Heapsort Algorithm

Eric & Bruce

Q: What is “Heapsort”?

• It is a sorting algorithm which takes an array of values and sorts them in either descending order or ascending order.

Q: How Fast is heapsort?

• Heapsort is guaranteed to run in O(n log n) time.

• Avg Run time: O(n log n)• Worst Case run time: O(n log n)

• Compare the Big O run time with Quicksort

Q: How does Heapsort compare?

Source: http://www.azillionmonkeys.com/qed/sort.html

Athlon XP 1.620Ghz Power4 1Ghz

Intel C/C++ WATCOM C/C++ GCC MSVC CC

Heapsort 2.09 4.06 4.16 4.12 16.91

Quicksort 2.58 3.24 3.42 2.80 14.99

Mergesort 3.51 4.28 4.83 4.01 16.90

Part 1: Average Run time test results, by compiler and chipset

Q: How does Heapsort compare?

Source: http://www.azillionmonkeys.com/qed/sort.html

Part 2: Comparisons and Read/Writes

Algorithm Comparisons Reads/Writes

Heapsort 61045.4 40878.2 Quicksort 22037.8 16322.5 Mergesort 31755.0 31225.0

Runtime Conclusions:

• Quicksort is faster then Heapsort in the Average case.

• Heapsort is faster then Quicksort when Quicksort hits its worst case scenario [O(n^2)]

• Best Algorithm: Hybrid of Quicksort and Heapsort

• Final Note: Use quicksort. The performance is slightly better then Heapsort.

Q: How does Heapsort work?

• General Outline:Step 1: Get an unsorted arrayStep 2: “Heapify” the arrayStep 3: Pop off the root node of the heapStep 4: Adjust Heap to maintain its heapish propertiesStep 5: Pop off roots until none exist

Properties of a Heap• Each node has a left and right child• All the leaves are at the bottom level or the

bottom 2 levels • Nodes are filled from left to right• All levels are completely filled with nodes

(exception: bottom level)• Min-Heap: Each child node is greater then the

parent node• Max-Heap: Each child node is less then the

parent node

Almost Complete Binary Tree

V should be pushed to the left

Level 3 should be completely filled

This is a Heap.

Heap as an Array

Heap

Array

Lets see some L33t codeParent(CI) = (CI - 1) / 2;

RightChild(CI) = 2 * (CI + 1);

LeftChild(CI) = 2 * CI + 1;

Inserting into a Heap

• Insert “E”

Inserting into a HeapStart by filling “E” into the next available node

This insertion may cause our tree to not be a heap anymore!!!

Move “E” up the tree until it’s parent is smaller

Inserting “D” into the heapStart by filling “D” into the next available node

This insertion may cause our tree to not be a heap anymore!!!

“D” is smaller then “L”. So, Swap positions

“D” is smaller then “H”. So, Swap positions

“D” is not smaller then “C” so the heap is complete

Removing from the Heap

Always take the root node.

Move the last node “L” into “C’s” old place

Since the smaller Child of “L” is “D” -> Swap “D” and “L”

Since the smaller Child of “L” is “H” -> Swap “L” and “H”

The heap is complete again!

Final Conclusions

1. Heapsort doesn’t require much memory space!

2. Heapsort is very fast!3. Quicksort > Heapsort4. Mergesort > Heapsort5. Heapsort + Quicksort = Introsort6. Introsort > (Quicksort || Heapsort)

Check out these cool links

• http://tide4javascript.com/?s=Heapsort• http://cis.stvincent.edu/html/tutorials/swd/h

eaps/heaps.html• http://en.wikipedia.org/wiki/Heapsort• SWEET animation:

http://www2.hawaii.edu/~copley/665/HSApplet.html