Penggunaan Pohon Biner Sebagai Struktur DatauntukPencarian
Penggunaan Pohon Biner Sebagai Struktur DatauntukPencarianRitaWijaya/13509098Program Studi TeknikInformatikaSekolah Teknik Elektro danInformatikaInstitutTeknologiBandung,II.Ganesha1t1Bandung411132,Indonesia13509098@std.stei.itb.ac.idAbstract—Cohen adalah graf tak-berarah terhubungyangtidak mengandung sirkuit. Pohon biner terurut danpohonseimbang adalah dua dari berbagai jenis pohon yangpernahkita pelajari. Kedua jenis pohon ini memilikimanfaatnyamasing-masing. Dalam makalah ini, kitaakanmembandingkan keefektifan pencarian menggunakanpohonbiner terurut seimbang dibandingkan pohon binerterurutlainnya.Index Terms—Pencarian, Pohon biner, Pohonbinerseimbang,Pohonbinerterurut.lPENDAHULUANBagi orang yang berkecimpung dalamduniakeinformatikaan, terutama programmer, pemilihanjenisstruktur data yang akan digunakan dalamprogrammembaca dampak yang besar. Kesalahan pemilihanjenisstruktur data dapat mengakibatkan buruknyakinerjaprogram karena program yang efisien menuntutpemilihanstruktur data yangtepat.Salah satu hal yang paling sering dilakukan dalamsuatuprogram adalah proses pencarian dan sorting.Strukturdata yang berbeda tentu saja memerlukan teknikpencarianyang berbeda. Hal ini mengakibatkan waktuyangdiperlukan untuk pencarian juga berbeda. Struktur dataitudapat berupa tabel, list berkait, termasukpohon.Kata pohon pasti sudah tidak asing lagi di telingakitaNamun, pohon sendiri memiliki berbagaipengertiantergantung konteks pembicaraan yang ada. Pohonyangakan kita bahas di sini bukanlah pohon yangmerupakantumbuhan dengan batang kayu, yang hidup dandapattumbuh. Pohon yang dimaksud dalam makalahinimerupakan salah satu dari aplikasi graf. Pohon ini adalahgraf tak-berarah terhubung yang tidakmengandungsirkuit.Pohon sendiri dikelompokkan lagi menjadiberbagaikategori, di antaranya pohon biner, pohon binerterurut,pohon n-ary, dan pohon seimbang. Masing-masingjenispohon memiliki kelebihan dan kekurangannyasendiri.Untuk pencarian dan sorting yang lebih cepat,umumnyadigunakan pohon biner terurut. Namun, apakahpencarianpada pohon biner terurut ini sudah merupakan yangpalingmangkus? Jika tidak, maka jenis pohon apa yanglebihefisien dalam hal pencarian ini? Pencarianmenggunakanstruktur data pohon inilah yang akan kita bahaslebihlanjut dalam makalahini.II. TEORIDASARGraf4035425864235Graf G didefinisikan sebagai pasangan himpunan(V,E)ditulis dengan notasi G=(V,E), yang dalam hal iniVadalah himpuynan tidak-kosong dari simpul-simpul(vertices atau node) dan E adalah himpunan sisi(edgesatauarcs)yangmenghubungkansepasangsimpul.Gambar 1 Graf dengan 6 Simpul dan 7sisiSebagai contoh definisi dari graf pada gambar 1 di atasadalah:U=(1,2,3,4,5,6}E —— [(1,2),(1,5),(2,3),(3,4),(4,5),(5,2),(4,6)}Berdasarkan orientasi arah pada sisi, maka secaraumumgraf dibedakan atas 2jenis:Graftak-berarahGraf tak berarah adalah graf yang sisinyatidakmemiliki orientasi arah. Graf pada gambar 1 diatasmerupakan salah satu contoh dari graftak-berarah.Pada grak tak-berarah, sisi (u,v) dan (v,u)adalahsama.GrafberarahGraf berarah adalah graf yang setiapsisinyadiberikan orientasi arah. Untuk busur (u,v) padagrakberarah, simpul u diamakan simpul asal dan simpulvdinamakan simpulterminal.Dalam graf, dikenal juga istilah lintasan. Lintasanyangpanjangnya n dari simpul awal ve ke simpul tujuan v,didalam graf G ialah barisan berselang-selingsimpul-simpuldansisi—sisiyangberbentukw. e,v,e,v,...,v,.,e„ v, sedemikian sehiflgga eJ (V , Vl). 2 '(VJ,V2)., e, — (v,-i v,) adalah sisi—sisi dari graf G. Lintasanyang berasal dan berakhir pada simpul yang samadisebutsirkuit atau siklus. Suatu graf tak-berarah G disebutgrafterhubung jika untuk setiap simpul u dan v didalamhimpunan V terdapat lintasan dari u ke v (yangjugaberarti ada lintasan dati v ke u). Jika tidak, maka Gdisebut graftak-terhubung.PohonPohon adalah graf tak-berarah terhubung yang tidakmengandung sirkuit.Gambar 2PohonMisalkan G = (V, E) adalah graf tak-berarahsederhanadan jumlah simpulnya n. Maka, semua pernyataandibawah ini adalahekivalen:G adalahpohon.Gambar 3 PohonBerakarBerikut beberapa istilah yang berkaitan denganpohonberakar:Anak danOrangtuaJika p merupakan simpul pada pohon berakar danqmerupakan anak dari simpul p (p orangtua dariq),maka harus terdapat sisi dari simpul p ke q.Padagambar 3, simpul A merupakan orangtua darisimpulB, C, D. Dengan kata lain, simpul B, C,Dmerupakan anak-anak dariA.LintasanLintasan antara simpul v dan simpul vadalahruntutan simpul-simpul dari v e v3, v4 vbsedemikian rupa sehingga v, adalah orang tuadari«+ › untuk 1 < i <k.KeturunanJika x dan y adalah simpul dalam pohon danterdapatlintasan dari x ke y, maka x disebut leluhur dari ydan y adalah keturunan darix.SaudarakandungSimpul-simpul yang memiliki orang tua yangsamadisebut saudarakandung.UpapohonYang dimaksud dengan upapohon dengan x(xadalah simpul di dalam pohon) sebagai akarnya ialahSetiap pasang simpul di dalam G terhubung denganupagraf T’) sedemikian sehingga V’lintasantunggal.G terhubung dan memiliki m = n-1 buahsisi.G tidak mengandung sirkuit dan memiliki m =n-1buahsisi.G tidak mengandung sirkuit dan penambahansatrusisipadagrafakanmembuathanyasatusirkuitG terhubung dan semua sisinya adalahjembatan(jembatan adalah sisi yang biladihapusmenyebabkan graf terpecah menjadiduakomponen.)Pohon yang sebuah simpulnya dianggap sebagaiakarsisi-sisinya diberi arah menjauh dari akardinamakanpohonberakar.mengandung x dan semua keturunannya danE’mengandung sisi-sisi dalam semua lintasanyangberasal darix.6.DerajatDerajat adalah jumlah anak pada simpultersebut.Simpul D pada gambar 3 berderajat 2 karenasimpulD memiliki 2 anak yaitu E danF.7. DaunDaun adalah simpul yang berderajat 0. Pohonpadagambar 3 memiliki 4 daun, yaitu B, C, G,F.8 SimpulDalamSimpuldalamialahsimpulyangmemilikianak.Aras atauTingkatJika akar dianggap memiliki aras = 0, makaarassimpullainnyamerupakanpanjanglintasandariakarke simpultersebut.Tinggi (height) atau Kedalaman(depth)Makalah II2092 Probabilitas dan Statistik — Sem. I Tahun2010/2011iVBORw0KGgoAAAANSUhEUgAAACUAAAAZCAIAAAA5RHCCAAAAAXNSR0IArs4c6QAAAARnQU1BAACx
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