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Counting Sort Non-comparison sort. Precondition: n numbers in the range 1..k. Key ideas: For each x count the number C(x) of elements ≤ x Insert x at output position C(x) and decrement C(x).

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An Example 71312457243 A[1..n] 1 2 3 4 5 6 7 8 9 10 11 21199864 C 1 2 3 4 5 6 7 for i = 2 to 7 do C[i] = C[i] + C[i–1] Input 2222210 C[1..k] 1 2 3 4 5 6 7 Auxiliary storage k two 1’s in A 6 elements ≤ 3

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Example (continued) 11246899 C 1 2 3 4 5 6 7 21199854 C 1 2 3 4 5 6 7 B[6] = B[C[3]] = B[C[A[11]]] = A[11] = 3 C[A[11]] = C[A[11]] – 1 A[1..n] 1 2 3 4 5 6 7 8 9 10 11 B[1..n] Output 3

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Example (cont’d) 11245899 B C 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 21199754 C 1 2 3 4 5 6 7 B[C[A[10]]] = A[10] = 4 C[A[10]] = C[A[10]] – 1 43

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Example (cont’d) 2 3445 7 10235789 B C 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 21098653 C 1 2 3 4 5 6 7 B[C[A[6]]] = A[6] = 4 C[A[6]] = C[A[6]] – 1

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Analysis of Counting Sort Counting-Sort(A, B, k) for i = 1 to k do C[i] = 0// (k) for j = 1 to length[A] do C[A[j]] = C[A[j]] + 1 // (n) // C[i] now contains the number of elements = i for i = 2 to k do C[i] = C[i] + C[i–1]// (k) // C[i] now contains the number of elements ≤ i for j = length[A] downto 1 do B[C[A[j]]] = A[j] C[A[j]] = C[A[j]] – 1// (n) Running time is (n+k) (or (n) if k = O(n))!

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Stability Counting sort is stable: Input order maintained among items with equal keys. Stability is important because data are often carried with the keys being sorted. radix sort (which uses counting sort as a subroutine) relies on it to work correctly. for j = length[A] downto 1 do B[C[A[j]]] = A[j] C[A[j]] = C[A[j]] – 1 How is stability realized?

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Radix Sort Sort a set of numbers in multiple passes, starting from the rightmost digit, then the 10’s digit, then the 100’s digit, etc. Example: sort 23, 45, 7, 56, 20, 19, 88, 77, 61, 13, 52, 39, 80, 2, 99 99 Pass 1: Pass 2: 2345 7 562019 88 77 61 13 52 39802 20 8061 52 2 23 45 56 77 7 88 19 39 99 13

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Analysis of Radix Sort Sort n d-digit numbers. Let k be the range of each digit. Each pass takes time (n+k). // use counting sort There are d passes in total. The running time for radix sort is (dn+dk). Linear running time when d is a constant and k = O(n). Correctness follows by induction on the number of passes.

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Breaking into Digits …… b bits r-bit digit Each digit in the range 0 -- 2 – 1 r b/r passes, each spending time (n+2 ). r Running time ((b/r) (n + 2 )). r b/r digits If b ≤ lg n, choose r = b (n) If b > lg n, choose r = lg n (bn/lg n) Key:

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Bucket Sort Assumption: input elements are uniformly distributed over [0, 1) In case the input range is larger than 1, normalize each element over the maximum first. n buckets are used for n input numbers. Insert A[i] into bucket nA[i] , 1 i n. Bucket sort runs in (n) expected time.

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An Example.78.17.26.94.12.39.68.72.23.21 0 1 2 3 4 5 6 7 8 9 Step 2: Concatenate all lists Step 1: equivalent of insertion sort within each list n inputs dropped into n equal-sized subintervals of [0, 1)..78.17.39.26.72.94.21.12.68.23 A[ ]

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Comparison of Sorting Methods (II) Method Space Average Max n=16 n=10000 Counting sort 2n + 1000 22n + 10010 22n 10362 32010 Radix sort n + (n+200) 32n 32n + 4838 4250 36838 Bucket sort n + (n+100).0175n + 18n 3.5n 645 35246 22 D. E. Knuth, “The Art of Computer Programming”, Vol 3, 2 nd ed., p.382, 1998. Implemented on the MIX Computer.

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Comparison of Sorting Algorithms Insertion sort: suitable only for small n. Merge sort: guaranteed to be fast even in its worst case; stable. Heapsort: requiring minimum memory and guaranteed to run fast; average and maximum time both roughly twice the average time of quicksort. Quicksort: most useful general-purpose sorting for very little memory requirement and fastest average time. (choose the median of three elements as pivot in practice :-) Counting sort: very useful when the keys have small range; stable; memory space for counters and for 2n records. Radix sort: appropriate for keys either rather short or with a lexicographic collating sequence. Bucket sort: assuming keys to have uniform distribution.

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Sorting in linear time Comparison sort: –Lower bound: (nlgn). Non comparison sort: –Bucket sort, counting sort, radix sort –They are possible in linear.

Sorting in linear time Comparison sort: –Lower bound: (nlgn). Non comparison sort: –Bucket sort, counting sort, radix sort –They are possible in linear.

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