An algorithm is a well-defined computational procedure that takes some value, or a set of values as input and produces some value, or a set of values, as output.

**What is a good algorithm?**

A good algorithm is measured by its running time and space utilization. In Laymen's terms:: "*The lesser the better*".

**Can we measure the run time and space utilization of any Algorithm? If Yes, how exactly?**

Yes, we can measure the efficiency of an algorithm by simply determining the Running Time/Memory Space needed by the algorithm to run to completion.

Now, this can be measured in 2 ways-

**Time Complexity**:: The time complexity of an algorithm is the amount of time it needs to run to completion.**Space Complexity**:: The space complexity of an algorithm is the amount of space it needs to run to completion

Also, along with this, we should be mindful of the efficiency of algorithms.

**Worst-case efficiency**: It is the maximum number of steps that an algorithm has to take for any collection of data values.**Best case efficiency**: It is the minimum number of steps that an algorithm has to take any collection of data values.**Average case efficiency**: It can be defined as the efficiency averaged on all possible inputs.

A graphical representation of the Worst, Best and Average cases of algorithms.

**Analysis of Algorithm**

Let's study different methods used in the analysis of algorithms.

**Asymptotic Analysis (Growth of function)**

It simply means a line that continuously approaches a given curve. However, never meets at any finite distance.

For example - Let's look at this picture, here X is asymptotic with X+1 as shown in the graph.

Asymptotic may also be defined as a way to describe the behaviour of functions in the limit or without bounds.

Let f(x) and g(x) be two functions of real numbers.We say that f and g are asymptotic and write f(x) g(x) if

**Asymptotic Notations**In Asymptotic notations, we ignore the smaller inputs and constants while figuring out complexity.

**Big-oh notation**It provides the upper bound for f(n) and gives the worst-case complexity i.e. the measure of the longest amount of time.We say that f(n) = O(g(n)), if and only if there are positive constants c and n0 such thatIf f(n) = O(g(n)), we say that g(n) is an upper bound on f(n).For Example -`F(n) = 3n+2 as 3n+24n for all n2 =O(n)F(n) = 3n+3 as 3n+34n for all n3 =O(n)`

**NOTE :: K*f(n) is O(f(n))** [i.e., constant coefficients can be dropped]G(n) = 7n4 is O(n4)That is, if we have a function multiplied by a constant, we can ignore the constant in the big-O.

**Big-Omega Notation ()**

It provides the lower bound for f(n) and gives the best-case complexity i.e. the measure of the shortest amount of time.We can say that f(n) = (g(n)), if and only if there are positive constants c and n0 such thatIf f(n) = (g(n)), we say that g(n) is a lower bound on f(n).

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