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CS221: Algorithms and Data Structures Lecture #3.5 Sorting Takes Priority

CS221: Algorithms and Data Structures Lecture #3.5 Sorting Takes Priority. Steve Wolfman 2011W2. Today’s Outline. Sorting with Priority Queues, Three Ways. Quick Review of Sorts.

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CS221: Algorithms and Data Structures Lecture #3.5 Sorting Takes Priority

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  1. CS221: Algorithms and Data StructuresLecture #3.5Sorting Takes Priority Steve Wolfman 2011W2

  2. Today’s Outline • Sorting with Priority Queues, Three Ways

  3. Quick Review of Sorts • Insertion Sort: Keep a list of already sorted elements. One by one, insert new elements into the right place in the sorted list. • Selection Sort: Repeatedly find the smallest (or largest) element and put it in the next slot. • Merge Sort: Divide the list in half, sort the halves, merge them back together. (Base case: length  1.)

  4. 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? F(7) E(5) D(100) A(4) B(6) deleteMin insert G(9) C(3)

  5. “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 • What sorting algorithm is this? • Insertion Sort • Selection Sort • Heap Sort • Merge Sort • None of these

  6. 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)

  7. “PQSort” deleteMaxes with Unsorted List MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 How long does inserting all of these elements take? And then the deletions…

  8. “PQSort” deleteMaxes with Unsorted List MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 PQ

  9. “PQSort” deleteMaxes with Unsorted List MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 PQ 5 9 4 8 1 6 10 12 13 2 3 14 7 20 PQ Result

  10. “PQSort” deleteMaxes with Unsorted List MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 PQ 5 9 4 8 1 6 10 12 13 2 3 14 7 20 PQ Result 5 9 4 8 1 6 10 12 13 2 3 7 14 20 PQ Result

  11. “PQSort” deleteMaxes with Unsorted List MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 PQ 5 9 4 8 1 6 10 12 13 2 3 14 7 20 PQ Result 5 9 4 8 1 6 10 12 13 2 3 7 14 20 PQ Result 5 9 4 8 1 6 10 12 7 2 3 13 14 20 PQ Result

  12. Two PQSort Tricks • Use the array to store both your results and your PQ. No extra memory needed! • 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.

  13. “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(n2)  What algorithm is this? • Insertion Sort • Selection Sort • Heap Sort • Merge Sort • None of these 5 9 4 8 1 6 10 12 7 2 3 13 14 20 PQ Result

  14. “PQSort” insertions with Sorted List MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 PQ 5 9 4 8 1 6 10 12 13 2 3 14 7 20 PQ 5 9 4 8 1 6 10 12 13 2 3 7 14 20 PQ 5 9 4 8 1 6 10 12 13 2 3 7 14 20 PQ

  15. “PQSort” insertions with Sorted List MAX-PQ How long does “build” take? Worst case: O(n2)  How long do the deletions take? What algorithm is this? • Insertion Sort • Selection Sort • Heap Sort • Merge Sort • None of these No time at all! 5 9 4 8 1 6 10 12 13 2 3 7 14 20 PQ

  16. “PQSort” Build with Heap MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 9 4 8 1 6 10 12 13 2 3 14 20 7 Floyd’s Algorithm 20 13 14 12 3 6 10 9 8 2 1 4 5 7 PQ Takes only O(n) time!

  17. “PQSort” deleteMaxes with Heap MAX-PQ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 20 13 14 12 3 6 10 9 8 2 1 4 5 7 PQ 14 13 10 12 3 6 7 9 8 2 1 4 5 20 PQ 13 12 10 9 3 6 7 5 8 2 1 4 14 20 PQ 12 9 10 8 3 6 7 5 4 2 1 13 14 20 PQ Totally incomprehensible as an array!

  18. “PQSort” deleteMaxes with Heap MAX-PQ 5 9 4 8 1 6 10 12 13 2 3 14 20 7 5 9 4 8 1 6 10 12 13 2 3 14 20 7

  19. “PQSort” deleteMaxes with Heap MAX-PQ 5 20 9 4 13 14 8 1 6 10 12 3 6 10 12 13 2 3 14 20 7 9 8 2 1 4 5 7 Build Heap Note: 9 ends up being perc’d down as well since its invariant is violated by the time we reach it.

  20. “PQSort” deleteMaxes with Heap MAX-PQ 13 14 20 12 10 13 10 9 3 6 7 12 3 6 7 13 14 20 14 20 5 8 2 1 4 9 8 2 1 4 5 12 3 6 10 9 10 12 8 7 9 7 9 10 9 8 2 1 4 5 7 5 3 6 1 8 3 6 1 8 3 6 7 10 12 13 14 20 12 13 14 20 13 14 20 2 4 5 4 2 5 4 2 1

  21. “PQSort” with Heap MAX-PQ 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? • Insertion Sort • Selection Sort • Heap Sort • Merge Sort • None of these 9 8 7 5 3 6 1 10 12 13 14 20 2 4 9 8 7 5 3 6 1 2 4 10 12 13 14 20 PQ Result

  22. “PQSort” • What sorting algorithm is this? • Insertion Sort • Selection Sort • Heap Sort • Merge Sort • None of these 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

  23. To Do • Homework! • Read: Epp Section 9.5 and KW Section 10.1, 10.4, and 10.7-10.10

  24. Coming Up • More sorting! • Midterm (Feb 15th)

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