Sorting  (Page 12/20)

• Like quicksort, merge sort on arrays has considerably better data cache performance, often outperforming heapsort on a modern desktop PC, because it accesses the elements in order.
• Merge sort is a stable sort .
• Merge sort parallelizes better ; the most trivial way of parallelizing merge sort achieves close to linear speedup , while there is no obvious way to parallelize heapsort at all.
• Merge sort can be easily adapted to operate on linked lists and very large lists stored on slow-to-access media such as disk storage or network attached storage . Heapsort relies strongly on random access , and its poor locality of reference makes it very slow on media with long access times.

An interesting alternative to Heapsort is Introsort which combines quicksort and heapsort to retain advantages of both: worst case speed of heapsort and average speed of quicksort.

Pseudocode

The following is the "simple" way to implement the algorithm, in pseudocode, where swap is used to swap two elements of the array. Notice that the arrays are zero based in this example.

function heapSort(a, count) is

input: an unordered array a of length count

(first place a in max-heap order)

heapify(a, count)

end := count - 1

while end>0 do

(swap the root(maximum value) of the heap with the last element of the heap)

swap(a[end], a[0])

(decrease the size of the heap by one so that the previous max value will

stay in its proper placement)

end := end - 1

(put the heap back in max-heap order)

siftDown(a, 0, end)

function heapify(a,count) is

(start is assigned the index in a of the last parent node)

start := count ÷ 2 - 1

while start ≥ 0 do

(sift down the node at index start to the proper place such that all nodes below

the start index are in heap order)

siftDown(a, start, count-1)

start := start - 1

(after sifting down the root all nodes/elements are in heap order)

function siftDown(a, start, end) is

input: end represents the limit of how far down the heap

to sift.

root := start

while root * 2 + 1 ≤ end do (While the root has at least one child)

child := root * 2 + 1 (root*2+1 points to the left child)

(If the child has a sibling and the child's value is less than its sibling's...)

if child<end and a[child]<a[child + 1] then

child := child + 1 (... then point to the right child instead)

if a[root]<a[child] then (out of max-heap order)

swap(a[root], a[child])

root := child (repeat to continue sifting down the child now)

else

return

The heapify function can be thought of as successively inserting into the heap and sifting up. The two versions only differ in the order of data processing. The above heapify function starts at the bottom and moves up while sifting down (bottom-up). The following heapify function starts at the top and moves down while sifting up (top-down).

function heapify(a,count) is

(end is assigned the index of the first (left) child of the root)

end := 1

while end<count

(sift up the node at index end to the proper place such that all nodes above

the end index are in heap order)

siftUp(a, 0, end)

end := end + 1

(after sifting up the last node all nodes are in heap order)