lect18-sorting-review(notcur)

# lect18-sorting-review(notcur) - Lecture Notes CMSC 251 Here...

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Lecture Notes CMSC 251 Here is an example. 576 49[4] 9[5]4 [1]76 176 494 19[4] 5[7]6 [1]94 194 194 95[4] 1[7]6 [2]78 278 296 = 57[6] = 2[7]8 = [2]96 = 296 278 29[6] 4[9]4 [4]94 494 176 17[6] 1[9]4 [5]76 576 954 27[8] 2[9]6 [9]54 954 The running time is clearly Θ( d ( n + k )) where d is the number of digits, n is the length of the list, and k is the number of values a digit can have. This is usually a constant, but the algorithm’s running time will be Θ( dn ) as long as k O ( n ) . Notice that we can be quite ﬂexible in the deﬁnition of what a “digit” is. It can be any number in the range from 1 to cn for some constant c , and we will still have an Θ( n ) time algorithm. For example, if we have d =2 and set k = n , then we can sort numbers in the range n * n = n 2 in Θ( n ) time. In general, this can be used to sort numbers in the range from 1 to n d in Θ( dn ) time. At this point you might ask, since a computer integer word typically consists of 32 bits (4 bytes), then doesn’t this imply that we can sort any array of integers in O ( n ) time (by applying radix sort on each of the d =4 bytes)? The answer is yes, subject to this word-length restriction. But you should be careful about attempting to make generalizations when the sizes of the numbers are not bounded. For example, suppose you have n keys and there are no duplicate values. Then it follows that you need at least B = d lg n e bits to store these values. The number of bytes is d = d B/ 8 e . Thus, if you were to apply radix sort in this situation, the running time would be Θ( dn )=Θ ( n log n ) . So there is no

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lect18-sorting-review(notcur) - Lecture Notes CMSC 251 Here...

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