1. CSC 213 – Large Scale Programming

2. Bucket Sort  Buckets, B, is array of Sequence  Sorts Collection, C, in two phases: 1. Remove each element v from C & add to B[v] 2. Move elements from each bucket back to C A B C

3. Bucket Sort Example

4. Bucket-Sort Algorithm Algorithm bucketSort( Sequence C) B = new Sequence [10] // & instantiate each Sequence // Phase 1 for each element v in C B[v].addLast(v) endfor // Phase 2 loc = 0 for each Sequence b in B for each element v in b C.set(loc, v) loc += 1 endfor endfor return C

5. Bucket Sort Properties  For this to work, values must be legal indices  Non-negative integers only can be used  Sorting occurs without comparing objects

6. Bucket Sort Properties  For this to work, values must be legal indices  Non-negative integers only can be used  Sorting occurs without comparing objects

7. Bucket Sort Properties

8.  For this to work, values must be legal indices  Non-negative integers only can be used  Sorting occurs without comparing objects  Example of a stable sort  Preserves relative ordering of objects with same value  (B UBBLE - SORT & M ERGE - SORT are other stable sorts)

9. Bucket Sort Extensions  Use Comparator for B UCKET - SORT  Get index for v using compare( v, null)  Comparator for booleans could return  0 when v is false  1 when v is true  Comparator for US states, could return  Annual per capita consumption of Jello  Consumption of jello overall, in cubic feet  State’s ranking by population

10. Bucket Sort Extensions  State’s ranking by population 1 California 2 Texas 3 New York 4 Florida 5 Illinois 6 Pennsylvania 7 Ohio 8 Michigan 9 Georgia

11. Bucket Sort Extensions  Extended B UCKET - SORT works with many types  Limited set of data needed for this to work  Need way to enumerating values of the set

12. Bucket Sort Extensions  Extended B UCKET - SORT works with many types  Limited set of data needed for this to work  Need way to enumerating values of the set enumeration is subtle hint

13. d -Tuples  Combination of d values such as ( k 1, k 2, …, k d )  k i is i th dimension of the tuple  A point ( x, y, z ) is 3-tuple  x is 1 st dimension’s value  Value of 2 nd dimension is y  z is 3 rd dimension’s value

14. Lexicographic Order  Assume a & b are both d-tuples  a = ( a 1, a 2, …, a d )  b = ( b 1, b 2, …, b d )  Can say a < b if and only if  a 1 < b 1 OR  a 1 = b 1 && ( a 2, …, a d ) < ( b 2, …, b d )  Order these 2-tuples using previous definition (3, 4) (7, 8) (3, 2) (1, 4) (4, 8)

15. Lexicographic Order  Assume a & b are both d-tuples  a = ( a 1, a 2, …, a d )  b = ( b 1, b 2, …, b d )  Can say a < b if and only if  a 1 < b 1 OR  a 1 = b 1 && ( a 2, …, a d ) < ( b 2, …, b d )  Order these 2-tuples using previous definition (3, 4) (7, 8) (3, 2) (1, 4) (4, 8) (1, 4) (3, 2) (3, 4) (4, 8) (7, 8)

16. Radix-Sort  Very fast sort for data expressed as d-tuple  Cheats to win  Cheats to win; faster than sorting’s lower bound  Sort performed using d calls to bucket sort  Sorts least to most important dimension of tuple  Luckily lots of data are d-tuples  String is d-tuple of char

17. Radix-Sort  Very fast sort for data expressed as d-tuple  Cheats to win  Cheats to win; faster than sorting’s lower bound  Sort performed using d calls to bucket sort  Sorts least to most important dimension of tuple  Luckily lots of data are d-tuples  Can sort an int by its digits

18. Radix-Sort  Very fast sort for data expressed as d-tuple  Cheats to win  Cheats to win; faster than sorting’s lower bound  Sort performed using d calls to bucket sort  Sorts least to most important dimension of tuple  Luckily lots of data are d-tuples  String is d-tuple of char  Can sort an int by its digits

19.  Represent int as a d-tuple of digits: 6210 = 111110 2 0410 = 000100 2  Decimal digits needs 10 buckets to use for sorting  Ordering using their bits needs 2 buckets  O (d∙ n ) time needed to run R ADIX - SORT  d is length of longest element in input  In most cases value of d is constant (d = 31 for int )  Radix sort takes O ( n ) time, ignoring constant Radix-Sort For int s

20. Radix-Sort In Action  List of 4-bit integers sorted using R ADIX - SORT 1001 0010 1101 0001 1110

21. Radix-Sort In Action  List of 4-bit integers sorted using R ADIX - SORT 1001 0010 1101 0001 1110 0010 1110 1001 1101 0001

22. Radix-Sort In Action  List of 4-bit integers sorted using R ADIX - SORT 1001 0010 1101 0001 1110 0010 1110 1001 1101 0001 1001 1101 0001 0010 1110

23. Radix-Sort In Action  List of 4-bit integers sorted using R ADIX - SORT 1001 0010 1101 0001 1110 0010 1110 1001 1101 0001 1001 1101 0001 0010 1110 1001 0001 0010 1101 1110

24. Radix-Sort In Action  List of 4-bit integers sorted using R ADIX - SORT 1001 0010 1101 0001 1110 0010 1110 1001 1101 0001 1001 1101 0001 0010 1110 1001 0001 0010 1101 1110 0001 0010 1001 1101 1110

25. Finding Value of a Digit  Value of the i th digit (base-10) of k : (( k / 10 i ) % 10)  Value of the i th bit (base-2) of k : (( k / 2 i ) % 2)  Value of the i th hex digit (base-16) of k : (( k / 16 i ) % 16)

26. Radix Sort Algorithm radixSort( Sequence C) for i = 0 to 31 B = new Sequence [2] // & instantiate each Sequence for each element v in C digit = (v / (1 << i)) % 2 B[digit].addLast(v) endfor loc = 0 for each Sequence b in B for each element v in b C.set(loc, v) loc += 1 endfor endfor endfor return C 26

27. For Next Lecture  Weekly assignment doubles-down on last week  Due at regular time tomorrow  Reviewing requirements for program #2  1 st Preliminary deadline is today  Spend time working on this: design saves coding  Reading on Graph ADT for Wednesday  Note: these have nothing to do with bar charts  What are mathematical graphs?  Why are they the basis of everything in CS?