Sorting algorithms

Topic Overview • Issues in Sorting on Parallel Computers • Sorting Networks • Bubble Sort and its Variants • Quicksort • Bucket and Sample Sort • Other Sorting Algorithms

Sorting: Overview • One of the most commonly used and well-studied kernels. • Sorting can be comparison-based or noncomparison-based. • The fundamental operation of comparison-based sorting is compare-exchange. • The lower bound on any comparison-based sort of n numbers is Θ(nlog n) . • We focus here on comparison-based sorting algorithms.

Sorting: Basics What is a parallel sorted sequence? Where are the input and output lists stored? • We assume that the input and output lists are distributed. • The sorted list is partitioned with the property that each partitioned list is sorted and each element in processor Pi's list is less than that in Pj's list if i < j.

Sorting: Parallel Compare Exchange Operation A parallel compare-exchange operation. Processes Pi and Pj send their elements to each other. Process Pi keeps min{ai,aj}, and Pj keeps max{ai, aj}.

Sorting: Basics What is the parallel counterpart to a sequential comparator? • If each processor has one element, the compare exchange operation stores the smaller element at the processor with smaller id. This can be done in ts + tw time. • If we have more than one element per processor, we call this operation a compare split. Assume each of two processors have n/p elements. • After the compare-split operation, the smaller n/p elements are at processor Pi and the larger n/p elements at Pj, where i < j. • The time for a compare-split operation is (ts+ twn/p), assuming that the two partial lists were initially sorted.

Sorting: Parallel Compare Split Operation A compare-split operation. Each process sends its block of size n/p to the other process. Each process merges the received block with its own block and retains only the appropriate half of the merged block. In this example, process Pi retains the smaller elements and process Pi retains the larger elements.

Sorting Networks • Networks of comparators designed specifically for sorting. • A comparator is a device with two inputs x and y and two outputs x' and y'. For an increasing comparator, x' = min{x,y} and y' = min{x,y}; and vice-versa. • We denote an increasing comparator by ⊕ and a decreasing comparator by Ө. • The speed of the network is proportional to its depth.

Sorting Networks: Comparators A schematic representation of comparators: (a) an increasing comparator, and (b) a decreasing comparator.

Sorting Networks A typical sorting network. Every sorting network is made up of a series of columns, and each column contains a number of comparators connected in parallel.

Sorting Networks: Bitonic Sort • A bitonic sorting network sorts n elements in Θ(log2 n) time. • A bitonic sequence has two tones - increasing and decreasing, or vice versa. Any cyclic rotation of such networks is also considered bitonic. ∀ 〈1,2,4,7,6,0〉 is a bitonic sequence, because it first increases and then decreases. 〈8,9,2,1,0,4〉 is another bitonic sequence, because it is a cyclic shift of 〈0,4,8,9,2,1〉. • The kernel of the network is the rearrangement of a bitonic sequence into a sorted sequence.

Sorting Networks: Bitonic Sort • Let s = 〈a0,a1,…,an-1〉 be a bitonic sequence such that a0 ≤ a1 ≤ ··· ≤ an/2-1 and an/2 ≥ an/2+1 ≥ ··· ≥ an-1. • Consider the following subsequences of s: s1 = 〈min{a0,an/2},min{a1,an/2+1},…,min{an/2-1,an-1}〉 s2 = 〈max{a0,an/2},max{a1,an/2+1},…,max{an/2-1,an-1}〉 (1) • Note that s1 and s2 are both bitonic and each element of s1 is less than every element in s2. • We can apply the procedure recursively on s1 and s2 to get the sorted sequence.

Sorting Networks: Bitonic Sort Merging a 16-element bitonic sequence through a series of log 16 bitonic splits.

Sorting Networks: Bitonic Sort • We can easily build a sorting network to implement this bitonic merge algorithm. • Such a network is called a bitonic merging network. • The network contains log n columns. Each column contains n/2 comparators and performs one step of the bitonic merge. • We denote a bitonic merging network with n inputs by ⊕BM[n]. • Replacing the ⊕ comparators by Ө comparators results in a decreasing output sequence; such a network is denoted by ӨBM[n].

Sorting Networks: Bitonic Sort A bitonic merging network for n = 16. The input wires are numbered 0,1, …, n - 1, and the binary representation of these numbers is shown. Each column of comparators is drawn separately; the entire figure represents a ⊕BM[16] bitonic merging network. The network takes a bitonic sequence and outputs it in sorted order.

Sorting Networks: Bitonic Sort How do we sort an unsorted sequence using a bitonic merge? • We must first build a single bitonic sequence from the given sequence. • A sequence of length 2 is a bitonic sequence. • A bitonic sequence of length 4 can be built by sorting the first two elements using ⊕BM[2] and next two, using ӨBM[2]. • This process can be repeated to generate larger bitonic sequences.

Sorting Networks: Bitonic Sort A schematic representation of a network that converts an input sequence into a bitonic sequence. In this example, ⊕BM[k] and ӨBM[k] denote bitonic merging networks of input size k that use ⊕ and Ө comparators, respectively. The last merging network (⊕BM[16]) sorts the input. In this example, n = 16.

Sorting Networks: Bitonic Sort The comparator network that transforms an input sequence of 16 unordered numbers into a bitonic sequence.

Sorting Networks: Bitonic Sort • The depth of the network is Θ(log2 n). • Each stage of the network contains n/2 comparators. A serial implementation of the network would have complexity Θ(nlog2 n).

Bubble Sort and its Variants The sequential bubble sort algorithm compares and exchanges adjacent elements in the sequence to be sorted: Sequential bubble sort algorithm.

Bubble Sort and its Variants • The complexity of bubble sort is Θ(n2 ). • Bubble sort is difficult to parallelize since the algorithm has no concurrency. • A simple variant, though, uncovers the concurrency.

Odd-Even Transposition Sequential odd-even transposition sort algorithm.

Odd-Even Transposition Sorting n = 8 elements, using the odd-even transposition sort algorithm. During each phase, n = 8 elements are compared.

Odd-Even Transposition • After n phases of odd-even exchanges, the sequence is sorted. • Each phase of the algorithm (either odd or even) requires Θ(n) comparisons. • Serial complexity is Θ(n2 ).

Parallel Odd-Even Transposition • Consider the one item per processor case. • There are n iterations, in each iteration, each processor does one compare-exchange. • The parallel run time of this formulation is Θ(n). • This is cost optimal with respect to the base serial algorithm but not the optimal one.

Parallel Odd-Even Transposition Parallel formulation of odd-even transposition.

Parallel Odd-Even Transposition • Consider a block of n/p elements per processor. • The first step is a local sort. • In each subsequent step, the compare exchange operation is replaced by the compare split operation. • The parallel run time of the formulation is

Parallel Odd-Even Transposition • The parallel formulation is cost-optimal for p = O(log n). • The isoefficiency function of this parallel formulation is Θ(p2p ).

Shellsort • Let n be the number of elements to be sorted and p be the number of processes. • During the first phase, processes that are far away from each other in the array compare-split their elements. • During the second phase, the algorithm switches to an odd-even transposition sort.

Parallel Shellsort • Initially, each process sorts its block of n/p elements internally. • Each process is now paired with its corresponding process in the reverse order of the array. That is, process Pi, where i < p/2, is paired with process Pp-i-1. • A compare-split operation is performed. • The processes are split into two groups of size p/2 each and the process repeated in each group.

Parallel Shellsort An example of the first phase of parallel shellsort on an eight-process array.

Parallel Shellsort • Each process performs d = log p compare-split operations. • With O(p) bisection width, each communication can be performed in time Θ(n/p) for a total time of Θ((nlog p)/p). • In the second phase, l odd and even phases are performed, each requiring time Θ(n/p). • The parallel run time of the algorithm is:

Sorting algorithms

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Sorting algorithms