ترتيب إنتقائي

ترتيب إنتقائي
{{{صورة}}}
بيانات عامّة
قسم: {{{قسم}}} تركيب البيانات: {{{بيانات}}} تعقيد الوقت: {{{وقت}}} تعقيد الفضاء: {{{فضاء}}} مثالي: {{{مثالي}}}

ملف:Selection-Sort-Animation.gif

صورة متحركة توضح عملية الترتيب الإنتقائي

خوارزمية الترتيب الإنتقائي نوع من أنواع خوارزميات الترتيب وبالتحديد خوارزميات الترتيب في المكان ووهذه الخوارزمية من الرتبةO n² وهو ما يجعلها طريقة مثلى في قوائم البيانات الطويلة وعموما هي أسوأ من قرينتها من الترتيب الإدخالي. الترتيب الإنتقائي مشهور بسهولته وكذلك أداءه مقارنة بقرنائه الأكثر تعقيدا خصوصا عند توافر ذاكرة محدودة.

خوارزمية

هذه الخوارزمية تعمل كما يلي:

1. إوجد العنصر الأقل قيمة في القائمة
2. بدل هذا العنصر مع العنصر الأول في القائمة
3. كرر الخطوتان السابقات ولكن هذه المرة إبدأ من العنصر التالي.

عمليا ستكون القائمة منقسمة إلى قسمين الجزء الأول في القائمة سيكون مرتبا بالفعل وهو الجزء الذي يتم بناءه أو بأول عن طريق إضافة العنصر الأصغر في البداية. والجزء الآخر هو الجزء محل عملية التريتب والذي يتضاءل حجمه مع الوقت.

وإليكم مثال على عملية الترتيب حيث نقوم بترتيب خمسة عناصر

64 25 12 22 11

11 25 12 22 64

11 12 25 22 64

11 12 22 25 64

11 12 22 25 64   

(نلاحظ عدم وجود تغيير لإن أو رقمين هم بالفعل أصغر رقمين)

يمكن استخدام القائمة المترابطة linked list من أجل إضافة وحذف أسرع وعلى سبيل المثال:

64 25 12 22 11

11 64 25 12 22

11 12 64 25 22

11 12 22 64 25

11 12 22 25 64

<source lang="C"> /* a[0] to a[n-1] is the array to sort */ int iPos; int iMin;

/* advance the position through the entire array */ /* (could do iPos < n-1 because single element is also min element) */ for (iPos = 0; iPos < n; iPos++) {

   /* find the min element in the unsorted a[iPos.. n-1] */

   /* assume the min is the first element */
   iMin = iPos;
   /* test against all other elements */
   for (i = iPos+1; i < n; i++) {
       /* if this element is less, then it is the new minimum */  
       if (a[i] < a[iMin]) {
           /* found new minimum; remember its index */
           iMin = i;
       }
   }

   /* iMin is the index of the minimum element. Swap it with the current position */
   if (iMin != iPos) {
       swap(a, iPos, iMin);
   }

} </source>

Mathematical definition

Let $L$ be a non-empty set and $f:L\to L$ such that $f(L)=L^{\prime }$ where:

1. $L^{\prime }$ is a permutation of $L$,
2. $e_i\le e_{i+1}$ for all $e\in L^{\prime }$ and $i\in \mathbb{\mathbb{N}}$,
3. $f(L)=\{\begin{array}{ll}L,& \text{if }|L|=1\\ \{s\}\cup f(L_s),& \text{otherwise}\end{array}$,
4. $s$ is the smallest element of $L$, and
5. $L_s$ is the set of elements of $L$ without one instance of the smallest element of $L$.

Analysis

Selection sort is not difficult to analyze compared to other sorting algorithms since none of the loops depend on the data in the array. Selecting the lowest element requires scanning all n elements (this takes n − 1 comparisons) and then swapping it into the first position. Finding the next lowest element requires scanning the remaining n − 1 elements and so on, for (n − 1) + (n − 2) + ... + 2 + 1 = n(n − 1) / 2 ∈ Θ(n²) comparisons (see arithmetic progression). Each of these scans requires one swap for n − 1 elements (the final element is already in place).

Comparison to other sorting algorithms

Among simple average-case Θ(n²) algorithms, selection sort almost always outperforms bubble sort and gnome sort, but is generally outperformed by insertion sort. Insertion sort is very similar in that after the kth iteration, the first k elements in the array are in sorted order. Insertion sort's advantage is that it only scans as many elements as it needs in order to place the k + 1st element, while selection sort must scan all remaining elements to find the k + 1st element.

Simple calculation shows that insertion sort will therefore usually perform about half as many comparisons as selection sort, although it can perform just as many or far fewer depending on the order the array was in prior to sorting. It can be seen as an advantage for some real-time applications that selection sort will perform identically regardless of the order of the array, while insertion sort's running time can vary considerably. However, this is more often an advantage for insertion sort in that it runs much more efficiently if the array is already sorted or "close to sorted."

While selection sort is preferable to insertion sort in terms of number of writes (Θ(n) swaps versus Ο(n²) swaps), it almost always far exceeds (and never beats) the number of writes that cycle sort makes, as cycle sort is theoretically optimal in the number of writes. This can be important if writes are significantly more expensive than reads, such as with EEPROM or Flash memory, where every write lessens the lifespan of the memory.

Finally, selection sort is greatly outperformed on larger arrays by Θ(n log n) divide-and-conquer algorithms such as mergesort. However, insertion sort or selection sort are both typically faster for small arrays (i.e. fewer than 10–20 elements). A useful optimization in practice for the recursive algorithms is to switch to insertion sort or selection sort for "small enough" sublists.

Variants

Heapsort greatly improves the basic algorithm by using an implicit heap data structure to speed up finding and removing the lowest datum. If implemented correctly, the heap will allow finding the next lowest element in Θ(log n) time instead of Θ(n) for the inner loop in normal selection sort, reducing the total running time to Θ(n log n).

A bidirectional variant of selection sort, called cocktail sort, is an algorithm which finds both the minimum and maximum values in the list in every pass. This reduces the number of scans of the list by a factor of 2, eliminating some loop overhead but not actually decreasing the number of comparisons or swaps. Note, however, that cocktail sort more often refers to a bidirectional variant of bubble sort.

Selection sort can be implemented as a stable sort. If, rather than swapping in step 2, the minimum value is inserted into the first position (that is, all intervening items moved down), the algorithm is stable. However, this modification either requires a data structure that supports efficient insertions or deletions, such as a linked list, or it leads to performing Θ(n²) writes.

In the bingo sort variant, items are ordered by repeatedly looking through the remaining items to find the greatest value and moving all items with that value to their final location. Like counting sort, this is an efficient variant if there are many duplicate values. Indeed, selection sort does one pass through the remaining items for each item moved. Bingo sort does two passes for each value (not item): one pass to find the next biggest value, and one pass to move every item with that value to its final location. Thus if on average there are more than two items with each value, bingo sort may be faster.

References

• Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison–Wesley, 1997. ISBN 0-201-89685-0. Pages 138–141 of Section 5.2.3: Sorting by Selection.
• Anany Levitin. Introduction to the Design & Analysis of Algorithms, 2^nd Edition. ISBN 0-321-35828-7. Section 3.1: Selection Sort, pp 98–100.
• Robert Sedgewick. Algorithms in C++, Parts 1–4: Fundamentals, Data Structure, Sorting, Searching: Fundamentals, Data Structures, Sorting, Searching Pts. 1–4, Second Edition. Addison–Wesley Longman, 1998. ISBN 0-201-35088-2. Pages 273–274

وصلات خارجية

• Animated Sorting Algorithms: Selection Sort – graphical demonstration and discussion of selection sort

قالب:Sorting

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