Question
Sorting networks are an interesting class of parallel sorting algorithms with applications in multi-processor computers and switching networks.
(a) Prove or disprove that adding another comparator on a sorting network at
(i) the end,
(ii) the front, or
(iii) anywhere

does not destroy the sorting property.

(b) Show via a counter-example that a network which contains comparators between every pair of wires is not necessarily a sorting network.

(c) Prove that there are some sorting networks cannot be used backwards, i.e. use the output wires for input and vice versa.

(d) Give a counter example to show that the following network is not a sorting network.
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Problem
a) i)
This does not change anything, because the input is already sorted before being processed by the comparator which is added to the end. Thus the conclusion is true, the property of the network is not destroyed.

ii). The conclusion is again true, because the comparator which is added at the front (before the sorting network) might change the original input only slightly. Or, this does not contradict the sorting property of the original sorting network which will process the modified input accordingly.
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      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • creating: site.layouts.app (531μs)
      • composing: site.layouts.app (25μs)
      • creating: components.canonical (552μs)
      • composing: components.canonical (111μs)
      • creating: components.open-graph (362μs)
      • composing: components.open-graph (103μs)
      • creating: site.headers.header (325μs)
      • composing: site.headers.header (78μs)
      • Illuminate\Cache\Events\CacheHit (1.85ms)
      • creating: components.footer (90μs)
      • composing: components.footer (100μs)
      • Illuminate\Cache\Events\CacheHit (988μs)
      • Illuminate\Cache\Events\CacheMissed (452μs)
      • Illuminate\Database\Events\StatementPrepared (767μs)
      • Illuminate\Database\Events\QueryExecuted (1.28ms)
      • eloquent.retrieved: App\Models\SubjectCat (103μs)
      • Illuminate\Cache\Events\KeyWritten (756μs)
      • Illuminate\Cache\Events\CacheHit (290μs)
      • Illuminate\Cache\Events\CacheHit (148μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (297μs)
      • Illuminate\Cache\Events\CacheHit (548μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • creating: components.forms.contact-us (290μs)
      • composing: components.forms.contact-us (184μs)
      • creating: components.forms.get-started (193μs)
      • composing: components.forms.get-started (92μs)
      • creating: components.forms.free-tool-download (150μs)
      • composing: components.forms.free-tool-download (80μs)
      • creating: components.forms.claim-free-worksheet (146μs)
      • composing: components.forms.claim-free-worksheet (78μs)
      • creating: components.forms.tutor-subscription-waitlist (146μs)
      • composing: components.forms.tutor-subscription-waitlist (77μs)
      • creating: components.forms.tutor-subscription-join (147μs)
      • composing: components.forms.tutor-subscription-join (76μs)
      • creating: components.forms.tutor-support (143μs)
      • composing: components.forms.tutor-support (73μs)
      • 312 x Illuminate\Cache\Events\CacheHit (17.05%)
        73.03ms
        19 x Illuminate\Database\Events\StatementPrepared (8.04%)
        34.43ms
        19 x Illuminate\Database\Events\QueryExecuted (4.53%)
        19.40ms
        10 x Illuminate\Cache\Events\CacheMissed (1.92%)
        8.22ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.02%)
        4.37ms
        10 x Illuminate\Cache\Events\KeyWritten (1.01%)
        4.31ms
        1 x Illuminate\Routing\Events\Routing (0.32%)
        1.35ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.25%)
        1.06ms
        11 x eloquent.retrieved: App\Models\SubjectCat (0.22%)
        960μs
        1 x creating: components.canonical (0.13%)
        552μs
        1 x Illuminate\Routing\Events\RouteMatched (0.12%)
        531μs
        1 x creating: site.layouts.app (0.12%)
        531μs
        1 x creating: components.open-graph (0.08%)
        362μs
        1 x creating: site.headers.header (0.08%)
        325μs
        1 x eloquent.booting: App\Models\SubjectCat (0.07%)
        301μs
        1 x creating: components.breadcrumbs (0.07%)
        296μs
        1 x creating: components.forms.contact-us (0.07%)
        290μs
        6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        257μs
        1 x composing: homework.show (0.06%)
        241μs
        2 x eloquent.retrieved: App\Models\Subject (0.05%)
        231μs
        1 x creating: components.forms.get-started (0.05%)
        193μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        189μs
        1 x creating: homework.show (0.04%)
        185μs
        1 x composing: components.forms.contact-us (0.04%)
        184μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        171μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        171μs
        1 x eloquent.booted: App\Models\Subject (0.04%)
        163μs
        1 x creating: components.forms.free-tool-download (0.04%)
        150μs
        1 x creating: components.forms.tutor-subscription-join (0.03%)
        147μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        146μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        146μs
        1 x creating: components.forms.tutor-support (0.03%)
        143μs
        1 x composing: components.breadcrumbs (0.03%)
        128μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        111μs
        1 x composing: components.canonical (0.03%)
        111μs
        1 x composing: components.open-graph (0.02%)
        103μs
        1 x composing: components.footer (0.02%)
        100μs
        1 x composing: components.forms.get-started (0.02%)
        92μs
        1 x creating: components.footer (0.02%)
        90μs
        1 x composing: components.forms.free-tool-download (0.02%)
        80μs
        1 x composing: site.headers.header (0.02%)
        78μs
        1 x composing: components.forms.claim-free-worksheet (0.02%)
        78μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.02%)
        77μs
        1 x composing: components.forms.tutor-subscription-join (0.02%)
        76μs
        1 x composing: components.forms.tutor-support (0.02%)
        73μs
        1 x eloquent.booting: App\Models\Subject (0.01%)
        58μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        45μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        42μs
        1 x composing: site.layouts.app (0.01%)
        25μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      19 statements were executed, 5 of which were duplicates, 14 unique. Show only duplicated39.28ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '45011' limit 1
        18.34mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 45011
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (228)
        1.17mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (45011)
        1.19mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.1mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 45011 and `subject` = 228 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        2.12mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 45011
        • 1: 228
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (228)
        1.21mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 45011 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        950μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 45011
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 45011 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        830μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 45011
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 45011 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        900μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 45011
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.03mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        980μstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.07mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.03mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.5mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.12mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.34mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      11SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      7HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      6HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          WYfWSsPoVtU8mWYmoOdUvhb7WMY7z0dka53Rhhub
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Parallel-Computing/45011
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Computer-Scienc...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
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          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
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          request_cookies
          []
          
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