LECTURE 25:BUCKET SORT & RADIX SORT

CSC 213 – Large Scale Programming

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

Bucket-Sort Algorithm

Algorithm bucketSort(Sequence<Integer> C)B = new Sequence<Integer>[10] // & instantiate each Sequence

// Phase 1 for each element v in C

B[v].addLast(v) // Assumes each number in C between 0 & 9endfor

// Phase 2loc = 0for each Sequence b in B

for each element v in bC.set(loc, v)loc += 1

endforendfor

return C

Bucket Sort Properties

For this to work, values must be legal indices Non-negative integers only can be used Sorting occurs without comparing objects

Bucket Sort Properties

For this to work, values must be legal indices Non-negative integers only can be used Sorting occurs without comparing

objects

Bucket Sort Properties

For this to work, values must be legal indices Non-negative integers only can be used

Sorting occurs without

comparing objects

Bucket Sort Properties

For this to work, values must be legal indices Non-negative integers only can be used Sorting occurs without comparing objects

Stable sort describes any sort of this type Preserves relative ordering of objects with

same value (BUBBLE-SORT & MERGE-SORT are other

stable sorts)

Bucket Sort Extensions

Use Comparator for BUCKET-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

Bucket Sort Extensions

State’s ranking by population

1 California2 Texas3 New York4 Florida5 Illinois6

Pennsylvania

7 Ohio8 Michigan9 Georgia

Bucket Sort Extensions

Extended BUCKET-SORT works with many types Limited set of data needed for this to work Need way to enumerate values of the set

Bucket Sort Extensions

Extended BUCKET-SORT works with many types Limited set of data needed for this to work Need way to enumerate values of the set

enumerateis subtle

hint

d-Tuples

Combination of d values such as (k1, k2, …, kd) ki is ith dimension of the tuple

A point (x, y, z) is 3-tuple x is 1st dimension’s value Value of 2nd dimension is y z is 3rd dimension’s value

Lexicographic Order

Assume a & b are both d-tuples a = (a1, a2, …, ad) b = (b1, b2, …, bd)

Can say a < b if and only if a1 < b1 OR a1 = b1 && (a2, …, ad) < (b2, …, bd)

Order these 2-tuples using previous definition (3 4) (7 8) (3 2) (1 4) (4 8)

Lexicographic Order

Assume a & b are both d-tuples a = (a1, a2, …, ad) b = (b1, b2, …, bd)

Can say a < b if and only if a1 < b1 OR a1 = b1 && (a2, …, ad) < (b2, …, bd)

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)

Radix-Sort

Very fast sort for data expressed as d-tuple 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“L E T T E R S”“L I N G E R S”

Radix-Sort

Very fast sort for data expressed as d-tuple 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

Digits of an int can be used for sorting, also

1 0 0 1 3 7 2 91 0 0 9 2 2 1 0

Radix-Sort For Integers

Represent int as a d-tuple of digits:621010 = 1111102 041010 =

0001002 Decimal digits needs 10 buckets to use for

sorting Ordering using their bits needs 2 buckets

O(d∙n) time needed to run RADIX-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 In Action

List of 4-bit integers sorted using RADIX-SORT100

10010

1101

0001

1110

Radix-Sort In Action

List of 4-bit integers sorted using RADIX-SORT001

01110100111010001

1001

0010

1101

0001

1110

Radix-Sort In Action

List of 4-bit integers sorted using RADIX-SORT 100

11101000100101110

00101110100111010001

1001

0010

1101

0001

1110

Radix-Sort In Action

List of 4-bit integers sorted using RADIX-SORT 100

10001001011011110

10011101000100101110

00101110100111010001

1001

0010

1101

0001

1110

Radix-Sort In Action

List of 4-bit integers sorted using RADIX-SORT 000

10010100111011110

10010001001011011110

10011101000100101110

00101110100111010001

1001

0010

1101

0001

1110

Radix-Sort

Algorithm radixSort(Sequence<Integer> C) // Works from least to most significant value for bit = 0 to 30 C = bucketSort(C, bit) // Sort C using the specified bitendfor

return C

What is big-Oh complexity for Radix-Sort? Call in loop uses each element twice Loop repeats once per digit to complete

sort

Radix-Sort

Algorithm radixSort(Sequence<Integer> C) // Works from least to most significant value for bit = 0 to 30 C = bucketSort(C, bit) // Sort C using the specified bitendfor

return C

What is big-Oh complexity for Radix-Sort? Call in loop uses each element twice

O(n) Loop repeats once per digit to complete

sort * O(1)

O(n)

Radix-Sort

Algorithm radixSort(Sequence<Integer> C) // Works from least to most significant value for bit = 0 to 30 C = bucketSort(C, bit) // Sort C using the specified bitendfor

return C

What is big-Oh complexity for Radix-Sort? Call in loop uses each element twice

O(n) Loop repeats once per digit to complete

sort * O(1) O(log n) times (?)

O(n log n)

For Next Lecture

Review requirements for program #2 1st Preliminary deadline is Monday 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?