Overview of Sorting Algorithms

Unit 7 Sorting Algorithms „ Simple Sorting algorithms „ Quicksort „ Improving Quicksort Overview of Sorting Algorithms „ Given a collection of items we want to arrange them in an increasing or decreasing order. „ You probably have seen a number of sorting algorithms including ¾ selection sort ¾ insertion sort ¾ bubble sort ¾ quicksort ¾ tree sort using BST's „ In terms of efficiency: ¾ average complexity of the first three is O(n2) ¾ average complexity of quicksort and tree sort is O(n lg n) ¾ but its worst case is still O(n2) which is not acceptable „ In this section, we ¾ review insertion, selection and bubble sort ¾ discuss quicksort and its average/worst case analysis ¾ show how to eliminate tail recursion ¾ present another sorting algorithm called heapsort Unit 7- Sorting Algorithms 2Selection Sort „ Assume that data ¾ are integers ¾ are stored in an array, from 0 to size-1 ¾ sorting is in ascending order Algorithm for i=0 to size-1 do x = location with smallest value in locations i to size-1 swap data[i] and data[x] end Complexity „ If array has n items, i-th step will perform n-i operations „ First step performs n operations second step does n-1 operations ... last step performs 1 operatio. „ Total cost : n + (n-1) +(n-2) + ... + 2 + 1 = n*(n+1)/2 . 2„ Algorithm is O(n ). Unit 7- Sorting Algorithms 3Insertion Sort Algorithm for i = 0 to size-1 do temp = data[i] x = first location from 0 to i with a value greater or equal to temp shift all values from x to i-1 one location forwards data[x] = temp end Complexity „ Interesting operations: comparison and shift „ i-th step performs i comparison and shift operations „ Total cost : 1 + 2 + ... + (n-1) + n = n*(n+1)/2. „ Algorithm is O(n2). Unit 7- Sorting Algorithms 4Bubble Sort „ n passes „ each pass swaps out of order adjacent elements Algorithm for end = size-1 to 0 do for i = 0 to end-1 do if data[i] > data[i+1] swap data[i] and data[i+1] end end Complexity „ Each step in inside for-loop performs end # of operations. „ Therefore, the total cost of the algorithm is n + (n-1) + ... + 1 = n*(n+1)/2. „ Algorithm is O(n2). Unit 7- Sorting Algorithms 5Tree Sort „ Insert each element in a BST or AVL tree. „ Traverse the tree inorder and place the elements back into the array. Algorithm tree = an empty BST or AVL tree for i = 0 to size-1 do insert data[i] in tree end for i = 0 to size-1 do get the next inorder item in tree store the item in data[i] end Complexity „ Inserting n items in the bst or AVL tree: O(n log(n)), on the average. „ Traversing the tree and inserting back the items in the array : O(n) time. „ Average cost : O(n log(n) + n) = O(n log(n)). „ If BST's are used, worst case cost: O(n2). Problem: „ algorithm needs an additional O(n) space for the tree. Unit 7- Sorting Algorithms 6Quicksort „ Quicksort is a 'Divide and Conquer' (recursive method) Algorithm – pick one element of the array as the pivot – partition the array into two regions: • the left region that has the items less than the pivot • the right region that has the items greater or equal to pivot – apply quicksort to the left region – apply quicksort to the right region Unit 7- Sorting Algorithms 7Quicksort Code typedef long int Item; // It returns the final position of the pivot void quicksort( Item a[], int first, int last ){ int partition( Item a[], int first, int last ) <ul style="display: flex;"><li style="flex:1">int pos; </li><li style="flex:1">// position of pivot </li></ul><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">Item pivot = a[first]; </li></ul>int left = first, right = last; // pivot if (first < last) <ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">// array has more than 1 item </li></ul>pos = partition( a, first, last ); quicksort(a, first, pos-1); quicksort(a, pos+1, last); while (left < right) { // search from right for item <= pivot while ( a[right] > pivot ) right--; } } //search from left for item >= pivot while ( left < right && a[left] <= pivot ) left++; // swap items if left and right have not cross if ( left < right ) { Item tmp =a[left]; a[left] = a[right]; a[right] = tmp; } }// place pivot in correct position a[first] = a[right]; a[right] = pivot; return right; }Unit 7- Sorting Algorithms 8Quicksort – Partition Algorithm „ The partition algorithm do the following: ¾ picks the first item as the pivot item ¾ divides the array into regions – the left region that contains all items that are <= pivot – the right region that contains all items that are > pivot ¾ places the pivot at the right slot (no need to move it any more) ¾ returns the pivot’s final position – after that, quicksort has to recursively sort the left region (the part before the pivot) and the right region (part after the pivot) „ Partition starts searching from the two ends of the array and swaps the items that are in the wrong region. ¾ When the two searches meet, the array is partitioned ¾ Then the pivot is placed in the right position Unit 7- Sorting Algorithms 9Quicksort - Example 1st partition <ul style="display: flex;"><li style="flex:1">initial: </li><li style="flex:1">17 </li></ul>17 32 32 l 68 68 16 16 14 14 15 15 r 44 44 22 22 17 <ul style="display: flex;"><li style="flex:1">15 </li><li style="flex:1">68 </li></ul>l<ul style="display: flex;"><li style="flex:1">16 </li><li style="flex:1">14 </li></ul>r<ul style="display: flex;"><li style="flex:1">32 </li><li style="flex:1">44 </li></ul>44 44 22 22 22 17 <ul style="display: flex;"><li style="flex:1">15 </li><li style="flex:1">14 </li><li style="flex:1">16 </li></ul>r 68 l 32 <ul style="display: flex;"><li style="flex:1">32 </li><li style="flex:1">16 </li></ul>16 15 15 14 14 l r 17 68 2nd partition 14 14 r 15 15 l16 3rd partition 4th partition 68 <ul style="display: flex;"><li style="flex:1">32 </li><li style="flex:1">44 </li><li style="flex:1">22 </li></ul>I r 22 22 r 32 32 l 44 44 68 5th partition 6th partition 22 32 32 r 44 44 lUnit 7- Sorting Algorithms 10 32 44 Quicksort – Time Complexity Average Time ¾ Partition is O(n): it's just a loop through the array ¾ Assuming that we split the array in half each time, we need: – 2 recursive calls of half the array, – 4 of 1/4 of the array, and so on. ¾ So the algorithm has at the most log(n) levels. ¾ Therefore, the average time complexity of the algorithm is O(n log n) Average Space ¾ It can be shown that the max number of frames on the stack is log n ¾ So average space is O(log n) Worst Time ¾ Array is sorted. Partition splits the array into an empty region and a region with n-1 items ¾ Then time = n + (n-1) + … + 2 + 1 which is O(n2) Worst Space ¾It will create n stack frames. So space is O(n) Unit 7- Sorting Algorithms 11 Improvements of Quicksort „ How can we improve the worst time case? ¾ select as pivot the median of the left, right and middle items ¾ significantly reduces the probability of the worst case „ How can we improve the worst space case? ¾ Remove tail recursion (recursive call done as the last operation) and replace it with iteration „ With these improvements, time/space complexity fof Quicksort is: <ul style="display: flex;"><li style="flex:1">Average </li><li style="flex:1">Worst (for time) </li></ul>2time <ul style="display: flex;"><li style="flex:1">O(n lg n) </li><li style="flex:1">O(n ) </li></ul>space O(lg n) O(1) „ Further improvements: ¾ Quicksort is inefficient on small arrays ¾ Stop using Quicksort when partition size is small (i.e. < 50) use insertion sort for this part of the array) Unit 7- Sorting Algorithms 12

Overview of Sorting Algorithms

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