“Zeitkomplexität” Code-Antworten

Zeitkomplexität

Trivial Runtime Analysis
**************************************

If there is no input, then it’s called a constant time algorithm. For example:

for (int i = 0; i < 1000000; i ++)
      x++;

above is O(1)
------------------------------------------------------------------------------
Let’s go through some code samples and analyze their runtime complexity.

for (int i = 0; i < N; i ++)
      x++;
All we need to do is count the number of times the statement x++ will execute.
Clearly, it’s N, so the time complexity is O(N), also called linear.
------------------------------------------------------------------------------
for (int i = 0; i < N; i++) 
    for (int j = 0; j < i; j++) 
        x++;
How many times the statement x++ executes:
So the time complexity is O(N^2), also called quadratic.
---------------------------------------------------------------------------

Logarithmic Runtime
**************************************************

Iterating powers of a number #
Let’s analyze the loop below where we iterate over all powers of 2
for (int i = 1; i <= N; i *= 2)
    x++;
    
In Big-O notation, the time complexity is O(logN)

A similar analysis gives O(logN) runtime for the loop below.
for (int i = N; i >= 1; i /= 2)
      x++;
---------------------------------------------------------------------------
Harmonic series #
Consider the piece of code below:

for (int i = 1; i <= N; i++)
    for (int j = i; j <= N; j += i)
        x++;

Therefore, the time complexity is O(NlogN).
---------------------------------------------------------------------------

Non Trivial Runtime
********************************************************

Sum of powers #
Take the code sample below:

for (int i = 1; i <= N; i *= 2)
    for (int j = 1; j <= i; j++)
        x++;
So, the run-time complexity is actually linear - O(N)
-----------------------------------------------------------------------------

Amortized Analysis
*****************************************************************

Consider this algorithm: We start with an array of size 2 
and each operation adds one element to the array, we do this operation N times. 
If the array is full, we see the current size of array say sz. 

Adding to the array if it’s not empty: O(1)
Copying array of size sz to a new location: O(sz)

Total number of operations:

1 + 1 + (1 + 2) + 1 + (1 + 4) + 1 + 1 + 1 + (1 + 8) + 1 + 1…

=> (1 + 1 + ... + 1) N times + (2 + 4 + 8 + ... ) < N + 2N<=3N

So, the complete algorithm runs in O(N) time
Allocate the 2*sz memory and copy the array to its location so we have space for the new sz elements.
ap_Cooperative_dev

Zeitkomplexitätsdefinition

Time complexity is the amount of time taken by an algorithm to run, as a function of the length of the input. It measures the time taken to execute each statement of code in an algorithm.
Envious Echidna

Ähnliche Antworten wie “Zeitkomplexität”

Fragen ähnlich wie “Zeitkomplexität”

Durchsuchen Sie beliebte Code-Antworten nach Sprache

Durchsuchen Sie andere Codesprachen