Sorting in Linear Time Lecture 5 Asst. Prof. Dr. İlker Kocabaş

Sorting in Linear TimeSorting in Linear Time

Lecture 5

Asst. Prof. Dr. İlker Kocabaş

How fast can we sort?How fast can we sort?

Decision-tree exampleDecision-tree example






Lower Bound for decision-tree Lower Bound for decision-tree sortingsorting

Lower bound for comparison Lower bound for comparison sortingsorting

Sorting in linear timeSorting in linear time

• Counting something, like histograms, can help us to sort an array. – clr = getcolor(x,y)– count[ clr ]++; Limitation: count[0..255]

Sorting in linear timeSorting in linear time

Counting sortCounting sort

COUNTING-SORT(A,B,k)

// Initialize aux. store

// Count how many times occurred.

// Cumulative sums of aux. store

// ???

Counting-sort exampleCounting-sort example















AnalysisAnalysis

Running TimeRunning Time

Stable sortingStable sorting

Counting sort array size Counting sort array size limitation problemlimitation problem

• If we need sort 32 bits integers, size of count array 232 = 4GB !

• Unsorted array size is usually smaller than this (1KB << 4GB)

• Solution... Use smaller parts like digits and sort these parts particularly : Radix sort

Radix sortRadix sort

Operation of radix sortOperation of radix sort

Analysis of radix sort Analysis of radix sort

• Given n d-digit number in which each digit can take on up to k-possible values, the running time of RADIX-SORT sorts these numbers is

(d*(n+k))

Analysis of radix sortAnalysis of radix sort

•Assume counting sort is the auxilary sort.•Sort n computer words of b-bits each.•Each word can be viewed as having d=b/r (base-2r) digits.

Example: Each key having d-digits of r bits. Let d=4, r=8 (b=32) then each digit is an integer in the

range 0 to 2r-1 = 28-1=255.

r=8 r=8 r=8 r=8

)}2)(/{()}2({)}({)( rr nrbndkndnT

Analysis of radix sort: Choosing Analysis of radix sort: Choosing rr

• How many passes should we make?• Each word of length b=rd bit is broken into r-bit

pieces.• For a given word length of b, the running time of an

array of length n that is T(r|n,b) can be minimized with respect to r.



ConclusionsConclusions

What about floating points?What about floating points?

• Integer can countable or divisible to parts.

• How can speed up sorting array of floating point numbers.

• We can group into k-parts– For numbers beetween 0.0 ≤ f ≤ 1.0– k*f have an integer part like 8.71– int(k*f) is part number.– k can be length of array.

Bucket SortBucket Sort

orderin together 1,...,1,0 lists theeconcatanet 6

sortinsertion with list sort 5

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Order StatisticsOrder Statistics

Hairy recurrence (cont.)Hairy recurrence (cont.)

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Sorting in Linear Time Lecture 5 Asst. Prof. Dr. İlker Kocabaş