Time Complexity
Definition:
“Time Complexity of an algorithm signifies the total time required by the program to run to completion.The Time complexity of algorithm 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.”
Big O Notation:
Big O notation is an upper bound, the worst-case time; Big O notation is required to run an algorithm on various inputs.
Example: if the time required by an algorithm on all inputs of size n is at most 5n3 + 3n.
Big Omega Notation:
Big Omega notation is a lower bound, the best-case time is proportional to the function multiplied by some constant factor.
Example: Quick sort
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Example: exchange the value of a and b if necessary so that a=<b^n is called constant time.
Linear Time:
An algorithm is said to be linear if its time complexity is O (n)
Function needs amount of time that is directly proportional to the number of input elements.
Example: Finding small values of n.
Quadratic Time: An algorithm is said to be quadratic if its time complexity is O (n^2)
Function needs amount of time that is quadratic proportional to the number of input elements.
Example: bubble sort algorithm, insertion sort algorithm, and selection sort algorithm.
Sub-Quadratic Time:
An algorithm is said to be Sub-quadratic time if T (n) =O (n^2)
Example: Shell Sort
Logarithmic time:
An algorithm is said to be Logarithmic time if T (n) =O (logn). When we performed operation on binary trees so we take logarithmic time.
Example: very easy example is that an algorithm that cuts a string in half.That is take O (logn) time (n is the length of the string)
Polylogarithmic time:
An algorithm is said to be Polylogarithmic time if T (n) = O ((logn) ^k)
Example: matrix chain
Robinson, E. P., & Sahin, F. (2001). Economic production lot sizing with periodic costs and overtime. Decision Sciences, 32(3), 423-452. Retrieved from http://search.proquest.com/docview/198075992?accountid=32521
This equation shifts from the parent function based on the equation f(x) = k+a(x-h) . In this equation, k shifts the parent function vertically, up or down, depending on the value of k. The h value shifts the parent function to the left or right. If h equals 1, it goes to the right 1 unit, if it is negative 1, it goes to the left 1 unit. If a is negative, the parent function is reflected on the x-axis. If x is negative, the parent function is reflected on the y-axis.
Precedence constraints are confinements on the request in which the work components can be performed. A few components must be done before others.
Thus total number of operations needed to execute the function for any given n, can be expressed as sum of 2 operations and the total number of operations needed to execute the function for n-1. Also when n=1, it just needs one operation to execute the function
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area for a certain amount of time - as long as this is for no more
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The time taken for this to happen is the measure of the rate of reaction. We must do this several times, and change the concentration of sodium thiosulphate. The rate of reaction is a measure of the change, which happens during a reaction in a single unit of time. The things that affect the rate of reaction are as follows. Surface area of the reactants Concentration of the reactants
where N_k are the constants and can be determined by the initial values of q_j and q ̇_j.