Unformatted text preview: CS 6643 F '11 Lec 22 – The latter case is preferred. So if i<j, the elements in Pi ≤
elements
elements in Pj • Output: sorted sequence is placed in the memory of one
process or distributed evenly among all processes – We assume the latter • Input: stored in one the memory of one process or
distributed among the memories of multiple processes Input and Output (Some slides are adapted from M. Quinn’s slides for Parallel Prog. in
C with MPI and OpenMP, 2004) Rajendra V. Boppana
UT San Antonio Sorting 3 Performing Comparisons CS 6643 F '11 Lec 22 – Use compareandexchange (actually, exchangeandcompare)
operation
– (ts + tw) time; time to compare is assumed too small compared to time to
communication • One element per process CS 6643 F '11 Lec 22 » W = Θ(n) – Noncomparisonbased: use certain properties of the elements
such as range of values to avoid comparison of elements with
one another » W = seq. work = Θ(n log n) – Comparisonbased: use compareandexchange operation as
the basic operation » We look at internal sorting techniques – Internal, all data fit into the available main memory, or external,
data are too large and stored in disk memory • Given S=<a1, …, an>, sorting permutes S into
S’ = <b1, … bn> such that
each bi is some ai and b1 ≤ b2 ≤ … ≤ bn
• Sorting techniques Sorting 4 2 CS 6643 F '11 Lec 22 Given
Given a seq. of elements
A[1..N]
Divide
Divide A into two smaller
nonempty sets such that all
elements
elements in one set are no
bigger
bigger than a pivot element
and
and all the elements in the
second set are greater than
the pivot element.
Repeat
Repeat the divide step on the
two subsets until the subsets
are of size 1.
Divide
Divide and conquer technique
Selection
Selection of the pivot is the
key Basic
Basic idea but
but Ο(n2) worstcase
worst Average
Average # of comparecompareexchanges = 1.4n log n Quicksort CS 6643 F '11 Lec 22 Here the first element of
the list is the pivot. – Use compareandsplit (actually, exchangemergeandsplit) operation
– (ts + m tw) time for m elements • Multiple elements Performing Comparisons 7 5 14 65 4 22 63 11 CS 6643 F '11 Lec 22 Unordered list of values 17 Sequential Quicksort Parallel quicksort
Hyperquicksort
Parallel sorting by regular sampling Quicksort 8 14 65 4 22 63 11 14 17 CS 6643 F '11 Lec 22 11 65 22 63 Recursively
apply quicksort
to low list 4 Sequential Quicksort CS 6643 F '11 Lec 22 Choose pivot value 17 Sequential Quicksort 11 9 4 11 22 63 High list
(> 17) 17 65 CS 6643 F '11 Lec 22 4 11 63 65 Recursively
apply quicksort
to high list 14 17 22 Sequential Quicksort CS 6643 F '11 Lec 22 Low list
(≤ 17) 14 Sequential Quicksort 12 10 11 14 17 22 63 65 CS 6643 F '11 Lec 22 Recursive
Recursive sorts of low, high lists can be done in
parallel Natural
Natural concurrency Generally
Generally recognized as fastest sort in average
case
When
When possible, design parallel algorithm based
on the fastest sequential algorithm Speed
Speed Parallel Sorting Based on Quicksort CS 6643 F '11 Lec 22 Sorted list of values 4 Sequential Quicksort 15 13 CS 6643 F '11 Lec 22 How
How it is done determines the overall time
complexity The
The last step is easy to parallelize
The
The first step is inherently sequential
Splitting
Splitting can be done in parallel, but may
need to combine the partial splits by each
process Selecting
Selecting a pivot
Splitting
Splitting an array into two
Sorting
Sorting the two subarrays Key
Key steps Parallelizing Quicksort CS 6643 F '11 Lec 22 Example:
Example: “Median of 3” technique Can
Can make worstcase less probable by using
worstsampling
sampling to choose pivot value Occurs
Occurs when low, high lists maximally
unbalanced at every partitioning step AverageAveragecase time complexity: Θ(n log n)
WorstWorstcase time complexity: Θ(n2) Attributes of Sequential Quicksort 16 14 P0 CS 6643 F '11 Lec 22 P2
P3 50, 12, 47, 72, 65, 54, 66, 22 83, 66, 67, 0, 70, 98, 99, 82 20, 40, 89, 47, 19, 61, 86, 85 Exchange “lower half” and “upper half” values” P1 75, 91, 15, 64, 21, 8, 88, 54 19 CS 6643 F '11 Lec 22 Upper
“half” Lower
“half” P3 P2 P1 P0 After exchange step 89, 86, 85 P3 P2 50, 12, 47, 72, 65, 54, 66,
22, 20, 40, 47, 19, 61
83, 98, 99, 82, 91, 88 P1 75, 15, 64, 21, 8, 54, 66, 67, 0, 70 Parallel Quicksort CS 6643 F '11 Lec 22 20, 40, 89, 47, 19, 61, 86, 85 P3 20, 40, 89, 47, 19, 61, 86, 85 17 83, 66, 67, 0, 70, 98, 99, 82 P2 83, 66, 67, 0, 70, 98, 99, 82 Parallel Quicksort CS 6643 F '11 Lec 22 50, 12, 47, 72, 65, 54, 66, 22 P1 50, 12, 47, 72, 65, 54, 66, 22 P0 Process P0 chooses and broadcasts
randomly chosen pivot value 75, 91, 15, 64, 21, 8, 88, 54 P0 Parallel Quicksort 75, 91, 15, 64, 21, 8, 88, 54 Parallel Quicksort 20 18 P0 P2
P3 50, 12, 47, 72, 65, 54, 66,
22, 20, 40, 47, 19, 61 83, 98, 99, 82, 91, 88 89, 86, 85 21 CS 6643 F '11 Lec 22 Upper
“half” Lower
“half” CS 6643 F '11 Lec 22 P3 23 98, 99 Upper “half”
of upper “half” Exchange values 83, 82, 91, 88, 89, 86, 85 Lower “half”
of upper “half” P2 P1 50, 47, 72, 65, 54, 66, 22, 40,
47, 61, 75, 64, 54, 66, 67, 70 Upper “half”
of lower “half” CS 6643 F '11 Lec 22 98, 99 82, 83, 85, 86, 88, 89, 91 22, 40, 47, 47, 50, 54, 54, 61,
64, 65, 66, 66, 67, 70, 72, 75 Each processor sorts values it controls Upper “half”
of upper “half” Lower “half”
of upper “half” Upper “half”
of lower “half” Lower “half”
of lower “half” 15, 21, 8, 0, 12, 20, 19 Lower “half”
of lower “half” 0, 8, 12, 15, 19, 20, 21 Parallel Quicksort Exchange values 89, 86, 85 83, 98, 99, 82, 91, 88 50, 12, 47, 72, 65, 54, 66,
22, 20, 40, 47, 19, 61 75, 15, 64, 21, 8, 54, 66, 67, 0, 70 Parallel Quicksort CS 6643 F '11 Lec 22 Processes P0 and P2 choose and
broadcast randomly chosen pivots P1 75, 15, 64, 21, 8, 54, 66, 67, 0, 70 Parallel Quicksort P0 Upper
“half” Lower
“half” Parallel Quicksort P3 P2 P1 P0 P3 P2 P1 P0 24 22 CS 6643 F '11 Lec 22 Execution
Execution time dictated by when last
process completes
Algorithm
Algorithm likely to do a poor job balancing
number of elements sorted by each process
Cannot
Cannot expect pivot value to be true median
Can
Can choose a better pivot value Analysis of Parallel Quicksort CS 6643 F '11 Lec 22 Time
Time complexity:
Tp = local sort time + splitsendmerge time + pivot selection
local
splitsend= Θ(n/p log(n/p)) + Θ(n/p log p) + Θ(log2 p)
p) Nodes
Nodes that have the same bit pattern for bits (log p 1) … (i+1) will form a subcube
(exception:
(exception: all nodes are used for i= log p 1)
Use allUse allreduce for pivot selection and dissemination
in
in each subcube
subcube
Each
Each process compares its elements with pivot and
sends elements greater than (less than or equal to)
pivot to neighbor in dimension i if that neighbor has
higher (lower) id n/p
n/p elements per process
Step
Step i = log p 1 … 0 Quicksort on Hypercube 27 25 CS 6643 F '11 Lec 22 First,
First, each process sorts its sublist
To
To complete sorting, processes exchange
values
Process
Process can use median of its sorted list as
the
the pivot value
This
This is much more likely to be close to the
true median Hyperquicksort CS 6643 F '11 Lec 22 In
In sequential version: a bad pivot increases
the number of recursive steps by 1 For
For example, all elements may be distributed
among processes 0, …, p/21 after the first step
p/2perform
perform quicksort with 2n/p elements per
process
process In
In parallel version: each time a bad pivot is
slected, the time to finish the rest of the
steps is increased by a factor of 2 Impact of Pivot Selection 28 26 CS 6643 F '11 Lec 22 29 CS 6643 F '11 Lec 22 19, 20, 40, 47, 61, 85, 86, 89 P3 19, 20, 40, 47, 61, 85, 86, 89 31 0, 66, 67, 70, 82, 83, 98, 99 P2 0, 66, 67, 70, 82, 83, 98, 99 P3 P2 P1 P0 P3 P2 P1 P0 Processes will exchange “low”, “high” lists 12, 22, 47, 50, 54, 65, 66, 72 P1 12, 22, 47, 50, 54, 65, 66, 72 Process P0 broadcasts its median value 8, 15, 21, 54, 64, 75, 91, 88 8, 15, 21, 54, 64, 75, 91, 88 Hyperquicksort P0 CS 6643 F '11 Lec 22 19, 20, 40, 47, 61, 85, 86, 89 P3 20, 40, 89, 47, 19, 61, 86, 85 Hyperquicksort CS 6643 F '11 Lec 22 0, 66, 67, 70, 82, 83, 98, 99 P2 83, 66, 67, 0, 70, 98, 99, 82 Each process sorts values it controls 12, 22, 47, 50, 54, 65, 66, 72 P1 50, 12, 47, 72, 65, 54, 66, 22 Number of processors is a power of 2 8, 15, 21, 54, 64, 75, 88, 91 P0 Hyperquicksort 75, 91, 15, 64, 21, 8, 88, 54 Hyperquicksort 32 30 CS 6643 F '11 Lec 22 33 CS 6643 F '11 Lec 22 83, 85, 86, 88, 89, 91, 98, 99 P3 61, 65, 66, 72, 85, 86, 89 After exchangeandmerge step 61, 64, 65, 66, 66, 67, 70, 72, 75, 82 P2 64, 66, 67, 70, 75, 82, 83, 88, 91, 98, 99 Communication pattern for second exchange 19, 20, 21, 22, 40, 47, 47, 50, 54, 54 P1 12, 19, 20, 22, 40, 47, 47, 50, 54 35 0, 8, 12, 15 P0 Hyperquicksort 0, 8, 15, 21, 54 CS 6643 F '11 Lec 22 61, 65, 66, 72, 85, 86, 89 P3 61, 65, 66, 72, 85, 86, 89 P3 P2 P1 P0 P3 P2 P1 P0 Processes P0 and P2 broadcast median values. 64, 66, 67, 70, 75, 82, 83, 88, 91, 98, 99 P2 64, 66, 67, 70, 75, 82, 83, 88, 91, 98, 99 Processes merge kept and received values. 12, 19, 20, 22, 40, 47, 47, 50, 54 P1 12, 19, 20, 22, 40, 47, 47, 50, 54 Hyperquicksort CS 6643 F '11 Lec 22 0, 8, 15, 21, 54 P0 Hyperquicksort 0, 8, 15, 21, 54 Hyperquicksort 36 34 CS 6643 F '11 Lec 22 The
The value of C determines the scalability.
Scalability depends on ratio of communication speed
to computation speed. M ( p C ) / p = p C / p = p C −1 Memory
Memory scalability: Asymptotic rate of n depends on C
For C≤1, the parallel system is highly scalable n log n = C n log p ⇒ log n = C log p ⇒ n = pC Sequential
Sequential time complexity: Θ(n log n)
Parallel
Parallel overhead: Θ(n log p)
Isoefficiency
Isoefficiency relation: Isoefficiency Analysis CS 6643 F '11 Lec 22 AverageAveragecase analysis
Lists
Lists stay reasonably balanced
Communication
Communication time dominated by message
transmission
transmission time, rather than message
latency
latency Complexity Analysis Assumptions 39 37 CS 6643 F '11 Lec 22 Our
Our analysis assumes lists remain balanced
As
As p increases, each processor’s share of list
decreases
Hence
Hence as p increases, likelihood of lists
becoming unbalanced increases
Unbalanced
Unbalanced lists lower efficiency
Would
Would be better to get sample values from
all processes before choosing median Another Scalability Concern CS 6643 F '11 Lec 22 Total
Total communication time for log p exchange
steps:
steps: Θ((n/p) log p) Initial
Initial quicksort step has time complexity
Θ((n/p) log (n/p))
Total
Total comparisons needed for log p merge
steps: Θ((n/p) log p) Complexity Analysis 40 38 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 43 CS 6643 F '11 Lec 22 0, 66, 67, 70, 82, 83, 98, 99 P2 0, 66, 67, 70, 82, 83, 98, 99 P2 P1 Each process chooses p regular samples. 12, 22, 47, 50, 54, 65, 66, 72 P1 12, 22, 47, 50, 54, 65, 66, 72 Each process sorts its list using quicksort. 8, 15, 21, 54, 64, 75, 88, 91 PSRS Algorithm P0 P2 83, 66, 67, 0, 70, 98, 99, 82 Number of processors does not
have to be a power of 2. P1 50, 12, 47, 72, 65, 54, 66, 22 P0 41 P0 75, 91, 15, 64, 21, 8, 88, 54 PSRS Algorithm 8, 15, 21, 54, 64, 75, 88, 91 PSRS Algorithm CS 6643 F '11 Lec 22 Each
Each process sorts its share of elements
Each
Each process selects regular sample of sorted list
One
One process gathers and sorts samples, chooses
pivot values from sorted sample list, and broadcasts
these pivot values
Each
Each process partitions its list into p pieces, using
pivot
pivot values
Each
Each process sends partitions to other processes
Each
Each process merges its partitions Parallel Sorting by Regular Sampling (PSRS
Algorithm) 44 42 CS 6643 F '11 Lec 22 45 CS 6643 F '11 Lec 22 One process sorts p2 regular samples. 47 One process chooses p1 pivot values.
CS 6643 F '11 Lec 22 0, 66, 67, 70, 82, 83, 98, 99 P2 0, 66, 67, 70, 82, 83, 98, 99 P2 P1 P0 P2 P1 P0 One process broadcasts p1 pivot values. 15, 22, 50, 54, 65, 66, 70, 75, 83 12, 22, 47, 50, 54, 65, 66, 72 P1 12, 22, 47, 50, 54, 65, 66, 72 15, 22, 50, 54, 65, 66, 70, 75, 83 8, 15, 21, 54, 64, 75, 88, 91 8, 15, 21, 54, 64, 75, 88, 91 PSRS Algorithm P0 One process collects p2 regular samples. PSRS Algorithm CS 6643 F '11 Lec 22 0, 66, 67, 70, 82, 83, 98, 99 P2 0, 66, 67, 70, 82, 83, 98, 99 15, 22, 50, 54, 65, 66, 70, 75, 83 12, 22, 47, 50, 54, 65, 66, 72 P1 12, 22, 47, 50, 54, 65, 66, 72 15, 54, 75, 22, 50, 65, 66, 70, 83 8, 15, 21, 54, 64, 75, 88, 91 P0 PSRS Algorithm 8, 15, 21, 54, 64, 75, 88, 91 PSRS Algorithm 48 46 P0 CS 6643 F '11 Lec 22 P2 54, 54, 64, 65, 66, 66 67, 70, 72, 75, 82, 83, 88, 91, 98, 99 Each process merges p partitions. P1 0, 8, 12, 15, 21, 22, 47, 50 PSRS Algorithm CS 6643 F '11 Lec 22 49 51 P2 P1 P0 CS 6643 F '11 Lec 22 Each
Each process ends up merging close to n/p elements
Experimental
Experimental results show this is a valid assumption
Processor
Processor interconnection network supports p
simultaneous message transmissions at full speed
4ary hypertree is an example of such a network Assumptions CS 6643 F '11 Lec 22 75, 88, 91 72 67, 70, 82, 83, 98, 99 P2 0, 66, 67, 70, 82, 83, 98, 99 Each process sends partitions to
correct destination process. 54, 64 54, 65, 66 66 P1 12, 22, 47, 50, 54, 65, 66, 72 Each process divides list, based on pivot
values. 8, 15, 21 12, 22, 47, 50 0 P0 PSRS Algorithm 8, 15, 21, 54, 64, 75, 88, 91 PSRS Algorithm 52 50 CS 6643 F '11 Lec 22 Parallel
Parallel quicksort and hyperquicksort: log p / 2
hyperquicksort:
PSRS
PSRS algorithm: (p1)/p Average
Average number of times each key moved: Parallel quicksort:
Parallel quicksort: poor
Hyperquicksort:
Hyperquicksort: better
PSRS
PSRS algorithm: excellent Three
Three parallel algorithms based on quicksort
Keeping
Keeping list sizes balanced Summary of Quicksort Methods CS 6643 F '11 Lec 22 Gather
Gather samples, broadcast pivots: Θ(log p)
Alltoall exchange: Θ(n/p)
AlltoOverall:
Overall: Θ(n/p) Communications
Communications Initial
Initial quicksort: Θ((n/p)log(n/p))
Sorting
Sorting regular samples: Θ(p2 log p)
Merging
Merging sorted sublists: Θ((n/p)log p
Overall:
Overall: Θ((n/p)(log n + log p) + p2log p)
p) Computations
Computations Time Complexity Analysis 55 53 CS 6643 F '11 Lec 22 Sorting Networks Sequential
Sequential time complexity: Θ(n log n)
Parallel
Parallel overhead: Θ(n log p)
Scalability
Scalability function same as for
hyperquicksort
Scalability
Scalability depends on ratio of
communication
communication to computation speeds Isoefficiency Analysis 54 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 Typical Sorting Network • Special networks with Omega network like topologies
using columns of twoelement comparators
• Sorting algorithm is embedded into the network through
the connection pattern
• Elements to be sorted are fed to the network at the left
and sorted sequence will appear at the right of the
network
network
• Time taken is proportional to the depth or number of
columns of comparators Sorting Networks 59 57 Bitonic Sorting Network CS 6643 F '11 Lec 22 » <1,3,5,7,8,6,4,2,0> is a bitonic sequence with i=5
» <7,8,6,4,2,0,1,3,5> is a bitonic sequence since rightrotating it by
three elements yields seq. in the first example
» Any sorted sequence is trivially bitionic – There exists an index 1≤i≤n such that <a1,…,ai> is monotonically
increasing and <ai, …, an> is monotonically decreasing, or
– There exists a cyclic shift or indices such that the above
condition
condition is satisfied
– Examples: • Proposed by Batcher in 1968
• Bitonic sequence is a sequence of <a1, …, an> such that
either CS 6643 F '11 Lec 22 • Two types: increasing comparator and decreasing comparator Comparators 60 58 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 Bitonic Merging Network – Given two bitonic sequences, create a larger bitonic sequence
consisting of elements in both sequences
– Given a bitonic sequence, obtain a sorted sequence • Two key operations Bitonic Sorting Network 63 61 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 Column of 2element
comparators Bitonic Merging Network – Denoted as BM(n)
– Has depth log n • Use bitonic splits repeatedly (log n) times to transform a
bitonic sequence into a sorted sequence
• A network that achieves this is called a bitionic merging
network – Let S=<a1, …, an> be a bitonic sequence. Then
S1 = <min{a1, an/2+1}, …,min{an/2,an}> and
S2 = <max{a1, an/2+1}, …,max{an/2,an}> are bitonic sequences
such that each element of S1 ≤ any element of S2
– Splitting S into S1 and S2 as given above is called a bitonic split • Given a bitonic sequence, obtain a sorted sequence
• Bitonic split Bitonic Merge 64 62 CS 6643 F '11 Lec 22 Bitonic Sorting Network for 16 Elements CS 6643 F '11 Lec 22 – Divide the given sequences into subsequences of size 2, which
are trivially bitonic
– Use BM networks so sort the subsequences into either
ascending
ascending or descending order as needed
– Concatenate adjacent two subsequences to create larger bitonic
sequences
– Repeat and until all n numbers form a bitonic sequence
– Use a BM(n) to finish the sorting • Given an arbitrary sequence of n elements, create a
bitonic sequence of n elements
• How Bitonic Sorting Network (BSN) 67 65 Bitonic Sorting Network CS 6643 F '11 Lec 22 d(n) = d(n/2) + log n
⇒ d(n) = [ (log n) (logn+1)]/2 • BM(n) can be implemented by an nxn Omega network
with 2element comparators instead of 2x2 switching
elements
• Time taken = depth of BSN, d(n) CS 6643 F '11 Lec 22 Bitonic Sequence Creating Network 68 66 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 Dimnesions used for Bitonic Sorting • 1 element per process
• Careful observation of BSN(n) indicates that elements at
distances that are power of 2 are compared and
exchanged
• So each compareandexchange operation done by
BSN(n) is near neighbor communication on a hypercube Bitonic Sorting on a Hypercube 71 69 Bitonic Sorting on a Hypercube CS 6643 F '11 Lec 22 • Add a prefix of 0 to lablel. This will be the bit used for step 5 when
i=d1.
• Tp = Ο(log2n)
P Tp = Ο(nlog2n), which is not costoptimal CS 6643 F '11 Lec 22 Example of Bitonic Merging on a Hypercube 72 70 Rowmajor snakelike order Rowmajor shuffled order Bitonic Sorting on a Mesh CS 6643 F '11 Lec 22 • Use n>>p, # of sorting elements >> # processes, for
costoptimal parallel sorting
• The idea is minimize the impact of the cost of
communication and excess comparisons inherent in the
underlying algorithm with enough work in each process
• With n=p, each process performs Ο(log2n) comparisons – Not optimal w.r.t. seq. work
– But it is the best that can be done on a mesh
– Distance argument • Tp = Ο(n1/2)
• Parallel cost,
PTp = Ο(n3/2) CS 6643 F '11 Lec 22 Rowmajor order • Apply hypercube/BSN algorithm
• How are elements mapped to processes?
• Majority of compare and exchanges will take multiple hops and thus
cause link contention Bitonic Sorting on a 2D Mesh 75 73 Multiple Elements Per Process CS 6643 F '11 Lec 22 • Let n>>p
• Each process has n/p elements
• Initially each process sorts the elements using a
sequential sorting algorithm
• Each process participates in bitonic sorting using
exchange
exchangemergesplit as the primitive operations CS 6643 F '11 Lec 22 Rowmajor shuffled order BM(16) on Mesh 76 74 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 OddEven Sorting Example • Tp = local sort time + merger & split time + exchange time
= Θ(n/p log(n/p)) + Θ(n/p log2 p) + Θ(n/p log2 p)
• For efficient operation, p = Ο(2(log n)1/2) Multiple Elements Per Process (Hypercube) 79 77 CS 6643 F '11 Lec 22 CS 6643 F '11 Lec 22 Parallel OddEven Sorting – Odd and even phases with n/2
comparisons in each phase
– In the Odd phase, elements in odd
positions are compared and
exchanged, if necessary, with the
next elements
– In the Even phase, even elements
are compared and exchanged with
the next elements
– n steps
– Easy to parallelize • Oddeven transposition sort for
even # of elements – Also it is difficult to parallelize the
key compareandexchange step
since each step is influenced by
the previous step • Bubble sort has Θ(n2) sequential
time complexity Sorting using General Purpose Processes 80 78 Enumeration Sort CS 6643 F '11 Lec 22 • Rank each element such that rank of ai is number of elements smaller than
ai in the original sequence
• Elements are permuted using the rank
• Using Ο(n2) processes give Ο(1) sorting time CS 6643 F '11 Lec 22 – Initially, processes that are far away from each other compare
and split their elements (log p steps)
– Then oddeven sort is performed as long as processes are
exchanging data • Shell sort will be useful in such cases – For almost sorted sequences with a few elements O(n) positions
away from the final sorted position will require O(n) iterations in
the seq. case • Oddeven sort moves elements only one position at a
time Shell Sort 83 81 CS 6643 F '11 Lec 22 Shell Sort Example 82 ...
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 Sort, Insertion Sort, LEC

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