Linear Sorts. EECS 2011 Prof. J. Elder - 1 -

Transcription

1 Linear Sorts - -

2 Linear Sorts? Comparison sorts are very general, but are W( nlog n) Faster sorting may be possible if we can constrain the nature of the input. - -

3 Ø Counting Sort Ø Radix Sort Ø Bucket Sort Linear Sorting Algorithms - -

4 Linear Sorts: Learning Outcomes Ø From understanding this lecture you should be able to: q Explain the difference between comparison sorts and linear sorting methods. q Identify situations when linear sorting methods can be applied and know why. q Explain and code any of the linear sorting algorithms we have covered

5 Ø Counting Sort Ø Radix Sort Ø Bucket Sort Linear Sorting Algorithms - 5 -

6 Example. Counting Sort Ø Invented by Harold Seward in 954. Ø Counting Sort applies when the elements to be sorted come from a finite (and preferably small) set. Ø For example, the elements to be sorted are integers in the range [ k-], for some fixed integer k. Ø We can then create an array V[ k-] and use it to count the number of elements with each value [ k-]. Ø Then each input element can be placed in exactly the right place in the output array in constant time

7 Counting Sort Input: Output: Ø Input: N records with integer keys between [ ]. Ø Output: Stable sorted keys. Ø Algorithm: q Count frequency of each key value to determine transition locations q Go through the records in order putting them where they go

8 CountingSort Input: Output: Index: Stable sort: If two keys are the same, their order does not change. Thus the 4 th record in input with digit must be the 4 th record in output with digit. It belongs at output index 8, because 8 records go before it ie, 5 records with a smaller digit & records with the same digit Count These! - 8 -

9 Input: Output: CountingSort Index: Value v: # of records with digit v: 5 9 N records. Time to count? θ(n) - 9 -

10 Input: Output: CountingSort Index: Value v: # of records with digit v: # of records with digit < v: N records, k different values. Time to count? θ(k) - -

11 CountingSort Input: Output: Index: Value v: # of records with digit < v: 5 = location of first record with digit v

12 CountingSort Input: Output: Index:? Value v: Location of first record with digit v Algorithm: Go through the records in order putting them where they go. - -

13 Loop Invariant Ø The first i- keys have been placed in the correct locations in the output array Ø The auxiliary data structure v indicates the location at which to place the i th key for each possible key value from [..k-]. - -

14 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go

15 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go

16 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go

17 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go

18 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go

19 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go

20 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go. - -

21 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go. - -

22 CountingSort Input: Output: Index: Value v: Location of next record with digit v Algorithm: Go through the records in order putting them where they go. - -

23 CountingSort Input: Output: Index: Value v: Location of next record with digit v. 4 9 Algorithm: Go through the records in order putting them where they go. - -

24 CountingSort Input: Output: Index: Value v: Location of next record with digit v. 4 9 Algorithm: Go through the records in order putting them where they go

25 CountingSort Input: Output: Index: Value v: Location of next record with digit v. 5 9 Algorithm: Go through the records in order putting them where they go

26 CountingSort Input: Output: Index: Value v: Location of next record with digit v. 5 9 Algorithm: Go through the records in order putting them where they go

27 CountingSort Input: Output: Index: Value v: Location of next record with digit v Time = θ(n) Total = θ(n+k) - 7 -

28 Ø Counting Sort Ø Radix Sort Ø Bucket Sort Linear Sorting Algorithms - 8 -

29 Input: An array of N numbers. Each number contains d digits. Each digit between [ k-] Output: Sorted numbers. Digit Sort: Select one digit Example. RadixSort Separate numbers into k piles based on selected digit (e.g., Counting Sort). Stable sort: If two cards are the same for that digit, their order does not change

30 RadixSort Sort by which digit first? The most significant Sort by which digit Second? The next most significant All meaning in first sort lost. - -

31 RadixSort Sort by which digit first? The least significant Sort by which digit Second? The next least significant

32 RadixSort Sort by which digit first? The least significant Sort by which digit Second? The next least significant Is sorted by least sig. digits.

33 RadixSort i+ Is sorted by first i digits. Sort by i+st digit Is sorted by first i+ digits. These are in the correct order because sorted by high order digit

34 RadixSort i+ Is sorted by first i digits. Sort by i+st digit Is sorted by first i+ digits. These are in the correct order because was sorted & stable sort left sorted

35 Loop Invariant Ø The keys have been correctly stable-sorted with respect to the i- least-significant digits

36 Running Time Running time is Q ( d( n + k)) Where d = n = k = # of digits in each number # of elements to be sorted # of possible values for each digit - 6 -

37 Ø Counting Sort Ø Radix Sort Ø Bucket Sort Linear Sorting Algorithms - 7 -

38 Example. Bucket Sort Ø Applicable if input is constrained to finite interval, e.g., real numbers in the range [ ). Ø If input is random and uniformly distributed, expected run time is Θ(n)

39 Ø Given A[..n]: Bucket Sort q Create new table B of length n q Insert A[i] into B na[i] - 9 -

40 PseudoCode Expected Running Time Q() n Q() Q( n) Q( n) n - 4 -

41 Loop Invariants Ø Loop q The first i- keys have been correctly placed into buckets of width /n. Ø Loop q The keys within each of the first i- buckets have been correctly stable-sorted

42 Ø Counting Sort Ø Radix Sort Ø Bucket Sort Linear Sorting Algorithms - 4 -

43 Linear Sorts: Learning Outcomes Ø From understanding this lecture you should be able to: q Explain the difference between comparison sorts and linear sorting methods. q Identify situations when linear sorting methods can be applied and know why. q Explain and code any of the linear sorting algorithms we have covered