{"id":3945,"date":"2025-03-12T19:57:08","date_gmt":"2025-03-12T18:57:08","guid":{"rendered":"https:\/\/maximini.eu\/work\/?p=3945"},"modified":"2025-03-13T14:16:15","modified_gmt":"2025-03-13T13:16:15","slug":"bubble-sort","status":"publish","type":"post","link":"https:\/\/maximini.eu\/work\/bubble-sort\/","title":{"rendered":"Bubble-Sort"},"content":{"rendered":"

Bubble-Sort<\/a> ist ein Sortieralgorithmus, der eine Liste von Elementen sortiert, indem er benachbarte Elemente vergleicht und gegebenenfalls vertauscht.<\/p>\n

Schrittfolge des Algorithmus <\/h2>\n

Der Algorithmus beginnt beim ersten Element eines unsortierten Arrays<\/a> und vergleicht die benachbarten Elemente von links nach rechts. Falls das linke Element gr\u00f6\u00dfer ist als das rechte werden beide vertauscht. Beim Tauschen wird das zu tauschende Element in der tempor\u00e4ren Hilfsvariable t zwischengespeichert. Die gr\u00f6\u00dferen Elemente steigen im Array auf wie Blasen (Bubbles).Der Algorithmus durchl\u00e4uft zwei verschachtelte Schleifen. Die erste For-Schleife<\/a> mit der Z\u00e4hlvariable i iteriert \u00fcber die Zeilen des Arrays und l\u00e4uft bis zum letzten Element (n – 1). Die zweite For-Schleife mit der Z\u00e4hlvariablen j iteriert innerhalb jeder Zeile \u00fcber die Spalten. Nach n -1 Durchl\u00e4ufen sind alle Elemente im Array sortiert.<\/p>\n

******** Pseudocode Bubble-Sort *********\r\n\r\nInput: unsorted array A\r\nOutput: sorted array A\r\n\r\nFor i = 1 to n-1 do             \/\/ \u00c4u\u00dfere Schleife\r\n  For j = 1 to n-i do           \/\/ Innere Schleife\r\n     if A[j-1]> A[j] then\r\n           t = A[j]\r\n           A[j] = A[j-1]\r\n           A[j-1] = t\r\n     End if\r\n   End For\r\nEnd For\r\nreturn A\r\n<\/b><\/pre>\n
Komplexit\u00e4t des Algorithmus (Laufzeitanalyse)<\/strong><\/h2>\n
Best-Case: O(n2<\/sup>)<\/span><\/strong>, Worst Case: O(n2<\/sup>)<\/span><\/strong>, Average Case: O(n2<\/sup>)<\/span><\/strong><\/h3>\n
Der Bubble-Sort-Algorithmus hat eine konstante Speicherplatzkomplexit\u00e4t O(1). Unabh\u00e4ngig davon, wie viele Elemente die Liste enth\u00e4lt, ben\u00f6tigt er immer eine konstante Menge an zus\u00e4tzlichem Speicher f\u00fcr die Schleifenvariablen und eine tempor\u00e4re Hilfsvariable. Seine Laufzeitkomplexit\u00e4t ist aber in allen drei F\u00e4llen quadratisch O(n\u00b2), da er n-mal die \u00e4u\u00dfere und n-mal die innere Schleife<\/a> durchl\u00e4uft (n \u00b7 n = n\u00b2).<\/p>\n
\n
Beispiel: Bubble-Sort-Algorithmus im Array A<\/h2>\n
Der Algorithmus sortiert die sechs Zahlen des Arrays A [ 11, 13, 4, 17, 8, 2 ] in aufsteigender Reihenfolge.<\/p>\n
Initialisierung
\n<\/b>Array A [ 11, 13, 4, 17, 8, 2 ], Anzahl Elemente n = 6 
\nZ\u00e4hlvariable i = 1
\nZ\u00e4hlvariable j = 1
\nHilfsvariable t = 0<\/p>\n
\n
Erster Durchlauf, i = 1, Array A [ 11, 13, 4, 17, 8, 2 <\/span>]<\/strong>
\nIm ersten Durchlauf wird die gr\u00f6\u00dfte Zahl 17 in f\u00fcnf Schritten (n – 1) ans Ende des Arrays gebubbelt. Der Algorithmus vergleicht f\u00fcnfmal das linke Element A [ j – 1] mit dem rechten Element A [ j ] und f\u00fchrt drei Vertauschungen durch.<\/p>\n
\n
erster Schritt <\/strong>(j = 1, t = 0)
\nArray A [11<\/strong><\/span>, 13<\/span><\/strong>, 4, 17, 8, 2 ]
\nVergleich A [ j – 1] = A[0] mit A[ j ] = A[1]: 11 < 13, kein Tausch<\/p>\n
zweiter Schritt <\/strong>(j = 2, t = 0)
\nArray A [11,<\/span> 13<\/strong><\/span>, 4<\/span><\/strong>, 17, 8, 2<\/span>] 
\nVergleich A [ j – 1] = A[1] mit A[ j ] = A[2]:  13<\/strong><\/span> ><\/strong> 4<\/span><\/strong>, 1. Tausch<\/strong>
\nt = A[ j ] = A[2] = 4<\/span>
\nA[ j ] = A[ j – 1] = A[2 – 1] = A[1] = 13 = t<\/span><\/p>\n
dritter Schritt <\/strong>(j = 3, t = 13)
\nArray A [11,<\/span> 4, 13<\/span><\/strong>, 17<\/span><\/strong>, 8, 2<\/span>]  
\nVergleich A [j -1] = A[2] mit A[j] = A[3]:  13 < 17<\/span>, kein Tausch<\/p>\n
vierter Schritt<\/strong>(j = 4, t = 13)
\nArray A [11,<\/span> 4, 13, 17<\/span><\/strong>, 8<\/span><\/strong>, 2<\/span>]  
\nVergleich A [j -1] = A[3] mit A[j] = A[4]:  17<\/span><\/strong><\/span> ><\/strong> 8<\/span><\/strong><\/span>, 2. Tausch<\/strong>
\nt = A[j] = A[4] = 8<\/span>
\nA[j] = A[ j – 1] = A[4 – 1] = A[3] = 17 = t<\/span><\/p>\n
f\u00fcnfter Schritt <\/strong>(j = 5, t = 17)
\nArray A [11,<\/span> 4, 13, 8, 17<\/span><\/strong>, 2<\/span><\/strong><\/span>]  
\nVergleich A [j -1] = A[4] mit A[j] = A[5]:  17<\/span><\/strong><\/span> ><\/strong> 2<\/span><\/strong><\/span>, 3. Tausch<\/strong>
\nt = A[j] = A[5] = 2<\/span>
\nA[j] = A[j-1] = A[5 – 1] = A[4] = 17 = t<\/span><\/p>\n
\n
Zweiter Durchlauf, i = 2, Array A [ 11, 4, 13, 8, 2, <\/span> 17 ]
\n<\/strong>Im zweiten Durchlauf wird die zweitgr\u00f6\u00dfte Zahl 13 in f\u00fcnf Schritten (n – 1) nach hinten gebubbelt. Der Algorithmus vergleicht f\u00fcnfmal das linke Element A [ j – 1] mit dem rechten Element A [ j ] aus und f\u00fchrt drei Vertauschung durch.<\/p>\n
\n
erster Schritt<\/strong> (j = 1, t = 17)
\nArray A [11<\/span><\/strong>, 4<\/span><\/strong>, 13, 8, 2, 17<\/span>]  
\nVergleich A [j-1] = A[0] mit A[j] = A[1]:  11 <\/span>><\/strong> 4<\/span><\/strong>, 4. Tausch<\/strong>
\nt = A[j] = A[1] = 11
\nA[j] = A[j-1] = A[1 – 1] = A[0] = 11 = t<\/p>\n
zweiter Schritt <\/strong>(j = 2, t = 11)
\nArray A [4, 11<\/span><\/strong>, 13<\/span><\/strong>, 8, 2, 17<\/span>] 
\nVergleich A [j -1] = A[1] mit A[j] = A[2]:  11 < 13, kein Tausch<\/p>\n
dritter Schritt <\/strong>(j = 3, t = 11)
\nArray A [4, 11, 13<\/span><\/strong>,<\/span> 8<\/span><\/strong>, 2, 17<\/span>]  
\nVergleich A [j -1] = A[2] mit A[j] = A[3]:  13 ><\/strong> 8, 5. <\/strong>Tausch<\/strong>
\nt = A[j] = A[3] = 8
\nA[j] = A[j-1] = A[3 – 1] = A[2] = 13 = t<\/p>\n
vierter Schritt<\/strong> (j = 4, t = 13)
\nArray A [4, 11, 8, 13<\/span><\/strong>, <\/span>2<\/span><\/strong>, 17<\/span>]  
\nVergleich A [j -1] = A[3] mit A[j] = A[4]:  13 ><\/strong> 2, 6. <\/strong>Tausch<\/strong>
\nt = A[j] = A[4] = 2
\nA[j] = A[j-1] = A[4 – 1] = A[3] = 13 = t<\/p>\n
f\u00fcnfter Schritt <\/strong>(j = 5, t = 13)
\nArray A [4, 11, 8, 2, 13<\/span><\/strong>, <\/span>17<\/span><\/strong>]  
\nVergleich A [j -1] = A[4] mit A[j] = A[5]:  13 < 17, kein Tausch<\/p>\n
\n
Dritter Durchlauf, i = 3, Array A [ 4, 11, 8, 2,<\/span> 13, 17 ]
\n<\/strong>Im dritten Durchlauf wird die drittgr\u00f6\u00dfte Zahl 11 in f\u00fcnf Schritten (n – 1) nach hinten gebubbelt. Der Algorithmus f\u00fchrt f\u00fcnf Vergleiche zwischen linkem A [ j – 1] und rechtem Element A [ j ] ) sowie zwei Vertauschungen durch.<\/p>\n
\n
erster Schritt<\/strong> ( j = 1, t = 13)
\nArray A [4<\/span><\/strong>, 11<\/span><\/strong>, 8, 2, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[0] mit A[ j ] = A[1]:  4 < 11, kein Tausch<\/p>\n
zweiter Schritt <\/strong>(j = 2, t = 13)
\nArray A [4, 11<\/span><\/strong>, 8<\/span><\/strong>, 2, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[1] mit A[ j ] = A[2]:  11 <\/span><\/strong><\/span>><\/strong> 8<\/span><\/strong><\/span>, 7. <\/strong>Tausch<\/strong>
\nt = A[ j ] = A[2] = 8
\nA[ j ] = A[ j – 1] = A[2 – 1] = A[1] = 11 = t<\/p>\n
dritter Schritt<\/strong> ( j = 3, t = 11)
\nArray A [4, 8, 11<\/span><\/strong>, 2<\/span><\/strong>, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[2] mit A[ j ] = A[3]:  11 <\/span><\/strong><\/span>><\/strong> 2<\/span><\/strong><\/span>, 8. <\/strong>Tausch<\/strong>
\nt = A[ j ] = A[3] = 2
\nA[ j ] = A[ j – 1] = A[3 – 1] = A[2] = 11 = t<\/p>\n
vierter Schritt <\/strong>(j = 4, t = 11)
\nArray A [4, 8, 11, 2<\/span><\/strong>, 13<\/span><\/strong>, 17] <\/span>
\nVergleich A [ j – 1] = A[3] mit A[ j ] = A[4]:  2 < 13, kein Tausch<\/p>\n
f\u00fcnfter Schritt<\/strong> (j = 5, t = 11)
\nArray A [4, 8, 2, 11, 13<\/span><\/strong>, 17<\/span><\/strong>] <\/span>
\nVergleich A [ j – 1] = A[4] mit A[ j ] = A[5]: 13 < 17, kein Tausch<\/p>\n
\n
Vierter Durchlauf, i = 4, Array A [ 4, 8, 2,<\/span> 11, 13, 17 ]
\n<\/strong>Im vierten Durchlauf wird die viertgr\u00f6\u00dfte Zahl 8 in f\u00fcnf Schritten (n – 1) nach hinten gebubbelt. Der Algorithmus f\u00fchrt wieder f\u00fcnf Vergleiche und zwei Vertauschungen durch.<\/p>\n
\n
erster Schritt <\/strong>( j = 1, t = 13)
\nArray A [4<\/span><\/strong>, 8<\/span><\/strong>, 2, 11, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[0] mit A[ j ] = A[1]:  4 < 8, kein Tausch<\/p>\n
zweiter Schritt <\/strong>(j = 2, t = 13)
\nArray A [4, 8<\/strong><\/span>, 2<\/strong><\/span>, 11<\/span>, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[1] mit A[ j ] = A[2]:  8<\/strong><\/span><\/span> ><\/strong> 2<\/strong><\/span><\/span>, 9. <\/strong>Tausch<\/strong>
\nt = A[ j ] = A[2] = 8
\nA[ j ] = A[ j – 1] = A[2 – 1] = A[1] = 11 = t<\/p>\n
dritter Schritt<\/strong> ( j = 3, t = 11)
\nArray A [4, 8, 11<\/span><\/strong>,<\/span> <\/b>2<\/span><\/strong>, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[2] mit A[ j ] = A[3]:  11<\/span><\/strong><\/span> ><\/strong> 2<\/span><\/strong><\/span>, 10. <\/strong>Tausch<\/strong>
\nt = A[ j ] = A[3] = 2
\nA[ j ] = A[ j – 1] = A[3 – 1] = A[2] = 11= t<\/p>\n
vierter Schritt <\/strong>( j = 4, t = 11)
\nArray A [4, 8, 2, 11<\/span><\/strong>,<\/span> 13<\/strong><\/span>, 17] <\/span>
\nVergleich A [ j – 1] = A[3] mit A[ j ] = A[4]:  11 < 13, kein Tausch<\/p>\n
f\u00fcnfter Schritt <\/strong>( j = 5, t = 11)
\nArray A [4, 8, 2, 11,<\/span> 13<\/span><\/strong>, 17<\/span><\/strong>] <\/span>
\nVergleich A [ j – 1] = A[4] mit A[ j ] = A[5]:  13 < 17, kein Tausch<\/p>\n
\n
F\u00fcnfter Durchlauf, i = 4, Array A [ 4, 2,<\/span> 8, 11, 13, 17 ]
\n<\/strong>Im letzten Durchlauf wird die f\u00fcnftgr\u00f6\u00dfte Zahl 4 in f\u00fcnf Schritten (n – 1) nach hinten gebubbelt. Der Algorithmus f\u00fchrt wieder f\u00fcnf Vergleiche und noch eine Vertauschung durch.<\/p>\n
\n
erster Schritt <\/strong>( j = 1, t = 11)
\nArray A [4<\/span><\/strong>, 2<\/span><\/strong>, 8, 11, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[0] mit A[ j ] = A[1]:  4<\/span><\/strong><\/span> ><\/strong> 2<\/span><\/strong><\/span> , 11. <\/strong>Tausch
\n<\/strong>t = A[ j ] = A[1] = 2
\nA[ j ] = A[ j – 1] = A[1 – 1] = A[0] = 4 = t<\/p>\n
zweiter Schritt <\/strong>( j = 2, t = 4)
\nArray A [2, 4<\/span><\/strong>, 8<\/span><\/strong>, 11, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[1] mit A[ j ] = A[2]:  4 < 8, kein Tausch
\n<\/strong><\/p>\n
dritter Schritt <\/strong>( j = 3, t = 4)
\nArray A [2, 4, 8<\/strong><\/span>, 11<\/strong><\/span>, 13, 17] <\/span>
\nVergleich A [ j – 1] = A[2] mit A[ j ] = A[3]:  8 < 11, kein Tausch<\/p>\n
vierter Schritt <\/strong>( j = 4, t = 4)
\nArray A [2, 4, 8, 11<\/strong><\/span>, 13<\/strong><\/span>, 17] <\/span>
\nVergleich A [ j – 1] = A[3] mit A[ j ] = A[4]:  11 < 13, kein Tausch<\/p>\n
f\u00fcnfter Schritt <\/strong>( j = 5, t = 4)
\nArray A [2, 4, 8, 11, 13<\/strong><\/span>, 17<\/span><\/strong>] <\/span>
\nVergleich A [ j – 1] = A[4] mit A[ j ] = A[5]:  13 < 17, kein Tausch<\/p>\n
\n
Output  Array A  [ 2, 4, 8, 11, 13, 17 ]<\/strong><\/p>\n
Der Algorithmus terminiert nach 5 (n – 1) Durchl\u00e4ufen und gibt das sortierte Array A [ 2, 4, 8, 11, 13, 17 ]  <\/span>aus. Insgesamt hat er \u00fcber alle Durchg\u00e4nge 25 Vergleiche und 11 Vertauschungen vorgenommen.<\/p>\n
**** Pseudocode Bubble-Sort A [11, 13<\/strong>, 4<\/strong>,<\/span> 17, 8, 2<\/span>] ****\r\n\r\ninput: A  [11, 13<\/strong>, 4<\/strong>,<\/span> 17, 8, 2<\/span>] \r\ndef t = 0\r\nFor i = 1 to n-1 do           \/\/ \u00c4u\u00dfere Schleife\r\n<\/b>  For j = 1 to n-i do         \/\/ Innere Schleife \r\n    if A[j-1] > A[j] then \r\n       t = A[j] \r\n       A[j] = A[j-1] \r\n       A[j-1] = t \r\n    End if \r\n  End For \r\nEnd For \r\nreturn A<\/b><\/pre>\n","protected":false},"excerpt":{"rendered":"
Bubble-Sort ist ein Sortieralgorithmus, der eine Liste von Elementen sortiert, indem er benachbarte Elemente vergleicht und gegebenenfalls vertauscht. Schrittfolge des Algorithmus  Der Algorithmus beginnt beim…<\/p>\n
weiterlesenBubble-Sort<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":3972,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[94,35],"tags":[92,103,102],"class_list":["post-3945","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algorithmen","category-informatik","tag-algorithmen","tag-bubblesort","tag-sortieralgorithmus","entry"],"yoast_head":"\nBubble-Sort - Katja Maximini | work<\/title>\n<meta name=\"description\" content=\"BubbleSort ist ein Sortieralgorithmus, der ein Array sortiert, indem er benachbarte Elemente vergleicht und gegebenenfalls vertauscht.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/maximini.eu\/work\/bubble-sort\/\" \/>\n<meta property=\"og:locale\" content=\"de_DE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bubble-Sort - 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