As software developers, we constantly strive to write efficient, well-performing code. We are constantly looking for ways to optimize our code and make it run faster. But how can we measure the efficiency of our code?
This is where the concept of order of growth comes in.
What is Order of Growth?
Order of growth is a measure of how quickly an algorithm or code grows relative to its input size. It is a way of quantifying the complexity of algorithms and code. It can be used to compare different algorithms or approaches to solving a problem and determine which one will be faster and more efficient. It is a way of expressing the rate at which a function grows as its inputs increase.
Mathematical Notations
Order of growth is essentially a way of measuring how well a function performs under different conditions. Mathematically, we can say that a function f(x) will grow faster than g(x) when x increases and the ratio of g(x)/f(x) is 0. This means that both f(x) and g(x) must be greater than or equal to 0, and the exponent n must also be greater than or equal to 0.
Mathematical notations such as big O notation, little O notation, big omega notation and little omega notation are used to describe the order of growth of an algorithm. These notations are used to express the rate at which a function grows as its inputs increase
How to Calculate Order of Growth?
Order of growth is typically expressed as a function of input size, such as n², n³, etc. The larger the number, the more complex the algorithm is. For example, an algorithm that is of order n² will take twice as long to run on a problem of size n as an algorithm of order n.
To calculate the order of growth:
Ignore all the lower-order terms.
Ignore the constant with the highest-order term.
For example, if the function is 5n² + 6n + 5, we would ignore the 6n and 5, leaving only the highest-order term 5n². This means that the order of growth is n², which is a quadratic order.
Comparison Scale
When comparing different algorithms, the comparison scale is used to rank the order of growth from lowest to highest. The comparison scale is C <log(log n)< log n < n⅓ < n½ < n < n² < n³ < n⁴< 2ⁿ < nⁿ, where C is a constant.
Best, Average, and Worst case
Best, average, and worst case are the three cases of a program that is used to analyze and compare the performance of a code.
Generally, the best case is the fastest case in which the code performs its task with the least amount of time and resources.
The average case is the case in which the code performs its task in an average amount of time and resources.
The worst case is the slowest case in which the code performs its task with the most amount of time and resources.
The best, average, and worst cases are primarily used to measure the performance of a code, and to determine the order of growth of the code.
Example
Now that we have a basic understanding of the order of growth, let’s look at an example. Consider the following code:
int getSum(int array, int x)
{
int sum = 0;
for (int i = 0; i < n; i++)
{
sum += array[i];
}
return sum;
}
This code takes an array of integers and returns the sum of all the elements in the array. The order of growth of this algorithm is linear (n), meaning it takes the same amount of time to run regardless of the size of the input.
Conclusion
In conclusion, the order of growth is a measure used to compare the efficiency of different algorithms. It is calculated by ignoring the lower-order terms and the leading constants. The comparison scale is used to rank the order of growth from lowest to highest. The best, average, and worst case of an algorithm must be considered to find its order of growth.