DS Analysis Terms-Time Complexity

Analysis Terms

In computer science, the analysis of algorithms is the determination of the amount of resources (such as time and storage) necessary to execute them. Most algorithms are designed to work with inputs of arbitrary length. Usually, the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps (time complexity) or storage locations (space complexity)

Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem. These estimates provide an insight into reasonable directions of search for efficient algorithms

In theoretical analysis of algorithms it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. Big O notation, Big-omega notation and Big-theta notation are used to this end. For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the sorted list being searched, or in O(log(n)), colloquially "in logarithmic time". Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency. However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called a hidden constant.

Exact (not asymptotic) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called model of computation. A model of computation may be defined in terms of an abstract computer, e.g., Turing machine, and/or by postulating that certain operations are executed in unit time. For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log2 n + 1 time units are needed to return an answer.

Time Complexity of Algorithms

Time complexity of an algorithm signifies the total time required by the program to run to completion. The time complexity of algorithms is most commonly expressed using the big O notation.

Time Complexity is most commonly estimated by counting the number of elementary functions performed by the algorithm. And since the algorithm's performance may vary with different types of input data, hence for an algorithm we usually use the worst-case Time complexity of an algorithm because that is the maximum time taken for any input size.

Calculating Time Complexity

Now lets tap onto the next big topic related to Time complexity, which is How to Calculate Time Complexity. It becomes very confusing some times, but we will try to explain it in the simplest way.

Now the most common metric for calculating time complexity is Big O notation. This removes all constant factors so that the running time can be estimated in relation to N, as N approaches infinity.

In general you can think of it like this Syntax:
statement;

Above we have a single statement. Its Time Complexity will be Constant. The running time of the statement will not change in relation to N.

Syntax:

for(i=0; i < N; i++)
{
  statement;
}

The time complexity for the above algorithm will be Linear. The running time of the loop is directly proportional to N. When N doubles, so does the running time.

Syntax:

for(i=0; i < N; i++) 
{
  for(j=0; j < N;j++)
  { 
    statement;
  }
}

This time, the time complexity for the above code will be Quadratic. The running time of the two loops is proportional to the square of N. When N doubles, the running time increases by N * N.

Logarithmic Tme Complexity

Syntax:

while(low <= high) 
{
  mid = (low + high) / 2;
  if (target < list[mid])
    high = mid - 1;
  else if (target > list[mid])
    low = mid + 1;
  else break;
}

This is an algorithm to break a set of numbers into halves, to search a particular field(we will study this in detail later). Now, this algorithm will have a Logarithmic Time Complexity. The running time of the algorithm is proportional to the number of times N can be divided by 2(N is high-low here). This is because the algorithm divides the working area in half with each iteration.

Syntax:

void quicksort(int list[], int left, int right)
{
    int pivot = partition(list, left, right);
    quicksort(list, left, pivot - 1);
    quicksort(list, pivot + 1, right);
}

Taking the previous algorithm forward, above we have a small logic of Quick Sort(we will study this in detail later). Now in Quick Sort, we divide the list into halves every time, but we repeat the iteration N times(where N is the size of list). Hence time complexity will be N*log( N ). The running time consists of N loops (iterative or recursive) that are logarithmic, thus the algorithm is a combination of linear and logarithmic.


Types of Notations for Time Complexity

Now we will discuss and understand the various notations used for Time Complexity.
  1. Big Oh denotes "fewer than or the same as" <expression> iterations.
  2. Big Omega denotes "more than or the same as" <expression> iterations.
  3. Big Theta denotes "the same as" <expression> iterations.
  4. Little Oh denotes "fewer than" <expression> iterations.
  5. Little Omega denotes "more than" <expression> iterations.

Understanding Notations of Time Complexity with Example

O(expression) is the set of functions that grow slower than or at the same rate as expression.

Omega(expression) is the set of functions that grow faster than or at the same rate as expression.

Theta(expression) consist of all the functions that lie in both O(expression) and Omega(expression).

Suppose you've calculated that an algorithm takes f(n) operations, where,

f(n) = 3*n^2 + 2*n + 4.   // n^2 means square of n

Since this polynomial grows at the same rate as n^2, then you could say that the function f lies in the setTheta(n^2). (It also lies in the sets O(n^2) and Omega(n^2) for the same reason.)

The simplest explanation is, because Theta denotes the same as the expression. Hence, as f(n) grows by a factor of n^2, the time complexity can be best represented as Theta(n^2). 



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