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Claudio Franciscus

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Outline Heap

Definisi

Operasi max_heapify

build_max_heap

Heapsort

Algoritma Kompleksitas

Implementasi

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Referensi Cormen, Thomas E, Charles E Leiserson, Ronald L

Rivest, and Clifford Stein. "Heapsort." Introduction toAlgorithm. 3rd ed. Cambridge: MIT, 2009. pp 151-169.

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Claudio Franciscus

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Heap

Definisi Sifat:

Max-heap

Min-heap

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Heap

Tree root > index pertama

parent(i) = i/2 > index parent

left(i) = 2i > index child kiri right(i) = 2i+1 > index child kanan

leaf > n/2+1 n > node tanpa child

heap_size

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Heap

Operasi build_max_heap

max_heapify

heapsort

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Max_Heapify

Algoritma Max_Heapify(A,i)

Asumsi: subtree > root left(i), right(i) > Max_Heap

Jika A[i] melanggar sifat Max_Heap,swap A[i], child berelement terbesar

panggil Max_Heapify(A,child)

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Max_Heapify

Pseudocodel= left(i)

r = right(i)

if (l A[i])then largest = l else largest = i

if (r A[largest])

then largest = r

if largest i

then exchange A[i] and A[largest]

Max_Heapify(A, largest)

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Max_Heapify

Example heap_size(A) = 10 max_heapify(A,2)

l = left(2) = 4

r = right(2) = 5 l A[2]

largest = l

r heap_size

largest == i

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Max_Heapify

Kompleksitas Running time:

pada Level 1 > O(1)

pada Level 2 > O(2).

.

.

pada Level teratas O(h) > O(h)

2^ (h)

2^ (h - 1)

2^ (h - 2)

.

.

21

leaf

dengan h:

2h

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Build_Max_Heap Mengubah array tidak teratur menjadi Max_Heap

Build_Max_Heap(A)for i = n/2 downto 1

do Max_Heapify(A,i)

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Build_Max_Heap

Example build_max_heap(A)

max_heapify(A,5)

no change max_heapify(A,4)

swap A[4], A[8]

max_heapify(A,3)

swap A[3], A[7] max_heapify(A,2)

swap A[2], A[5]

swap A[5], A[10]

max_heapify(A,1)

swap A[1], A[2] swap A[2], A[4]

swap A[4], A[9]

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Build_Max_Heap

Kompleksitas Build_Max_Heap(A)

for i = n/2 downto 1

do Max_Heapify(A,i) Max_Heapify > level 1 > O(1)

> level 2 > O(2)

> Level l > O(l)

Jumlah node > Level 1 > n/4

> Level 2 > n/8

> Level 2log n> 1

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Build_Max_Heap

Kompleksitas Total running time dapat dijumlahkan:

n/4(1c) + n/8(2c) + n/16 (3c) + + 1(2log n c)

Jika n/4 = 2k

maka:c 2k( 1/20+ 2/21+ 3/22+ + (k+1)/2k)

Karena total jumlah di dalam kurung > konstan,maka, kompleksitas Build_Max_Heap > O(n)

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Claudio Franciscus

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HeapSort

Algoritma1. Lakukan Build_Max_Heap dari array yang tidak

teratur

2.Swap element A[1](terbesar) dan A[n](terakhir)3. Pisahkan elemen terakhir dengan mengurangi Heap-size

4. Karena root (A[1]) menyalahi aturan lakukan

Max_Heapify(A,1)5. Kembali ke Step 2 jika Heap-size 0

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HeapSort

Example

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HeapSort

Kompleksitas Running Time:

n iterasi > heap_size = 0

tiap iterasi > swap dan Max_Heapifykompleksitas / iterasi > O(2log n)

Keseluruhan > O(n 2log n)

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Claudio Franciscus

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Implementasi Java

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Claudio Franciscus