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

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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)

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“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

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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

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

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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

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

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“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

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

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“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

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

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

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

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“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.

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“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

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“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

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“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

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Coming Up More sorting! Programming Project #2 18

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