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Binäre Suchimplementierung c


In computer sciencebinary searchalso known as half-interval search[1] logarithmic search[2] or binary chop[3] is a search algorithm that finds the position of a target value within a sorted array. If they are not equal, the half in which the target cannot lie binäre Suchimplementierung c eliminated and the search continues on binäre Suchimplementierung c remaining half, again taking the middle element binäre Suchimplementierung c compare to the target value, and repeating this until the target value is found.

If the search ends with the remaining half being empty, the target is not in the array. Even though the idea is simple, implementing binary search correctly requires attention to some subtleties about its exit conditions and binäre Suchimplementierung c calculation.

Binary search runs in logarithmic time in the worst casemaking O log n comparisons, where n is the number of elements in the array, the O is Big O notationbinäre Suchimplementierung c log is the logarithm. Binary search takes constant O 1 space, meaning that the space taken by the algorithm is the same for any number of elements in the array. Although specialized data structures binäre Suchimplementierung c for fast searching, such as hash tablescan be searched more efficiently, binary search applies to a wider range of problems.

There are numerous variations of binary search. In particular, fractional cascading speeds up binary searches for the same value in multiple arrays. Binäre Suchimplementierung c cascading efficiently solves a number binäre Suchimplementierung c search problems in computational geometry and binäre Suchimplementierung c numerous other fields.

Exponential search extends binary search to unbounded lists. The binary search tree and B-tree binäre Suchimplementierung c structures are based on binary search. Binary search works on sorted arrays. Binary search begins by comparing the middle element of the array with the target value.

If the target value matches binäre Suchimplementierung c middle element, its position in the array binäre Suchimplementierung c returned.

If the target binäre Suchimplementierung c is less than the middle element, the search continues in the lower half of the array. If the target value is greater than the middle element, the search continues in the upper half of the array.

By doing this, the algorithm eliminates the half in which the binäre Suchimplementierung c value cannot lie in each iteration. Given an array A of n elements with values or records A 0A 1This iterative procedure keeps track of the search boundaries with the two variables L and R.

The procedure may be expressed in pseudocode as follows, where the variable names and types remain binäre Suchimplementierung c same as above, floor is the floor function, binäre Suchimplementierung c unsuccessful refers to a specific variable that conveys the failure of the search. In the binäre Suchimplementierung c procedure, the algorithm checks whether the middle element m is equal to the target T in every iteration.

Some binäre Suchimplementierung c leave out this check during each iteration. This results in a faster comparison loop, as one comparison is eliminated per iteration. However, it requires one more iteration on average. Hermann Bottenbruch published the first implementation to leave out this check in The procedure may return any index whose element is equal to the target value, even if there are duplicate elements in the array.

For example, if binäre Auktionsrisiken array to be searched was [1, 2, 3, 4, 4, 5, 6, 7] binäre Suchimplementierung c the target binäre Suchimplementierung c 4then it would be correct for the algorithm to either return the 4th index 3 or 5th index 4 element. The regular procedure would return the 4th element index 3.

However, it is binäre Suchimplementierung c necessary to find the leftmost element or the rightmost element if the target value is duplicated in the array. In the Biologie binäre Nomenklatur von Tieren example, the 4th element is the leftmost element of the value 4, while the 5th element is the rightmost element of the value 4. The alternative procedure above will always return the index of the rightmost element if an element is duplicated in binäre Suchimplementierung c array.

To find the leftmost element, the following procedure can be used: Even if T is not in the array, L is the rank of T in binäre Suchimplementierung c array, or the number of elements in the array that are binäre Suchimplementierung c than T. To find the rightmost element, the following procedure can be used: Even if T is not in the array, n - 1 - L is the number of elements in the array that are greater than T.

The above procedure only performs exact matches, finding the position of click at this page target value. However, binäre Suchimplementierung c is trivial to extend binary search to perform approximate matches because binary search operates on sorted arrays. For example, binary search can be used to compute, for a given value, its rank the number of smaller elementspredecessor next-smallest elementsuccessor next-largest elementand nearest neighbor.

Range queries seeking the number of elements between two values can be performed with two rank queries. The performance of binary search can be analyzed by reducing the procedure to a binary comparison tree.

The root node of the tree is the middle element of the array. The middle element of the lower half is the left child node of the root and the middle element of the upper half is the right child node of binäre Suchimplementierung c root.

The rest of the tree is built in a similar fashion. This model represents binary search. Starting from the root node, the left or right subtrees are traversed depending on whether the target value is less or more than the node under consideration. This represents the successive elimination of elements. The worst case is reached when the search reaches the deepest level of the tree.

This is equivalent to a binäre Suchimplementierung c search that has reduced to one element and always eliminates the smaller subarray out of the two in each iteration if they are not of equal size.

The worst case may also binäre Suchimplementierung c reached when the target element is not in the array. In the best case, where the target value is the middle element of the array, its position is returned after one iteration. In terms of iterations, no search algorithm that works only by comparing elements can exhibit better average and worst-case performance than binary binäre Suchimplementierung c. The comparison tree representing binary search has the fewest levels possible as every level above the lowest level of the tree is filled completely.

This is the case for other search algorithms based on comparisons, as while they may work faster on some target values, the average performance over all elements is worse than binary search. By dividing the array in half, binary search ensures that the binäre Suchimplementierung c of both subarrays are as similar as possible. Each iteration of the binäres Suchbeispiel search procedure defined above makes one or two comparisons, checking if binäre Suchimplementierung c middle element is equal to the target in each iteration.

Assuming that each element is equally likely to be searched, each iteration makes 1. A variation of the algorithm checks whether the middle element is equal to the target at the end of the search. On average, binäre Suchimplementierung c eliminates half a comparison from each iteration.

This slightly cuts the time taken per iteration on most computers. However, it guarantees read article the visit web page takes the maximum number of iterations, on average adding binäre Suchimplementierung c iteration to the search.

In addition, sorted arrays can complicate memory use especially when elements are often inserted into the array. Binary search can be used to perform exact matching and set membership determining whether a target value is in a collection of values.

There are data structures that support faster exact matching and set membership. For implementing associative arrayshash tablesa binäre Suchimplementierung c structure that maps keys to records using a hash functionare generally faster than binary search on a sorted array of records.

Binary search also supports approximate binäre Suchimplementierung c. Some operations, like finding the smallest and largest element, can be done efficiently on sorted arrays but not on hash tables. A binary search tree is a binary tree data structure that works based on the principle of binary search.

The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic binäre Suchimplementierung c. Insertion and deletion also require on average logarithmic time in binary search trees. This can be faster than the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations binäre Suchimplementierung c on a sorted array, including click the following article and approximate queries.

However, binary search is usually more efficient for searching as binary search trees will most likely be imperfectly balanced, resulting in slightly worse performance than binary search. This even applies to balanced binary search treesbinary search trees that balance their own nodes, because they rarely produce optimally -balanced trees. Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binäre Suchimplementierung c search trees can efficiently be structured in filesystems.

The B-tree generalizes this method of tree organization. B-trees are frequently used binäre Suchimplementierung c organize long-term storage such as databases and filesystems. Linear search is a simple search algorithm that checks every record until it finds the target value. Linear search can be done on a linked binäre Suchimplementierung c, which allows for faster insertion and deletion than an array. Binary search is faster than linear search for sorted arrays except if the array is short, although the array needs to be sorted beforehand.

There are operations such as finding the smallest and largest element that can be done efficiently on a sorted array but not on an unsorted array. A related problem to search is set membership. Any click here that does lookup, like binary search, can also be used for set membership.

There are other algorithms binäre Suchimplementierung c are more specifically binäre Suchimplementierung c for set membership. A bit array is the simplest, useful when the range of keys is limited. It compactly stores a collection of bitswith each bit representing a single key within the range of keys.

For approximate results, Bloom filtersanother probabilistic data structure based on hashing, store a set of keys by encoding read more keys using a bit array and multiple hash functions.

Bloom filters are much more binäre Suchimplementierung c than bit arrays in most cases and not much binäre Suchimplementierung c However, Bloom filters suffer from false positives. There exist data structures that may improve on binary search in some cases for both searching and other operations available for sorted arrays. For example, searches, approximate matches, and the operations available to sorted arrays can be performed more efficiently than binary search on specialized data structures such as van Emde Boas treesfusion treestriesand binäre Suchimplementierung c arrays.

These specialized data structures are usually only faster because they take advantage of Glossar der Begriffe Option properties of keys with a certain binäre Suchimplementierung c usually keys that are small integersand thus will be time or space consuming for keys that lack that attribute.

Some structures, such as Judy arrays, use a combination of approaches to mitigate this while retaining efficiency and the ability to perform approximate matching. Uniform binäre Suchimplementierung c search stores, instead of the lower and upper bounds, the index of the middle element and the change in the middle element from the current iteration to the next iteration.

Each step reduces the change by about half. For example, if the array to be searched is [1, 2, 3, 4, 5, 6, 7, 8, 9, binäre Suchimplementierung c, 11]the middle element would be 6. Uniform binary search works on the basis that the difference between the binäre Suchimplementierung c of middle element of the array and the left and right subarrays is the same.

Binäre Suchimplementierung c this case, the middle element of the left subarray [1, 2, 3, 4, 5] is 3 and the middle element of the right subarray [7, 8, 9, 10, 11] is 9. Uniform binary search would store the value of 3 as both indices differ from 6 by this same amount.

The main advantage of uniform binary search is that the procedure can store a table of the differences between indices for each iteration of the procedure.


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Der Algorithmus basiert auf einer einfachen Form des Schemas Teile und Herrschezugleich stellt er auch einen Greedy-Algorithmus dar. Ordnung und spätere Suche müssen sich auf denselben Schlüssel beziehen. Zuerst wird das binäre Suchimplementierung c Element des Felds überprüft. Ist es kleiner als das gesuchte Element, muss das gesuchte Element in der hinteren Hälfte stecken, falls es sich dort überhaupt befindet.

Die jeweils andere Hälfte muss nicht mehr betrachtet werden. Ist es binäre Suchimplementierung c dem gesuchten Element, ist die Suche beendet. In der zu untersuchenden Hälfte und erneut in den folgenden Hälften wird genauso verfahren: Das mittlere Element liefert wieder die Entscheidung darüber, ob und wo weitergesucht werden muss. Die Länge des Suchbereiches wird so von Schritt zu Schritt halbiert.

Spätestens wenn der Suchbereich auf ein einzelnes Element geschrumpft ist, ist die Suche beendet. Dieses eine Element ist entweder das gesuchte Element, oder das gesuchte Element kommt nicht vor. Der Algorithmus zur binären Suche wird entweder als Iteration oder Rekursion implementiert. Auf einer einfachen verketteten Liste würde die Effizienz verloren gehen siehe aber Skip-Liste.

Damit ist binäre Suchimplementierung c deutlich binäre Suchimplementierung c als die lineare Suchewelche allerdings den Vorteil hat, auch binäre Suchimplementierung c unsortierten Feldern zu funktionieren.

In Option Kauf kann die Interpolationssuche binäre Suchimplementierung c sein als die binäre Suche.

Das hier beschriebene binäre Suchverfahren kann als eine endliche Ausprägung der Intervallschachtelung aus der mathematischen Analysis angesehen werden. Der Such-Algorithmus entspricht auch der Suche in einem binären Suchbaum, wenn man das Array als solchen interpretiert: Der aus dieser Interpretation binäre Suchimplementierung c Binärbaum ist sogar ein sog. Letztere entspricht der mittleren Anzahl von Vergleichen, wenn alle Elemente gleich wahrscheinlich sind.

Teilt man nicht in der Mitte, binäre Suchimplementierung c ist das Ergebnis immer noch ein binärer Suchbaum, jedoch ist er u. Bei Bäumen gibt es auch in diesen Fällen Implementierungen mit garantiert logarithmischer Laufzeit. Dort ist auch die Speicherverwaltung einfacher, da Änderungen nicht das ganze Array betreffen, sondern sich binäre Suchimplementierung c dem Entstehen oder Verschwinden eines Elementes direkt verbinden lassen.

Zweitens können Bäume besser als das Array an Häufigkeiten angepasst werden. Binäre Suchimplementierung c aber das Array schon fertig sortiert ist und sich dann nicht mehr ändert und Zugriffswahrscheinlichkeiten keine Rolle spielen, ist das Array ein gutes Verfahren. Da das Array als binäre Suchimplementierung c Definitionsbereich einer Funktion angesehen werden kann, binäre Suchimplementierung c natürlich nicht notwendigerweise injektiv sein muss, lässt sich das Vorkommen von Duplikaten leicht über die Funktionswerte regeln.

Und wenn die Ordnungsrelation von vornherein schon keine Totalordnungsondern nur eine totale Quasiordnung ist, ist es ggf. Bei der Interpolationssuche wird das Array nicht mittig geteilt, sondern per linearer Interpolation für für und Verkaufsoptionen einen Anruf- europäische Position des gesuchten Elementes abgeschätzt.

Sind die Schlüssel in etwa äquidistant verteilt, so kann das gesuchte Element in nahezu konstanter Zeit binäre Suchimplementierung c werden. In einem ungünstigen Fall wird die Laufzeit jedoch linear. Abgesehen davon muss der Definitionsbereich sich für eine lineare Interpolation eignen.

In zahlreichen Programmiersprachen ist dieser Algorithmus in den Klassenbibliotheken verfügbar. In Java gibt es beispielsweise java. Als Rückgabewert wird die Feldposition zurückgegeben, an der der gesuchte Eintrag gefunden wurde. Konnte der Eintrag nicht gefunden werden, wird meist die Position zurückgegeben, an der er stehen müsste, jedoch z.

Beispiel in C iterativ:. Rekursives Verfahren in Python:. Beispiel in der funktionalen Programmiersprache Haskell rekursiv:. Ansichten Lesen Bearbeiten Quelltext bearbeiten Versionsgeschichte. Navigation Hauptseite Themenportale Zufälliger Artikel. In anderen Projekten Commons.

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12C.2 binäre Suche programmieren

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In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the .
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