ALGORITMA PARALEL UNTUK MENJUMLAHKAN N BUAH BILANGAN DAN PENGURUTAN MENGGUNAKAN ALGORITMA BUBBLESORT
OLEH :
SUPRIYANTO (G651090191)
Dosen : Dr. Yeni Herdiyeni, S.Kom, M.Kom
MARET 2010
DEPARTEMEN ILMU KOMPUTER SEKOLAH PASCA SARJANA INSTITUT PERTANIAN BOGOR BOGOR
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Sesungguhnya agama (yang diridhai) disisi Allah hanyalah Islam. Tiada berselisih orangorang yang telah diberi Al Kitab kecuali sesudah datang pengetahuan kepada mereka, karena kedengkian (yang ada) di antara mereka. Barangsiapa yang kafir terhadap ayatayat Allah maka sesungguhnya Allah sangat cepat hisab-Nya. (QS. Ali Imran :19)
Allah lah yang menciptakan tujuh langit dan seperti itu pula bumi. Perintah Allah berlaku padanya, agar kamu mengetahui bahwasanya Allah Maha Kuasa atas segala sesuatu, dan sesungguhnya Allah, ILMU-Nya benar-benar meliputi segala sesuatu. (QS. 65:12)
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KATA PENGANTAR
Puji syukur keharidat Allah SWT yang senantiasa memberikan limpahan rahmat dan hidayah-Nya kepada kita semua. Berkat rahmat Allah pulalah tugas perdana mata kuliah Pemrosesan Paralel ini dapat diselesaikan. Makalah ini akan membahas tentang cara kerja Algoritma Paralel penjumlahan n buah bilangan dan pengurutan menggunakan algoritma Bubble Sort. Penulis menyadari bahwa dalam penulisan makalah ini masih jauh dari kesempurnaan, namun demikian penulis berusaha maksimal untuk menyuguhkan tulisan terbaik. Akhir kata, kami mengucapkan terima kasih atas perhatian, saran, kritik dan masukan yang diberikan demi kemajuan di kemudian hari.
Bogor,
Maret 2010 Penulis,
Supriyanto
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DAFTAR ISI
BAB I. PENDAHULUAN PENDAHULUAN ............................ ............................................ ................................ ................................ .............................. ......................... ........... 1 A. Latar Belakang Belakang ................................ .............................................. .............................. .............................. .............................. ...................... ...... 1 B. Tujuan .............................. .............................................. ................................ ............................... ............................... ................................ .................. .. 1 BAB II. TINJAUAN PUSTAKA ........................................................................................... 2 A. Interconection Networks .................................................................................... 2 1.
Shared Versus Swithed Media..................................................................... 2
2.
Topologi Jaringan Switch ........................................................................... 2
B. Processor Processor Array ................................ ................................................. ............................... .............................. ................................ .................. .. 6 C. Multiprocesso Multiprocessorr ............................. ............................................. .............................. .............................. ................................. ........................ ....... 6 D. Multicomputer Multicomputer............................. ............................................. .............................. .............................. ................................. ........................ ....... 7 E. Klasifikasi Klasifikasi Flyn ................................ ................................................. ............................... .............................. ................................ .................. .. 8 BAB III. PEMBAHASAN ..................................................................................................... 9 A. Algortima Penjumlahan n Buah Bilangan .......................................................... 9 1.
Algoritma Penjumlahan Secara Sekuensial ................................................. 9
2.
Algoritma Penjumlahan Secara Paralel ....................................................... 9
.................................. .... 10 B. Pengurutan Dengan Menggunakan Algoritma Bublesort .............................. 1.
Pengurutan secara sekuensial .................................................................... 10
2.
Formulasi Paralel de ngan Pola Transposisi Ganjil-Genap G anjil-Genap Pada Bublesort . 11
BAB IV. KESIMPULAN ..................................................................................................... 15
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DAFTAR GAMBAR
Gambar 2.1. (a) shared medium dan (b) s witched medium (Quinn, 2004).......... .................... ........... ......... 2 Gambar 2.2. Fully Connected Network ................................................................................. 3 Gambar 2.3. Linier dan Ring ................................................................................................. 3 Gambar 2.4. Topologi Jaringan (a) Mesh (b) Torus 2D, dan (c) Torus 3D (Quinn, 2004) .... .. 3 Gambar 2.5. Jaringan Tree dan Hypertree (Quinn, 2004)................... .......... .................... ........... ..................... ........... .............. .... 4 Gambar 2.6. Butterflay Network dengan 8 prosesor (Quinn, 2004) .................... ......... ........... .................... ........... ......... 4 Gambar 2.7. Cube dan Hypercube ......................................................................................... 5 Gambar 2.8. Shuffle Exchange .............................................................................................. 5 Gambar 2.9. Contoh Arsitektur Prosesor ............................................... ................................. ...................... ...... 6 Prosesor Array.............................. Gambar 2.10. Centralized Processor(Quinn, 2004) ................................................................ 7 Gambar 2.11. Distributed Multiprocessor(Quinn, 2004) ........................................................ 7 Gambar 2.12. Symmetrical Multicomputer (Quinn, 2004) ..................................................... 7 Gambar 2.13. Asymmetrical Multicomputer (Quinn, 2004) ................................................... 8 Gambar 3.1. Gambaran Komputasi secara Paralel untuk Penjumalahan n buah bilangan ..... 10 Gambar 3.2. Ilustrasi Algoritma Bubblesort .............................. .............................................. ................................ .......................... .......... 11 Gambar 3.3. Ilustrasi Algoritma Ganjil-Genap pada Bubblesort (Grama dkk, 2003) ............ .......... .. 12
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BAB I. PENDAHULUAN
A. Latar Belakang Perancangan algoritma adalah komponen penting dalam pemecahan masalah menggunakan komputer. Suatu algoritma sekuensial pada dasarnya adalah sebuah teknik atau urut-urutan langkah dasar untuk memecahkan masalah dengan menggunakan satu prosesor. Demikian pula, algoritma parallel adalah teknik pemecahan masalah dengan menggunakan beberapa prosesor. Pemecahan masalah dilakukan dengan menggunakan teknologi paralel,dimana beberapa komputer bekerja secara paralle untuk menyelesaikan masalah komputasi. Algoritma paralel memungkinkan eksekusi terhadap berbagai operasi dilakukan secara bersamaan. Namun demikian dalam melakukan desain algoritma paralel perlu dilakukan pekerjaan-pekerjaan.Dalam prakteknya penyusunan algoritma parallel mencakup langkah-langkah berikut : • Mengidentifikasi bagian dari pekerjaan yang dapat dilakukan secara bersamaan. •
Memetakan pekerjaan-pekerjaan ke beberapa proses yang berjalan secara parallel
•
Mendistribusikan input, output, dan data intermediate yang terkait dengan program.
•
Mengelola teknik pengaksesan secara bersama oleh beberapa prosesor.
Sementara itu tujuan utama penggunaan algoritma paralel adalah untuk mempercepat pemrosesan suatu algoritma. Pada makalah ini akan dibahas teknik menganalisis dan mendesain algoritma paralel untuk kasus penjumlahan n buah bilangan dan pengurutan dengan menggunakan algoritma bublesort. Makalah ini menitikberatkan pada teknik mendekomposisi permasalahan agar dapat dikerjakan dengan menggunakan algoritma paralel.
B. Tujuan Tujuan dari penulisan makalah ini adalah : 1. Mempelajari cara kerja Algoritma Paralel penjumlahan n buah bilangan dan pengurutan dengan menggunakan algoritma Bubblesort . 2. Membuat pseudocode Algoritma diatas.
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BAB II. TINJAUAN PUSTAKA
Sekitar tahun 1960 sampai 1990-an para peneliti dan para insinyur mencari berbagai variasi dari arsitektur komputer paralel. Berbagai perusahaan mulai memproduksi komputer paralel untuk digunakan dalam berbagai keperluan komputasi. Namun demikian, performansi dari berbagai desain prosesor tidak hanya tergantung pada banyaknya prosesor. (Quinn, 2004). Pada tulisan ini akan dibahas tentang berbagai variasi dari interconection network (koneksi jaringan). Selain itu akan dibahas juga secara sekilas mengenai prosesor array, multiprosesor dan multicomputers, yang merupakan tiga jenis komputer paralel yang sangat populer. Selain itu akan diperkenalkan juga Flynn Taksonomi dari komputer parallel dan metode pipelining yang sangat banyak digunakan dalam berbagai masalah yang diselesaikan dengan algoritma paralel.
A. Interconection Networks 1.
Shared Versus Swithed Media Pengunaaan jaringan interkoneksi bertujuan untuk : • Menghubungkan prosesor kepada share memori • Menghubungkan satu prosesor dengan prosesor lainnya
Adapun media yang digunakan adalah • Shared medium, dan • Switch medium Gambar 2.1. memperlihatkan bagaimana cara kerja shared memori dan switch media
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Gambar 2.2. Fully Connected Network Pada kenyataanya sulit sekali mewujudkan koneksi seperti pada Gambar 2.1 ini, dikarenakan biaya investasi yang dibutuhkan sangat mahal. Adapun Arsitektur yang umum digunakan adalah :
Linier dan Ring
Gambar 2.3. Linier dan Ring
Mesh, Torus : 2D, 3D
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Tree dan Hypertree
Gambar 2.5. Jaringan Tree dan Hypertree (Quinn, 2004) Pada jaringan Tree hanya ada satu jalur untuk setiap dua simpul, semakin tinggi tree, maka semaik beresiko akan terjadi komunikasi bottleneck pada level-level yang lebih tinggi dari tree. Untuk menghindari hal tersebut maka pada ja ringan model Tree dapat dikembangkan menjadi hypertree, dimana node pada berbagai level dapat terhubung satu sama lain, dengan kata lain ada pembagian jalur koneksi.
Butterflay d Tipe butterflay adalah topologi tidak langsung, dimana n = 2 prosesor terkoneksi dengan n (log n + 1) node swith (Gambar 2.6). Node swith di bagi ke dalam log n + 1 baris, atau rangk yang berisi n node. Rank tersebut dilabelkan dengan 0 hinga log n, dimana rank 0 dan d bisa terhubung bisa tidak, yang mengakibatkan node pada switch terhubung ke 4 node yang lainnya.
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Cube dan Hypercube
Gambar 2.7. Cube dan Hypercube
Shuffle Exchange
Gambar 2.8. Shuffle Exchange Penjelasan-penjelasan di atas menggambarkan bahwa tidak satupun network optimal pada berbagai kondisi. Perlu ada pertimbangan-pertimbangan khusus dalam pemilihan tipe switch yang akan digunakan. Tabel 2.1. memperlihatkan network cost untuk masingmasing topologi jaringan. Tabel 2.1. Network Cost untuk berbagai topologi Jaringan
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B. Processor Array Prosesor array dapat digambarkan seperti komputer yang diimplementasikan layaknya pada sekuensial komputer yang dihubungkan satu sama lain secara identik, yang disinkronisasi untuk beroperasi mengolah berbagai macam data ( multiple data).
Gambar 2.9. Contoh Arsitektur Prosesor Prosesor Array
C. Multiprocessor Terdapat dua jenis arsitektur multiprosesor yang digunakan dalam pemrosesan paralel yaitu centralized processor (Gambar 2. 10) dan distributed multiprocessor (Gambar 2.11).
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Centralized Processor
Gambar 2.10. Centralized Processor(Quinn, 2004)
Distributed Multiprocessor
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Gambar 2.13. Asymmetrical Multicomputer (Quinn, 2004)
E. Klasifikasi Flyn Flyn (1966) mengklasifikasikan komputer berdarkan instruction stream dan data stream yaitu : • Single Intruction –Single Data Stream (SISD) • Multi Intruction –Single Data Stream (MISD) • Single Intstruction – Multiple Data Stream (MIMD) • Multiple Instruction Multiple Data (MIMD)
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BAB III. PEMBAHASAN
A. Algortima Penjumlahan n Buah Bilangan 1.
Algoritma Penjumlahan Secara Sekuensial Agoritma Penjumlahan secara sekuensial (SISD)
Begin sum a0 for i 1 to n-1 do sum sum + ai endfor end Waktu eksekusi algoritma di atas sebanding dengan banyaknya eksekusi loop for. Loop tersebut pasti akan dieksekusi sebanyak n-1 kali. Jadi waktu eksekusinya adalah O(n).
Misalkan algoritma tersebut dijalankan dengan input : ai= {1,2,3,4,5,6,7,8}, {1,2,3,4,5,6,7,8}, maka ilustrasi prosesnya dapat dilihat sebagai berikut.
Sum = a0 = 1 i = 1 sum = sum+ a1= 1 + 2 = 3 i = 2 sum = sum+ a2= 3 + 3 = 6 i = 1 sum = sum+ a3= 6 + 4 = 10
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Gambar 3.1. Gambaran Komputasi secara Paralel untuk Penj umalahan n buah bilangan Ilustrasi proses komputasi penjulahan yang dilakukan pada algoritma diatas dapat dilihat pada Gambar 3.1. Pada gambar tersebut tersebut terlihat bahwa penjumlahan penjumlahan bilangan dapat dilakukan secara paralel. Pada fase satu A[0] dijumlahkan dengan A[1], dan A[1] dijumlahkan dengan A[2], A[3] dijumlahkan dengan A[4] dan seterusnya. Sehingga pada fase dua, dijumlahkan dua angka yang berdekatan secara bersamaan, yang merupakan hasil penjumlahan pada fase sebelumnya. Begitupun fase-fase selanjutnya sehingga didapatkan jawaban akhir yang diinginkan.
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tibelakangnya lebih kecil, maka dilakukan penukaran. Dengan cara ini, bilangan terkecil pasti akan "naik" hingga posisi yang diinginkan. Iterasi pada setiap loop 2 terjadi dengan waktu Θ(n) kali, sehingga total waktu yang diperlukan adalah Θ(n ).
Gambar 3.2. Ilustrasi Algoritma Bubblesort 2.
Formulasi Paralel dengan Pola Transposisi Ganjil-Genap Pada Bublesort
Pola Transposisi Ganjil-Genap Pada Bublesort Pada transposisi ganjil-genap pada algoritma penguruan n elemen pada n fase (n genap), membutuhkan n/2 operasi pembandingan. Algoritma dibagi ke dalam dua fase, yang disebut dengan fase ganjil dan fase genap. Susun
menjadi urutan yang akan diurutkan. Selama fase ganjil, elemen-elemen dengan indeks ganjil
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pembandingan dan pertukaran dari masing-masing bagian dilakukan secara bersamaan (paralel). Pada kasus ini dapat dilihat bahwa n sebagai jumlah proses (sekaligus sebagai jumlah elemen yang akan diurutkan). Asumsikan bahwa proses-proses tersebut diatur dalam sebuah array satu dimensi. Elemen ai dinisiasi pada proses Pi untuk i = 1, 2, ..., n. Selama fase ganjil, setiap proses yang memiliki label genap dibandingkan dan jika memenuhi syarat maka dipertukarkan dengan tetangga yang ada sebelah kannnya. Demikian juga untuk fase genap, masing-masing proses dengan label ganjil dibandingkan dan jika memenuhi syarat dipertukarkan dengan tetangga sebelah kanannya. Maka perumusan aloritma parlalel untuk kasus bubbelsort adalah sebagai berikut :
1. 2. 3.
procedure ODD-EVEN_PAR (n) begin id := process's label
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Kompleksitas pada kasus pengurutan n buah bilangan secara sekuensial adalah Θ(n log n), dapat ditingkatkan efisiensinya dengan formulasi ini, seperti terlihat pada persamaan berikut :
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BAB IV. KESIMPULAN
Dari penulisan ini dapat disimpulkan bahwa untuk melakukan penjumlahan n buah bilangan secara sekuensial diperlulkan waktu eksekusi untuk n buah data sebanyak O(n). Hal itu dapat dipepercepat dengan mendokomposisi proses menjadi proses-proses parallel. Hasil dekomposisi ini diupayakan agar dapat dikerjakan di beberapa prosesor secara parallel. Sementara itu pada pengurutan dengan menggunakan buble sort secara sekuensial membutuhkan waktu eksekusi Θ(n log n). Dengan menggunakan ide indeks ganjil dan genap maka
pengurutan n buah bilangan dengan bubble sort dapat dilakukan.
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DAFTAR PUSTAKA
Grama, A; Gupta, A; Karypis, George; dan Kumar, V. 2003. Introduction to Parallel Computing, Second Edition - Chapter 9 . Sorting . Addison Wesley. USA. Petersan, W.P. dan Arbenz, P. 2004. Introduction to Parallel Computing, A Practical Guide With Examples in C . Oxford University. Quinn, Michael J. 2004. Parallel Programming in C with MPI and OpenMP. Mc Graw Hill.