dpr302
TRANSCRIPT
Asymptotes and Algorithms What You’ve Forgotten Since UniversityBy Gary ShortDeveloper EvangelistDeveloper Express
DPR302
Agenda
IntroductionWhat is an Algorithm?Why Optimization is ImportantOptimisation is Child’s PlayWhy do we Use Mathematics?One Tactic for Optimizing an AlgorithmBut .Net Saves me from all that, Right?Questions.
Introduction
Gary ShortDeveloper Evangelist DevExpressMicrosoft MVP in C#[email protected]@garyshort
What is an Algorithm?
… a finite list of well defined steps to complete a task or calculation.
There are simple ones…
There are complex ones…
Prove no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer
value of n greater than two
Why is optimization important?
http://www.flickr.com/photos/mushjazzlexima-leakpaalex/2237327967/
Optimization is Child’s Play
Take This Algorithm…
Stop current taskWash handsSit at table
Foreach item of food in MainCourseConsume food
Wait for dessertConsume dessertWait for meal completionForeach dish in dishes
Clear dish from table
Foreach dish in dishesWash dish
Return to previous task
Optimized by my 4 Year Old Daughter to…
Pause current gameSet velocity = MAX_INTMove to tableTakeSliceBread(2)Foreach item of food in MainCourse
Place item on BreadSlice(1)
Place BreadSlice(2) on topLeave tableResume current game
And it’s not a skill we lose in adulthood
Morning Algorithm
RiseShowerBoil kettleMake teaToast breadSpread butter on toastConsume breakfastBrush teethCommute to work
Optimized for Smallville…
RiseRush downstairsPut kettle on to boilPut toast in toasterRush back upstairsShowerRush downstairsMake tea and spread toastWatch SmallvilleConsume tea and toast
The Only Thing Hard About This is…
The language we chose to use to describe it
𝐴=𝜋 𝑟2
(𝑥+𝑎 )𝑛=∑𝑘= 0
𝑛
(𝑛𝑘)𝑥𝑘𝑎𝑛−𝑘
𝑓 (𝑥 )=𝑎0+∑𝑛=1
∞
(𝑎𝑛cos𝑛𝜋 𝑥𝐿
+𝑏𝑛sin𝑛𝜋 𝑥𝐿 )
𝑎 2+𝑏 2=𝑐 2
𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎
𝑒𝑥=1+ 𝑥1 !
+𝑥2
2 !+𝑥
3
3 !+…,−∞<𝑥<∞
𝐴=𝜋 𝑟2
𝐴=𝜋 𝑟2
𝑎 2+𝑏 2=𝑐 2
𝑎 2+𝑏 2=𝑐 2
𝑎2 +𝑏
2=𝑐2
𝐴=𝜋 𝑟2
(𝑥+𝑎)𝑛 =∑
𝑘=0
𝑛 (𝑛𝑘)𝑥𝑘 𝑎
𝑛−𝑘
So Why do we use Math?
http://www.flickr.com/photos/truelifeimages/2102930162/
http://www.flickr.com/photos/29019866@N00/4783131736/
Our Model Has…
Single processorRAMUnderstands arithmetical instructions
AddSubtractPopPush
All instructions executed sequentiallyAll instructions take constant time to complete.
demo
Running Time is Then…
Sum running times for each statementProduct of the cost and the timeSo…T(n) = c1n + c2(n-1) + c3(n-1) + c4(sum of tj for 1 <= j <= n) + c5(sum of tj -1 for 1 <= j <= n) + c6(sum of tj -1 for 1 <= j <= n) + c7(n-1).
Best Case Running time
When the collection is already sortedT(n) = c1n + c2(n-1) + c3(n-1) + c4(n-1) + c7(n-1-)T(n) = (c1+c2+c3+c4+c7)n – (c2+c3+c4+c7)Which is a function in the form an + bThus T(n) is a linear function of n.
Let’s Quickly Prove That…
Worst Case Running Time
Reverse sorted orderT(n) = c1n + c2(n-1) + c3(n-1) + c4(n(n+1)/2 -1) + c5(n(n-1)/2) + c6(n(n-1)/2) + c7(n-1)T(n) = (c4/2 + c5/2 + c6/2)n^2 + (c1 + c2 + c3 + c4/2 – c5/2 – c6/2 + c7)n – (c2 + c3 + c4 + c7)Which is a function in the form an^2 + bn + cThus T(n) is a quadratic function of n.
Again Let’s Quickly Prove That
So Now We Can Prove Speed
an + bIs faster thanan^2 + bn + c
Can we Make The Notation More Readable?
Asymptotic Notation
Big ‘O’ NotationFor the functions
an + ban^2 + bn + c
If n is sufficiently largeThen the largest term of n dominates the function
SoT(n) = an + b = O(n) T(n) = an^2 + bn + c + O(n^2).
So Now we Can Say…
Insertion Sort isBest Case
O(n)
Worst CaseO(n^2)
But we only care about the worst case
Let’s Optimize Our Search Algorithm
Let’s do What we Did When we Were Kids
Divide the problem up into smaller parts and conquer each of those smaller parts before combining the solutions.
Demo
Okay, so How Fast is That Algorithm?
Merge Sort in 3 Easy Parts
DivideComputes the middle of the arrayConstant time
D(n) = O(1)
ConquerRecursively solve two n/2 sized sub problems
Each contribute 2T(n/2)
CombineCombine step is linear
C(n) = O(n)
So worst case2T(n/2) + O(n)
Use The Master Method to Solve This Recursion
What is The Master Method?
‘Recipe’ for solving recurrences in the formT(n) = aT(n/b) + f(n)Where
The constants a >= 1 and b > 1And f(n) is asymptotically positive
Our function 2T(n/2) + O(n) is in this form
How Does it Work?
Recall it works with function of the formaT(n/b) + f(n)
We compare f(n) with n log baThe larger of the two determines the solutionIf n log ba is larger then
T(n) = O(nlogba)
If f(n) is larger thenT(n) = O(f(n))
If they are equal then we multiply by log factorT(n) = O(f(n) log n)
So in Our Example
We’re in case threeWorst case Merge Sort is
O(n log n)
Which is much better than Insertion SortO(n^2)
But .net Handles all this for me
List<T> - Add
List<T> - Growing
List<T> - EnsureCapacity
List<T> - Capacity
List<T> - AddRange
List<T> - AddRange
List<T> - InsertRange
So What are the Performance Implications?
Data Provider
Tester
Results
10 100 1000 10000 100000 10000000
5000
10000
15000
20000
25000
30000
Performance: Add Versus AddRange
AddAddRange
Number of Elements Added
Nu
mb
er
of
Tic
ks
What We’ve Learned
Performance is importantOptimization comes naturally to usDivide and conquerRunning time is the sum of product of cost and execution Maths syntax makes this all seem scaryAsymptotic notation simplifies languageThis stuff matters, even in the age of .net.
DPR Track Resources
http://www.microsoft.com/visualstudio http://www.microsoft.com/visualstudio/en-us/lightswitch http://www.microsoft.com/expression/http://blogs.msdn.com/b/somasegar/http://blogs.msdn.com/b/bharry/http://www.microsoft.com/sqlserver/en/us/default.aspxhttp://www.facebook.com/visualstudio
Resources
www.microsoft.com/teched
Sessions On-Demand & Community Microsoft Certification & Training Resources
Resources for IT Professionals Resources for Developers
www.microsoft.com/learning
http://microsoft.com/technet http://microsoft.com/msdn
Learning
http://northamerica.msteched.com
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