Understanding Big Omega (Ω), Big Theta (Θ), Small Oh (o) and Small Omega (ω) Notations

How to describe Big Omega(Ω) ?

If run time of an algorithm is of Ω(g(n)), it means that the running time of the algorithm (as n gets larger) is at least proportional to g(n). Hence it helps in estimating a lower bound of the number of basic operations to be performed.

More specifically, f(x) = Ω(g(x)) (big-omega) means that the growth rate of f(x) is asymptotically greater than or equal to the growth rate of g(x)

Mathematically, a function f(x) is equal to Big Omega of another function g(x), i,e f(x) = Ω(g(x) is true if and only if there exists two constants (C1 and C2)such that

a) C1 and C2 are always positive

b) 0<= C1*g(n) <= f(n) for any value of n => C2

How to describe Big Theta (Θ)?

If run time of an algorithm is of Θ(g(n)), it means that the running time of the algorithm (as n gets larger) is equal to the growth rate of g(n). Hence it helps in estimating a tight bound of the number of basic operations to be performed.

Hence f(x) = Θ(g(x)) (big - theta) means that the growth rate of f(x) is asymptotically equal to the growth rate of g(x)

Mathematically, a function f(x) is equal to Big Theta of another function g(x), i,e f(x) = Θ(g(x) is true if and only if there exists three constants (C1 and C2 and C3)such that

a) C1, C2 and C3 are always positive

b) 0<= C1*g(n) <= f(n) <= C2*g(n) for any value of n => C3

What are Small Oh and Small Omega?

f(x) = o(g(x)) (small-oh) means that the growth rate of f(x) is asymptotically less than to the growth rate of g(x).

Mathematically, a function f(x) is equal to Small Oh of another function g(x), i,e f(x) = o(g(x) is true if and only if there exists two constants (C1 and C2)such that

a) C1 and C2 are always positive

b) 0<= f(n) <C1*g(n) for any value of n => C2

So this gives a loose upper bound for complexities of f(x).

On the other hand, f(x) = ω(g(x)) (small-omega) means that the growth rate of f(x) is asymptotically greater than the growth rate of g(x).

Mathematically, a function f(x) is equal to Small Omega of another function g(x), i,e f(x) = ω(g(x) is true if and only if there exists two constants (C1 and C2)such that

a) C1 and C2 are always positive

b) 0<= C1*g(n) < f(n) for any value of n => C2

So this gives a loose lower bound for complexities of f(x).

So our discussions on Complexity Analysis of Algorithm gets to an end. Thanks for reading.

Introductory Discussions on Complexity Analysis

What is Complexity Analysis of Algorithm?

Complexity Analysis, simply put, is a technique through which you can judge about how good one particular algorithm is.  Now the term “good” can mean many things at different times.

Suppose you have to go from your home to the Esplanade! There are many ways from your home that may lead to Esplanade. Take any one, and ask whether this route is good or bad. It may so happen that this route is good if the time of travel is concerned (that is the route is short enough), but at the same time, it may be considered bad taking the comfort into considerations (This route may have many speed breakers leading to discomforts). So, the goodness (or badness as well) of any solution depends on the situations and whatever is good to you right now, may seem as bad if the situation changes. In a nutshell, the goodness/badness or the efficiency of a particular solution depends on some criteria of measurements.

So what are the criteria while analyzing complexities of algorithms?

Focusing only on algorithms, the criteria are Time and Space. The criteria Time, judges how fast or slow the algorithms run when executed; and the criteria Space judges how big or small amount of memory (on primary/hard disks) is required to execute the algorithm. Depending on these two measuring criteria, two type of Algorithm Analysis are done; one is called Time Complexity Analysis and the second one is Space Complexity Analysis.

Which one is more important over the other?

I am sorry! I do not know the answer; rather there is no straight forward answer to this question. Think of yourself. Thinking of the previous example of many solutions that you have for travelling from your home to Esplanade, which criteria is most important? Is it Time of Travel, or is it Comfort? Or is it Financial Cost? It depends actually. While you are in hurry for shopping at New Market, the Time Taken would probably be your choice. If you have enough time in your hand, if you are in jolly mood and if you are going for a delicious dinner with your friends, probably you would choose Comfort; and at the end of the month, when you are running short with your pocket money, the Financial Cost would be most important to you.  So the most important criterion is a dynamic notion that evolves with time.

Twenty or thirty years back, when the pace of advancement of Electronics and Computer Hardware was timid, computer programs were forced to run with lesser amount of memory.  Today you may have gigantic memory even as RAM, but that time, thinking of a very large hard disk was a day dreaming! So at that time, Space Complexity was much more important than the Time Complexity, because we had lesser memory but ample times.

Now the time has changed! Now a day, we generally enjoy large memories but sorry, we don’t have enough time with us. We need every program to run as quick as possible! So currently, Time Complexity wins over Space Complexity.  Honestly, both of these options are equally important from theoretical perspective but the changing time has an effect to these.         

In the next blog, I will discuss on Time Complexity Analysis. That blog will throw lights to the insights of Objective, Characteristics, General Assumptions and some examples of Time Complexity analysis.