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Analysis of Algorithm
SmoothSort
 Habib Ullah MS(CS)
Science of Computer Programming
Volume 1, Issue 3, Pages 223–233
Smoothsort, an alternative for sorting in situ
(Edsger Wybe)
http://www.sciencedirect.com/science/article/pii/0167642382900168
Contents
• What is Smoothsort
• Heapsort Analysis
• Idea: A family of heaps
• Leonardo Numbers & Trees
• Smoothsort operations
• Algo & Runtime Analysis
• Conclusion
Waybe
Why is Heapsort O(n lg n)?
Why is Heapsort O(n lg n)?
Why is Heapsort O(n lg n)?
Why is Heapsort O(n lg n)?
Building a Leonardo Heap
Insert Operation
Insert Operation
DE queuing from a Leonardo Heap
Remove Operation
Dequeue Operation
Algo & Runtime Analysis
• Insert
– If last two heaps are two consecutive leonardo numbers
• Add new element as there root
– Else if the rightmost is not of size 1
• New element becomes a new heap of size 1. This 1 is taken to be
L(1)
– Else
• New element becomes a new heap of size 1. This 1 is taken to be
L(0)
• Restore
– Set new element as "current" heap.
– While there is a heap to the left of the current heap and its root is
larger than the current root and both of its child heap roots
• Swap(left-root with current).[Now current is that left root]
– While the current heap has a size greater than 1 and either child heap
of the current heap has a root node greater than the root of the
current heap
• Swap the greater child root with the current root. That
child heap becomes the current heap.
C1
(Log N) times (Log N)
Log N
Dequeue
– Remove topmost node of rightmost heap.
– If it has no children, we're done.
– Otherwise:
– Fix up the left of the two heaps.
– Then fix up the right of the two heaps.
For N elements to dequeue running time is (N LogN) but
this is asymptotically far less than Heap sort.
C1
Log N
Conclusion
• Compared to Heap Sort
– Efficient in Best Case
– Equal in Worst Case but the asymptotic graph is
far less than Heap Sort
Smooth Sort

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Smooth Sort

  • 2. Science of Computer Programming Volume 1, Issue 3, Pages 223–233 Smoothsort, an alternative for sorting in situ (Edsger Wybe) http://www.sciencedirect.com/science/article/pii/0167642382900168
  • 3. Contents • What is Smoothsort • Heapsort Analysis • Idea: A family of heaps • Leonardo Numbers & Trees • Smoothsort operations • Algo & Runtime Analysis • Conclusion
  • 5.
  • 6. Why is Heapsort O(n lg n)?
  • 7. Why is Heapsort O(n lg n)?
  • 8. Why is Heapsort O(n lg n)?
  • 9. Why is Heapsort O(n lg n)?
  • 10.
  • 11.
  • 12.
  • 13.
  • 14. Building a Leonardo Heap Insert Operation
  • 16.
  • 17.
  • 18.
  • 19.
  • 20.
  • 21.
  • 22.
  • 23.
  • 24.
  • 25.
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  • 28.
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  • 31.
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  • 34.
  • 35.
  • 36.
  • 37.
  • 38.
  • 39.
  • 40.
  • 41.
  • 42.
  • 43.
  • 44.
  • 45. DE queuing from a Leonardo Heap Remove Operation
  • 47.
  • 48.
  • 49.
  • 50.
  • 51.
  • 52.
  • 53.
  • 54.
  • 55.
  • 56.
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  • 72.
  • 73.
  • 74.
  • 75.
  • 76.
  • 77.
  • 78.
  • 79.
  • 80.
  • 81.
  • 82. Algo & Runtime Analysis
  • 83. • Insert – If last two heaps are two consecutive leonardo numbers • Add new element as there root – Else if the rightmost is not of size 1 • New element becomes a new heap of size 1. This 1 is taken to be L(1) – Else • New element becomes a new heap of size 1. This 1 is taken to be L(0) • Restore – Set new element as "current" heap. – While there is a heap to the left of the current heap and its root is larger than the current root and both of its child heap roots • Swap(left-root with current).[Now current is that left root] – While the current heap has a size greater than 1 and either child heap of the current heap has a root node greater than the root of the current heap • Swap the greater child root with the current root. That child heap becomes the current heap. C1 (Log N) times (Log N) Log N
  • 84. Dequeue – Remove topmost node of rightmost heap. – If it has no children, we're done. – Otherwise: – Fix up the left of the two heaps. – Then fix up the right of the two heaps. For N elements to dequeue running time is (N LogN) but this is asymptotically far less than Heap sort. C1 Log N
  • 85.
  • 86.
  • 87.
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  • 103.
  • 104.
  • 105.
  • 106.
  • 107.
  • 108.
  • 109.
  • 110.
  • 111. Conclusion • Compared to Heap Sort – Efficient in Best Case – Equal in Worst Case but the asymptotic graph is far less than Heap Sort