1. CS221: Algorithms & Data Structures Lecture #3.5 Sorting Takes Priority (Sorting with PQs, Three Ways) Steve Wolfman 2010W2 1

2. How Do We Sort with a Priority Queue? You have a bunch of data. You want to sort by priority. You have a priority queue. WHAT DO YOU DO? 2 F(7) E(5) D(100) A(4) B(6) insert deleteMin G(9)C(3)

3. “PQSort” Sort(elts): pq = new PQ for each elt in elts: pq.insert(elt); sortedElts = new array of size elts.length for i = 0 to elts.length – 1: sortedElts[i] = pq.deleteMin return sortedElts 3 What sorting algorithm is this? a.Insertion Sort b.Selection Sort c.Heap Sort d.Merge Sort e.None of these

4. Reminder: Naïve Priority Q Data Structures Unsorted list: –insert: worst case O(1) –deleteMin: worst case O(n) Sorted list: –insert: worst case O(n) –deleteMin: worst case O(1) 4

5. “PQSort” deleteMaxes with Unsorted List MAX-PQ 5 9481610121323142075 123456789101112013 9481610121323147205 PQ Result 9481610121323714205 PQResult 9481610127231314205 PQResult

6. Two PQSort Tricks 1)Use the array to store both your results and your PQ. No extra memory needed! 2)Use a max-heap to sort in increasing order (or a min-heap to sort in decreasing order) so your heap doesn’t “move” during deletions. 6

7. “PQSort” deleteMaxes with Unsorted List MAX-PQ 7 9481610121323142075 123456789101112013 9481610121323147205 PQ Result 9481610121323714205 PQResult 9481610127231314205 PQResult

8. “PQSort” deleteMaxes with Unsorted List MAX-PQ How long does “build” take? No time at all! How long do the deletions take? Worst case: O(n 2 )  What algorithm is this? a.Insertion Sort b.Selection Sort c.Heap Sort d.Merge Sort e.None of these 8 9481610127231314205 PQResult

9. “PQSort” insertions with Sorted List MAX-PQ 9 9481610121323142075 123456789101112013 PQ 9481610121323147205 PQ 9481610121323714205 PQ 9481610121323714205 PQ

10. “PQSort” insertions with Sorted List MAX-PQ 10 94816 121323714205 PQ How long does “build” take? Worst case: O(n 2 )  How long do the deletions take? No time at all! What algorithm is this? a.Insertion Sort b.Selection Sort c.Heap Sort d.Merge Sort e.None of these

11. “PQSort” Build with Heap MAX-PQ 11 9481610121323142075 123456789101112013 14123610982145720 PQ Floyd’s Algorithm Takes only O(n) time!

12. “PQSort” deleteMaxes with Heap MAX-PQ 12 123456789101112013 14123610982145720 PQ 1310123679821452014 PQ 1210936758214142013 PQ 9108367542113142012 PQ Totally incomprehensible as an array!

13. “PQSort” deleteMaxes with Heap MAX-PQ 13 32 12 10618 49 5 94816 121323142075 7 14

14. “PQSort” deleteMaxes with Heap MAX-PQ 14 321312 10618 49 5 720141289 106312 1413 20 754 Build Heap Note: 9 ends up being perc’d down as well since its invariant is violated by the time we reach it.

15. “PQSort” deleteMaxes with Heap MAX-PQ 1289 106312 1413 20 754 1289 76312 1013 14 54 20 1285 7639 1012 13 4 1420 1245 7638 109 12 131420 245 1638 79 10 12131420 42 1635 78 9 1012131420

16. “PQSort” with Heap MAX-PQ 16 How long does “build” take? Worst case: O(n) How long do the deletions take? Worst case: O(n lg n) What algorithm is this? a.Insertion Sort b.Selection Sort c.Heap Sort d.Merge Sort e.None of these 42 1635 78 9 1012131420 8753612410121314209 PQResult

17. “PQSort” Sort(elts): pq = new PQ for each elt in elts: pq.insert(elt); sortedElts = new array of size elts.length for i = 0 to elements.length – 1: sortedElts[i] = pq.deleteMin return sortedElts 17 What sorting algorithm is this? a.Insertion Sort b.Selection Sort c.Heap Sort d.Merge Sort e.None of these

18. Coming Up More sorting! Programming Project #2 18