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ALGORITHMS AND DATA STRUCTURES

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CHAPTER P

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Putting all this together to search html chars we write

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h a l f = lookup("f r a c l 2 " htmlchars, NELEMS(htm1chars)) ;

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to find the array index of the character % Binary search eliminates half the data at each step The number of steps is therefore proportional to the number of times we can divide n by 2 before we're left with a single element Ignoring roundoff, this is logzn If we have 1000 items to search, linear search takes up to 1000 steps, while binary search takes about 10; if we have a million items linear takes a million steps and binary takes 20 The more items, the greater the advantage of binary search Beyond some size of input (which varies with the implementation), binary search is faster than linear search

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22 Sorting

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Binary search works only if the elements are sorted If repeated searches are going to be made in some data set, it will be profitable to sort once and then use binary search If the data set is known in advance, it can be sorted when the program is written and built using compile-time initialization If not, it must be sorted when the program is run One of the best all-round sorting algorithms is quicksort, which was invented in 1960 by C A R Hoare Quicksort is a fine example of how to avoid extra computing It works by partitioning an array into little and big elements: pick one element of the array (the "pivot") partition the other elements into two groups: "little ones" that are less than the pivot value, and "big ones" that are greater than or equal to the pivot value recursively sort each group When this process is finished, the array is in order Quicksort is fast because once an element is known to be less than the pivot value, we don't have to compare it to any of the big ones; similarly big ones are not compared to little ones This makes it much faster than the simple sorting methods such as insertion sort and bubble sort that compare each element directly to all the others Quicksort is practical and efficient; it has been extensively studied and myriad variations exist The version that we present here is just about the simplest implementation but it is certainly not the quickest This quicksort function sorts an array of integers:

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SECTION 22

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/ a quicksort: s o r t v[O] v[n-11 i n t o increasing order v o i d q u i c k s o r t ( i n t v[], i n t n)

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i f (n <= 1 /+ nothing t o do +/ ) return ; swap(v, 0, rand() % n) ; /+ move p i v o t elem t o vCO] a/ l a s t = 0: = ; /a p f o r (i 1 i < n; i++) a r t i t i o n a/ i f ( v [ i ] < v[O]) swap(v, ++last, i); swap(v, 0, l a s t ) ; /a restore p i v o t a/ quicksort(v, l a s t ) ; /a recursively s o r t a/ quicksort(v+last+l, n- last- 1) ; / a each p a r t a/

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The swap operation, which interchanges two elements, appears three times in quicksort, so it is best made into a separate function:

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swap: interchange v [ i l and vCj1 v o i d swap(int v[], i n t i, i n t j)

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i n t temp; temp = v [ i ] ; vCi1 = v [ j ] ; v C j l = temp;

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Partitioning selects a random element as the pivot swaps it temporarily to the front, then sweeps through the remaining elements, exchanging those smaller than the pivot ("little ones") towards the beginning (at location l a s t ) and big ones towards At the end (at location i) the beginning of the process, just after the pivot has been swapped to the front, 1a s t = 0 and elements i = 1through n-1 are unexamined:

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