100% tevredenheidsgarantie Direct beschikbaar na betaling Zowel online als in PDF Je zit nergens aan vast
logo-home
Summary ADS notes €7,99   In winkelwagen

Samenvatting

Summary ADS notes

 33 keer bekeken  0 keer verkocht

Notes based on slides and the book for the exams and to understand the theory. Very useful for the analysis exam

Voorbeeld 3 van de 24  pagina's

  • Nee
  • 3-11, some 12
  • 6 mei 2022
  • 24
  • 2020/2021
  • Samenvatting
book image

Titel boek:

Auteur(s):

  • Uitgave:
  • ISBN:
  • Druk:
Alle documenten voor dit vak (2)
avatar-seller
randomgirl
Algorithms & Data Structures

1. Complexity Analysis
Efficiency of an algorithm, measured by 2 elements:
 Time complexity - amount of time it takes for a given algorithm to execute
 Space complexity - mount of space a given algorithm uses at most during
execution


Empirical time analysis
Analysis method for time complexity - roughly equivalent to a brute force approach
when programming or a proof by exhaustion for Reasoning and Logic: execute and
measure the time:
long startTime = System.currentTimeMillis(); // record the
starting time
/* (run the algorithm) */
long endTime = System.currentTimeMillis(); // record the ending
time
long elapsed = endTime - startTime; // calculate the
elapsed time


Problems:
 Results may differ when different hardware / compiler / OS / etc. are used
 Experiments are restricted to a limited set of inputs
 Requires a full implementation of the algorithm
 A compiler may optimize your code or require a warm up time.


Theoretical complexity analysis for time
Way to analyse the efficiency of an algorithm without having to run additional code,
don’t need to execute your algorithm to analyse it.
Primitive operations are operations that have a run time of 1:
- Assigning a value to a variable
- Performing an arithmetic operation
- Comparing two numbers (and no more than two)
- Accessing a single element of an array by index
- Calling and returning from a method


Assign a mathematical function to the algorithm that describes the running time.
T algorithm ( n )=number of primitive operations performed for an input size n

,To set up T algorithm ( n ):
1. State the size of the problem n - length of an array.
2. Simplify the program to only using primitive operations (optional).
3. Express the running time by counting operations.
! Note: when expressing the running time, we consider the worst-case of our problem
n.
! Note: a + only represents an arithmetic operation when applied to numbers.


Theoretical complexity analysis for space
Space used can go down, while time cannot. Count the number of stack frames
used.
Salgorithm ( n )=maximum amount of memory needed at any point in the algorithm for an input size n

A stack frame is added or pushed onto the stack when a method is called and is only
removed or popped off the stack when you return from the method. With recursive
method it is possible to have multiple stack frames on the stack at once.


Big O Proof
f ( n ) is O ( g ( n ) ) if f ( n ) ≤ c∗g ( n ) , c ≥ 0 , n0 ≥n



2. Asymptotic growth comparison

The following functions are from ‘best’ asymptotic growth to ’worst’:
1. Constant: f ( n )=c
2. Logarithmic: f ( n )=log ⁡(n)
3. Linear: f ( n )=n
4. Linearithmic (n-log-n): f ( n )=n log ⁡(n)
5. Quadratic: f ( n )=n2
6. Cubic: f ( n )=n3
7. Polynomials: f ( n )=n a
8. Factorial: f ( n )=n !
9. Exponential: f ( n )=e n
10. Worst ever: f ( n )=n n

, 3. Often used tricks

Gauss's sum identity:

n
n(n+1)
∑ i=1+ 2+ 3+…+n= 2
i=1


Geometric sum:

a1 ( 1−r )
m m

∑ ai=1+a+ a2 +…+ an= 1−r
, r ≠1
i=0


where:
o m is the number of terms, in this case n
o a 1 is the first term, in this case 1
o r is the constant that each term is multiplied by to get the next
term, in this case a
Particular geometric series:
n

∑ 2i =2n+1−1
i=0




4. Finding time complexity of a recursive algorithm

1. State / define the size of the input n
2. State the recurrence equation by counting operations. The exact amount of
operations shouldn’t be counted, that depends on the hardware, compiler, etc.
It is more about counting loops, etc. Specific operations can be replaced by a
constant, often called c xwhere x is a number
3. Create a form of this recurrence equation where there is no T(n) in the
equation anymore. There are 2 methods to do so:
a. Making an educated guess and then proving this (by induction)
b. Unfolding the equation and then create a final equation based on that
4. Determine the complexity in Big O notation
5. Prove the complexity in Big O notation




5. Arrays

Voordelen van het kopen van samenvattingen bij Stuvia op een rij:

Verzekerd van kwaliteit door reviews

Verzekerd van kwaliteit door reviews

Stuvia-klanten hebben meer dan 700.000 samenvattingen beoordeeld. Zo weet je zeker dat je de beste documenten koopt!

Snel en makkelijk kopen

Snel en makkelijk kopen

Je betaalt supersnel en eenmalig met iDeal, creditcard of Stuvia-tegoed voor de samenvatting. Zonder lidmaatschap.

Focus op de essentie

Focus op de essentie

Samenvattingen worden geschreven voor en door anderen. Daarom zijn de samenvattingen altijd betrouwbaar en actueel. Zo kom je snel tot de kern!

Veelgestelde vragen

Wat krijg ik als ik dit document koop?

Je krijgt een PDF, die direct beschikbaar is na je aankoop. Het gekochte document is altijd, overal en oneindig toegankelijk via je profiel.

Tevredenheidsgarantie: hoe werkt dat?

Onze tevredenheidsgarantie zorgt ervoor dat je altijd een studiedocument vindt dat goed bij je past. Je vult een formulier in en onze klantenservice regelt de rest.

Van wie koop ik deze samenvatting?

Stuvia is een marktplaats, je koop dit document dus niet van ons, maar van verkoper randomgirl. Stuvia faciliteert de betaling aan de verkoper.

Zit ik meteen vast aan een abonnement?

Nee, je koopt alleen deze samenvatting voor €7,99. Je zit daarna nergens aan vast.

Is Stuvia te vertrouwen?

4,6 sterren op Google & Trustpilot (+1000 reviews)

Afgelopen 30 dagen zijn er 67474 samenvattingen verkocht

Opgericht in 2010, al 14 jaar dé plek om samenvattingen te kopen

Start met verkopen
€7,99
  • (0)
  Kopen