignores the results of previous comparisons.  Insertion (and Bubble) only compare adjacent items only move items one step at a time Doing better is possible, but we will need recursion... Why so slow? Can we do better? 17