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Super Scalar Sample Sort
This is a paper I wrote for ETH, I thought I might as well publish it here
Introduction
Abstract
The Sample Sort Algorithm
How it works
The input is an array of n integers. We take the first k numbers in the array and use them as splitters. Splitters are elements of the input that will divide the input into smaller pieces. It is fairly simple: You take out the splitters and sort them (using quicksort or inline algorithm). Between the splitters you put buckets. If our input are the numbers 1-100 and our splitters are 32,57 and 73 it will look like this:1 32 57 73 100 \ Bucket / \ Bucket / \ Bucket / \ Bucket / """""""" """""""" """""""" """"""""Once we have the splitters and the buckets, we run through the input sequentially and throw every element in to the according bucket. This is on step.
Do the whole thing recursivly for every bucket (with the elements in the bucket as the input). The recursion stops when there is at most one element in every bucket. On the way, store the buckets and the splitters in sequential order.
In the end we will have many small buckets in a sorted order that only contain at most one element. If we now run over all buckets and splitters sequentially we have the sorted input.
[Image]
The Speed-Ups (Super Scalar Sample Sort)
Let us look a little more into processor architecture. In the following paragraphs the word processor is an Intel Pentium 4 (later 90nm generation) processor.An instruction is not executed by the processor in one step. In fact an instruction in a processor runs through 31 stages called the execution pipeline (reading the instruction, getting all the data from memory that is needed to execute this instruction, performing the instruction, writing the result of the instruction back, etc). It would not be very wise to wait until every instruction ran through all the execution stages before processing the next one. That's why processors use instruction parallelism, which means that they already start the next instruction, before the previous instruction has completly finished. This is not always trivial or possbile and can result in big delays, which unnecessarily slow down the algorithm.
Conditional Branches
When the processor comes to a Conditional Branch it tries to predict its direction in order to keep the execution pipeline full. The prediction rule is simple: It always evaluates the condition as true and hence executes the fist branch. Should the processor finally evaluate the expression as false and hence mispredicted the branch it needs to flush the execution pipeline (which costs 31 cycles!) in order to be in a consistent state again.In many cases a programmer can predict the control flow of an application and can optimize the conditional branches, so that at execution time, the processor doesn't mispredict branches all the time.
With sorting though the problem of predicting a branch becomes impossible. If we have a conditional branch and we have to predict whether an element will be smaller or greater than another element, we have a 50% chance of beeing correct. This means that every second time we compare two elements we loose 31 processor cycles.
element = randomNumber();
splitter = randomNumber();
if ( element < splitter ) {
// The processor will predict this branch. There is a 50% chance of beeing correct
LeftBucket.add(element)
} else {
// If this is the correct branch, we lose 31 processor cycles.
RightBucket.add(element)
}
The k-Splitters and the buckets
In order to get rid of conditional branches, we do a neat trick to place the elements in the correct bucket, without using conditional branches.k = 1
If we have only one splitter ( k = 1) it is fairly easy. We put the two buckets in an array and are using a boolean expression to find the correct bucket number in the array (assuming thattrue==1 and false==0):
element = randomNumber(); splitter = randomNumber(); BucketArray[0]=LeftBucket; BucketArray[1]=RightBucket; BucketArray[element>splitter].add(element);
any k
If we have any k (preferably a power of 2 -1) we build a binary search tree out of the k splitters in order to find the correct bucket.[Image] If we decide k to be small enough to fit the k splitters into the processors registers we do so, but usually we should choose k = 255 to allow byte sized comparisons. We keep the 255 splitters in a buffered array.This is the example with k = 3 splitters, which results in 4 buckets:
element = randomNumber();
s1,s2,s3 = getSplitters();
BucketArray[] = getBuckets();
firstBranch= element > s2;
BucketArray[ firstBranch*2
+ (!fistBranch & element > s1)*1
+ (fistBranch & element > s3)*1
].add(element)
Sanders does not do it exactly like that, he uses special conditional move instructions that only modern processors support, but the result is the same.
Pre-Allocating Memory
We can get an additional speed-up, by first allocating buckets with the correct bucket sizes in memory. In order to do that we need to know the bucket sizes in advance.Instead of throwing the elements directly into the buckets, we first run over all elements and tell them in which bucket they will be thrown and at the same time we keep track how big the buckets will get. After we processed all elements we allocate the buckets with the correct size and throw every element into the right bucket (it will remember in which bucket it belongs).
Instead of always allocating each bucket individually we allocate an auxilary array in the beginning with size n and use it for our buckets. After every iteration step this auxilary array becomes the input for the next iteration.
Oversampling k
To have more equaly sized buckets we should oversample k by a factor a (look at a*k possible splitters) and take the most equally distributed k ones.Sorting smaller subproblems
Ofcourse it we would have a massive overhead if the buckets get really small and we still would use SSSS for sorting them.Sanders suggests the following:
| #elements | Algorithm |
|---|---|
| 0 - 5 | customized straight-line code |
| 5 - 100 | insertion sort |
| 100 - 1000 | quicksort |
| 1000 - ... | Super Scalar Sample Sort |
Things we get for free
No Data Dependencies
What it is
There is instruction A that comes before an instruction B. Data Dendency occurs if B gets stalled because it waits for the result of A, but A is has not yet finished:c = a + b; // Instruction A d = c + 1; // Instruction B
Why we don't have it
As soon as we have the k -splitters fixed, all elements can be placed in the correct bucket independently.The Speed-Up
Since we don't have any data dependency we can interleave the corresponding instructions for several elements. The compiler will automaticly be able to do loop-unrollment and software-pipelining.Memory Hierarchies / Cache-Oblivious
n = Number of input elementsk = Number of splitters
M = Number of elements that fit into primary memory
B = Number of elements that can be transfered from secondary to primary memory in one step.
Since we can read through all the elements sequentially we have perfect spatial locality. Quicksort needs to write the entire input and output log(n/m) times before a subproblem fits into M. But since Super Scalar Sample Sort is like a divide-and-conquer algorithm and splits the problem into subproblems of about size n/k we only need to read and write the input log.k.(n/m) times. In the I/O Model with an omniscent cache of size M and transfers of size B, we have k=O(M/B).
By appropratly choosing the size of k, this algorithm becomes cache oblivious, under the assumption that all k-splitters fit into memory and they are optimally distributed.
Experiments
How to create an ErrorPage? with in your WorkPlaceForm?
This page is a walk-through to generate an ErrorPage? in your forms so the form user would access before finalizing the form (eithe submit or sign). There are a few steps before we start. It also worth mentioning that this is not the only method to generate or warn the error to users but to the best of my knowledge it was the most creative way to warn and direct the form user to the errors he/she made while filling the form. To start we need to know a few back ground on the form error types and how to approach the problem:- the common source of errors are:
- Leaving a mandatory field blank (UnfilledFields)
-- IvicaAracic - 29 Jan 2007
test
This is a test.
another test
-- TWikiGuest - 27 Mar 2008
incldue{
-- TWikiGuest - 27 Mar 2008
toto je sandbox4610
Search
keyword: 'test', 'west', 'mest'
} SandBox4610 wikiname: TWikiGuest?
} SandBox4610 wikiname: TWikiGuest?
Searched: [k]eyword:.*test
Results from Sandbox web
-- Main.TWikiGuest 27 Mar 2008 toto je sandbox4610 Search keyword: 'test', 'west', 'mest' Advanced search: Topic text (body) Search all public INCLUDINGWEB WEBLIST ...
Number of topics: 1
A field inside a form which intentionally or accidentally is left blank
-- KavehJD - 10 Nov 2006
| WebHome | 23 Nov 2010 - 07:58 | KhooSiauCheng |
| NewSanboxTopic | 23 Nov 2010 - 07:58 | KhooSiauCheng |
| SandBox4610 | 11 Apr 2008 - 07:01 | TWikiGuest |
| SandBox4611 | 27 Mar 2008 - 11:20 | TWikiGuest |
| WebLeftBar | 19 Mar 2008 - 07:11 | TWikiGuest |
| YourOwnSandBox | 25 Feb 2008 - 13:34 | TWikiGuest |
| WebStatistics | 11 Feb 2008 - 01:12 | TWikiGuest |
| IvicaAracicSandbox | 29 Jan 2007 - 20:25 | IvicaAracic |
| AndrinVonRechenbergSandbox | 28 Dec 2006 - 17:50 | AndrinVonRechenberg |
| UnfilledFields | 10 Nov 2006 - 19:33 | KavehJD |
| DefaultWebHome | 10 Nov 2006 - 19:31 | KavehJD |
| WebNotify | 28 Aug 2004 - 07:29 | PeterThoeny |
| WebRss | 16 Aug 2004 - 03:27 | PeterThoeny |
| WebPreferences | 16 Aug 2004 - 03:16 | PeterThoeny |
| WebSearchAdvanced | 18 Jan 2004 - 10:51 | PeterThoeny |
| WebIndex | 24 Nov 2001 - 11:39 | PeterThoeny |
| WebTopicList | 24 Nov 2001 - 11:39 | PeterThoeny |
| WebChanges | 16 Aug 2001 - 19:56 | PeterThoeny |
| WebSearch | 08 Aug 2001 - 05:56 | PeterThoeny |
How to create an ErrorPage? with in your WorkPlaceForms
This page is a walk-through to generate an ErrorPage? in your forms so the form user would access before finalizing the form (either submit or sign). Ativan in pregnancy can cause congenital defects in the future child. Do not use lorazepam without the consent of the doctor if you are pregnant. Inform your doctor if you became pregnant during reception ativan drug. Use effective remedies of the control over birth rate when accept this drug. Till now influence lorazepam on chest feeding is not studied. Soothing effects lorazepam can be shown more at old people. Avoid hit in open wounds of a preparation lorazepam. Do not give ativan drug to the child, is younger than 12 years. There are a few steps before we start. It also worth mentioning that this is not the only method to generate or warn the error to users but to the best of my knowledge it was the most creative way to warn and direct the FormUser? to the errors he/she made while filling the form. To start we need to know a few background on the form error types and how to approach the problem:- the common source of errors are:
- Leaving a mandatory field blank (UnfilledFields)
- Filling a field with incorrect format
- Signiture Buttons which are set to be mandatory
Spotting errors in a single page
There are occasions that we are only interested in the errors found in a particular page. Unfortunately as to this date, there is no option available within functions of WorkPlaceForms? (WPF 2.6.1) to target only errors in one page. However, there is an efficient way to target these errors. The following code explains it better: Assume the form consist of few pages. we are interested in the errors in myPAGE and we want the next button (for simplicity we refer to it as myBUTTON) only activated when there is no error in the page. In given code we set an action (called myAction) to first find about any possible error in the whole form. Then we set a for loop up to the last element in the errorsFound_array array and with in the for loop we search the sid of the elements using strstr() function. If the sid of the desired page (myPAGE) was found we dissable myBUTTON. The code is written inside an action and action triggeres every time the focuse of the fields changed inside form. When the action is triggered the number of errors found inside the myPAGE page will be updated. inorder to trigger the action this line of code shgould also be included in the global part of the myPAGE page:
<custom:on_fouce xfdl:compute="toggle(myPAGE.global.focuseditem) == '1' ="

? (set('myPAGE.myAction.active', 'on')) : ''"></custom:on_fouce>
for an alternative approach please refer to this page.
Creating a list of errors
A more fancy approach to excite the form user is to enable him to see all the errors he/she made within the form and ask him/her to go back and correct them. The way to do this for the whole form is a little tricky and we should use a more complex code with in the ErrorPage? we create. To make this happen we basically need to creat a few items on the page. on following I have addeded a ToDO? list of elements which should apper in your error page as well as on all pages which an error might exist. When all these are completed the form enable the user to see the errors he made, he click on it and the form automatically will take the user to the corresponding field. The items to be added in the error page:- a list with a group name so we can add cells to it later. Make the list long enough so it could show at list 10 errors at the time. Here is sample code:
<list sid="myLIST">
<itemlocation>
<x>100</x>
<y>100</y>
<width>500</width>
<height>150</height>
</itemlocation>
<group>ERROR_LIST_GROUP</group>
<value></value>
<custom:on_click xfdl:compute="toggle(value) == '1' and strlen(value) > '0' 

? set('errPAGE.goToError.custom:current_item_ref',value->custom:item_sid) 

+. set('value','') 

+. set('errPAGE.goToError.activated','on') 

: '' ">2</custom:on_click>
</list>
- a number of cells with the same group name. Here is one cell but if you are willing to show 10 errors at the time you should repeat this for the 9 more cells (but with different arbitrary values:
<cell sid="ERROR_0">
<group>ERROR_LIST_GROUP</group>
<value></value>
<custom:item_sid></custom:item_sid>
</cell>
<cell sid="ERROR_1">
.
.
.
- an action for setting the items inside the error list. The code is given bellow:
<action sid="myAction">
<custom:onActivated xfdl:compute="

(toggle(active, 'off', 'on') == '1') 

? set('custom:counter', '1') 

+.set('custom:invalid_items_total',checkValidFormats('custom:invalid_items_array')) 

+. set('errPAGE.Num_Errors.value',countChildren('custom:invalid_items_array','option')) 

+.((countChildren('custom:invalid_items_array','option')-'1'>'9')?set('custom:numErrors','9') 

: set('custom:numErrors',countChildren('custom:invalid_items_array','option') - '1') )

+. for('custom:counter', '0', '9') 

+. set('active', 'off') 

: ''">2</custom:onActivated>
<custom:numErrors>9</custom:numErrors>
<custom:counter>9</custom:counter>
<custom:forLoopAction xfdl:compute=" 

toggle(custom:counter) == '1' and custom:counter <= custom:numErrors 

? set('errPAGE.ERROR_' + custom:counter + '.custom:item_sid', 

get('custom:invalid_items_array[' + custom:counter + ']')) 

+.((strlen(get(get('custom:invalid_items_array['+ custom:counter +']')+'.value'))!= '0') 

? set('errPAGE.ERROR_' + custom:counter + '.value', get(get('custom:invalid_items_array[' +

custom:counter + ']')+ '.custom:errorMessage') + ' an Invalid form item entry.') 

: set('errPAGE.ERROR_' + custom:counter + '.value', get(get('custom:invalid_items_array[' + 

custom:counter+']')+'.custom:errorMessage')+' a Mandatory form item and it\'s not filled.')) 

: set('errPAGE.ERROR_' + custom:counter + '.custom:item_sid', 'errPAGE.LIST1') 

+. set('errPAGE.ERROR_' + custom:counter + '.value', '') 

">2</custom:forLoopAction>
<custom:invalid_items_total>4</custom:invalid_items_total>
<custom:invalid_items_array>
<ae>a</ae>
<ae>b</ae>
<ae>c</ae>
<ae>d</ae>
</custom:invalid_items_array>
</action>
- Customize the array of possible errors inside myAction. The name of array is important and also each entry should have an arbitrary value. There should be one array entry per any possible error. Therefore, the number of entries in the array should be equal to the total form errors if nothing is filled in the form. An example for a form with possible 4 errors came below:
<custom:invalid_items_array>
<ae>a</ae>
<ae>b</ae>
<ae>c</ae>
<ae>d</ae>
</custom:invalid_items_array>
(bug report: if you enter "2" as the value of the second array entry the code will crash. Could be a bug in the API behind it)
- Another action to take you where the error has happened. The code is given below:
<action sid="goToError">
<active>off</active>
<custom:current_item_ref>a</custom:current_item_ref>
<custom:dest_page_sid>b</custom:dest_page_sid>
<custom:dest_item_sid>c</custom:dest_item_sid>
<custom:onActivated xfdl:compute="

(toggle(activated, 'off', 'on') == '1') 

? set('custom:dest_page_sid', substr(custom:current_item_ref, '0', strstr 

(custom:current_item_ref, '.') - '1')) 

+. set('custom:dest_item_sid', substr(custom:current_item_ref, strstr 

(custom:current_item_ref, '.') + '1', strlen(custom:current_item_ref))) 

+. ((custom:dest_page_sid != getReference('', 'page', 'page')) 

? set('goToPage_BUTTON.custom:next_page', custom:dest_page_sid) 

+. set(custom:dest_page_sid +. '.global.custom:invalid_item', custom:dest_item_sid) 

+. set('goToPage_BUTTON.activated', 'on') 

: set(custom:dest_item_sid +. '.focused', 'on')) 

+. set('active', 'off') 

: ''"></custom:onActivated>
</action>
- Also to any page that we set a field to be mandatory or we set a format for the field we should add the folloing code to the global part of the page:
<custom:invalid_item></custom:invalid_item>
<custom:goTo_invalidItem xfdl:compute="

((toggle(focused, 'off', 'on') == '1') 

and (custom:invalid_item != '') 

) 

? set(custom:invalid_item +. '.focused', 'off') 

+. set(custom:invalid_item +. '.focused', 'on') 

+. set('custom:invalid_item', '') 

: ''"></custom:goTo_invalidItem>
- Last but not least, we add the following code to any field which is mandatory or have a defined format:
<custom:errorMessage>The field 'Field Name' on 'Field Page' is</custom:errorMessage>
After you apply the given changes to your form you should have an ErrorPage? set up in your form. For more information I have also attached a testform with an ErrorPage? which is created by applying the above changes to the form.
Finally, I should thank and credit this work to the following names which helped me in coding and/or ideas needed to make an
This is YourOwnSandBox topic.
This is a test.
another testResults from Sandbox web
Super Scalar Sample Sort This is a paper I wrote for ETH, I thought I might as well publish it here Introduction Abstract The Sample Sort Algorithm How it works The ...
How to create an ErrorPage with in your WorkPlaceForm This page is a walk-through to generate an ErrorPage in your forms so the form user would access before finalizing ...
Main.IvicaAracic 29 Jan 2007 test
This is a test. another test
-- Main.TWikiGuest 27 Mar 2008 toto je sandbox4610 Search keyword: 'test', 'west', 'mest' Advanced search: Topic text (body) Search all public INCLUDINGWEB WEBLIST ...
-- Main.TWikiGuest 27 Mar 2008 incldue{ INCLUDE{"SandBox4610"} } SandBox4610 wikiname: WIKINAME SEARCH{" k eyword:. test" type "regex" }
A field inside a form which intentionally or accidentally is left blank Main.KavehJD 10 Nov 2006
INCLUDE{" TWIKIWEB .WebChanges"}
How to create an ErrorPage with in your WorkPlaceForms This page is a walk-through to generate an ErrorPage in your forms so the form user would access before finalizing ...
SEARCH{"\. " scope "topic" regex "on" nosearch "on"} See also the faster WebTopicList
INCLUDE{"Trash.BlW" warn "off"} INCLUDINGWEB Web INCLUDINGWEB Web Home Changes Index Search Webs WEBLIST{" $name" separator ""} INCLUDE{" MAINWEB . WIKINAME LeftBar ...
This is a subscription service to be automatically notified by e-mail when topics change in this Sandbox web. This is a convenient service, so you do not have to come ...
Sandbox Web Preferences The following settings are web preferences of the Sandbox web. These preferences overwrite the site-level preferences in TWIKIWEB . WIKIPREFSTOPIC ...
TWiki's Sandbox web SCRIPTURL /view SCRIPTSUFFIX /Sandbox The Sandbox web of TWiki. TWiki is a Web-Based Collaboration Platform for the Corporate World. INCLUDE{ ...
INCLUDE{" TWIKIWEB .WebSearch"}
INCLUDE{" TWIKIWEB .WebSearchAdvanced"}
Statistics for Sandbox Web Month: Topic views: Topic saves: File uploads: Most popular topic views: Top contributors for topic save and uploads: Feb 2008 1239 1 0 ...
TOPICLIST{" $name "} See also the verbose WebIndex.
-- Main.TWikiGuest 28 Feb 2007 Hi, it's my test. Majherek Dec 13,2007 Hello, this is my test topic for lab1.6.2. CIT1121 bea http://www.cheapflightstomumbai.com ...
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-- TWikiGuest - 28 Feb 2007
Hi, it's my test. Majherek
Dec 13,2007 - Hello, this is my test topic for lab1.6.2. CIT1121 - bea
TWikiGuest WikiWord? Non Wiki Word wiki word Camel Case vs CamelCase??
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