Lecture 25: BUCKET SORT & RADIX Sort

CSC 213 – Large Scale Programming Lecture 25:BUCKET SORT & RADIX Sort

Bucket-Sort • Buckets, B, is array of Sequence • Sorts Collection, C, in two phases: • Remove each elementv from C & add to B[v] • Move elements from each bucket back to C A B C

Bucket-Sort Algorithm AlgorithmbucketSort(Sequence<Integer>C)B=newSequence<Integer>[10] // & instantiate eachSequence // Phase 1 for each element v in CB[v].addLast(v) // Assumes each number in C between 0 & 9endfor// Phase 2loc = 0for each Sequenceb in Bfor each element v in bC.set(loc,v)loc+= 1endforendfor 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 • 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 vusing compare(v, null) • Comparatorfor booleans could return • 0when vis false • 1 when vis true • Comparator for US states, could return • Annual per capita consumption of Jello • Consumption of jellooverall, in cubic feet • State’s ranking by population

Bucket Sort Extensions • State’s ranking by population

Bucket Sort Extensions • Extended Bucket-sort works with many types • Limited set of data neededfor this to work • Need way to enumeratevalues of the set

Bucket Sort Extensions • Extended Bucket-sort works with many types • Limited set of data neededfor this to work • Need way to enumeratevalues 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 • xis1st dimension’s value • Value of 2nd dimension isy • zis3rd dimension’s value

Lexicographic Order • Assume a&bare both d-tuples • a= (a1,a2, …, ad) • b= (b1,b2, …, bd) • Can say a<bif and only if • a1< b1OR • 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&bare both d-tuples • a= (a1,a2, …, ad) • b= (b1,b2, …, bd) • Can say a<bif and only if • a1< b1OR • 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 intcan be used for sorting, also 1 0 0 1 3 7 2 9 1 0 0 9 2 2 1 0

Radix-Sort For Integers • Represent int as a d-tuple of digits:621010 = 1111102041010 = 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 dis 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-sort 1001 0010 1101 0001 1110

Radix-Sort In Action • List of 4-bit integers sorted using Radix-sort 1001 0010 0010 1110 1101 1001 0001 1101 0001 1110

Radix-Sort In Action • List of 4-bit integers sorted using Radix-sort 1001 0010 1001 0010 1110 1101 1101 1001 0001 0001 1101 0010 0001 1110 1110

Radix-Sort In Action • List of 4-bit integers sorted using Radix-sort 1001 0010 1001 1001 0010 1110 1101 0001 1101 1001 0001 0010 0001 1101 0010 1101 0001 1110 1110 1110

Radix-Sort In Action • List of 4-bit integers sorted using Radix-sort 1001 0010 1001 1001 0001 0010 1110 1101 0001 0010 1101 1001 0001 0010 1001 0001 1101 0010 1101 1101 0001 1110 1110 1110 1110

Radix-Sort AlgorithmradixSort(Sequence<Integer>C) // Works from least to most significant value for bit = 0 to 30C = 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 AlgorithmradixSort(Sequence<Integer>C) // Works from least to most significant value for bit = 0 to 30C = 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 AlgorithmradixSort(Sequence<Integer>C) // Works from least to most significant value for bit = 0 to 30C = 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 digitto 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?

Lecture 25: BUCKET SORT & RADIX Sort