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At the top of the f o r loop, elements 1 through l a s t are strictly less than the pivot, elements l a s t + l through i-1 are greater than or equal to the pivot, and elements i through n-1 have not been examined yet Until v [ i l >= vCO1, the algorithm may swap v [ i ] with itself; this wastes some time but not enough to worry about
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After all elements have been partitioned, element 0 is swapped with the l a s t element to put the pivot element in its final position; this maintains the correct ordering Now the array looks like this:
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The same process is applied to the left and right sub-arrays; when this has finished, the whole array has been sorted How fast is quicksort In the best possible case, the first pass partitions n elements into two groups of about n/2 each the second level partitions two groups, each of about n/2 elements, into four groups each of about n/4 the next level partitions four groups of about n/4 into eight of about n/8 and so on This goes on for about log, n levels, so the total amount of work in the best case is proportional to n + 2xn/2 + 4xn/4 + 8xn/8 (log2n terms), which is nlog2n On the average, it does only a little more work It is customary to use base 2 logarithms; thus we say that quicksort takes time proportional to nlogn This implementation of quicksort is the clearest for exposition, but it has a weakness If each choice of pivot splits the element values into two nearly equal groups our analysis is correct, but if the split is uneven too often, the run-time can grow more like n 2 Our implementation uses a random element as the pivot to reduce the chance that unusual input data will cause too many uneven splits But if all the input values are the same, our implementation splits off only one element each time and will thus run in time proportional to n ' The behavior of some algorithms depends strongly on the input data Perverse or unlucky inputs may cause an otherwise well-behaved algorithm to run extremely slowly or use a lot of memory In the case of quicksort, although a simple implementation like ours might sometimes run slowly, more sophisticated implementations can reduce the chance of pathological behavior to almost zero
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The standard libraries for C and C t e include sort functions that should be robust against adverse inputs, and tuned to run as fast as possible Library routines are prepared to son any data type, but in return we must adapt to their interface, which may be somewhat more complicated than what we showed above In C, the library function is named q s o r t , and we need to provide a comparison function to be called by q s o r t whenever it needs to compare two values Since
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the values might be of any type, the comparison function is handed two voi da pointers to the data items to be compared The function casts the pointers to the proper type, extracts the data values, compares them, and returns the result (negative, zero, or positive according to whether the first value is less than, equal to, or greater than the second) Here's an implementation for sorting an array of strings, which is a common case We define a function scmp to cast the arguments and call strcmp to do the comparison
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/* scmp: s t r i n g compare o f * p l and ap2 a/ i n t scmp(const void a p l , const void *pi!)
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char + v l , av2; v l = *(char a * ) p l ; v2 = *(char a*) p2; r e t u r n strcmp(v1, v2) ;
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We could write this as a one-line function, but the temporary variables make the code easier to read We can't use strcmp directly as the comparison function because qsort passes the address of each entry in the array, & s t r [i] type charaa), not s t r [i] type (of (of char*), as shown in this figure:
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array of N pointers:
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