Question

Show that a running time \(T(N)=a N+b\) is an \(O(N)\) running time. Hint: The only issue is the \(+b\). Assume \(N\) is always at least 1

Short answer

Question: Prove that the running time \(T(N) = aN + b\) is an \(O(N)\) running time. Answer: To prove that the running time \(T(N) = aN + b\) is an \(O(N)\) running time, we found a constant \(C\) such that \(T(N) \leq CN\) for all values of \(N \geq 1\). This demonstrates that the function \(T(N)\) is upper-bounded by a constant multiplied by N for large N, which satisfies the definition of \(O(N)\) running time.

Step-by-step solution

Rewrite \(T(N)\) using a constant

Since we want to show that \(a N + b\) is \(O(N)\), we will rewrite the given function \(T(N)\) using the constant \(C\). Recall that \(T(N) = aN + b\). Let us assume that \(N \geq 1\). Now we need to find a value of \(C\) such that: $$T(N) \leq C * N$$ for all \(N \geq N_0\).

Establish a relationship between \(T(N)\) and \(C\)

Let's consider some cases. If \(a \geq 0\) and \(b \geq 0\), then: $$T(N) = aN + b \leq aN + a \leq (a+1)N$$ Let \(C = a+1\). We can see that for all \(N \geq 1\), \(T(N) \leq CN\). Now, if \(a \geq 0\) and \(b < 0\), let's assume that \(a \geq |b|\) (i.e., \(a\) is greater than or equal to the absolute value of \(b\)). In this case, we can write: $$T(N) = aN + b \leq aN + a \leq 2aN$$ Let \(C = 2a\) (since a is a positive value). We can see that for all \(N \geq 1\), \(T(N) \leq CN\).

Conclusion

In both cases, we found a constant \(C\) such that \(T(N) \leq CN\) for all \(N \geq 1\). Therefore, the running time \(T(N) = aN + b\) is an \(O(N)\) running time.

Key concepts

Algorithmic Complexity

When it comes to understanding how efficiently an algorithm performs, we refer to its algorithmic complexity. This is a measure of how the resource requirements of an algorithm, such as time or memory, scale with the size of the input. Crucially, it gauges how an algorithm's resource consumption increases as the input size, denoted as 'N', grows larger.

For example, consider an algorithm that processes a list of data. If the time it takes to complete the process doubles with each additional data item, the algorithm has a high complexity. Conversely, if the algorithm's processing time increases linearly with the number of data items, it is deemed more efficient and has lower complexity. The goal in algorithm design is usually to minimize complexity to make the algorithm scalable and robust for large inputs.

Time Complexity Analysis

Time complexity analysis is a subset of algorithmic complexity that focuses on the computation time an algorithm takes. The aim is to quantify the time taken by an algorithm as a function of the input size, denoted as 'T(N)'. Time complexity gives us an insight into the performance of an algorithm by providing an upper limit on the time required, which is especially important for large inputs.

For instance, in the problem given where the running time is represented as \(T(N) = aN + b\), analysts want to understand how the execution time changes as 'N' increases. To ensure that the algorithm stays performant even as N scales, it's beneficial to describe its time complexity using Big O Notation, which brings us to the essence of asymptotic behavior and finding an appropriate Big O category for the given algorithmic time function.

Asymptotic Notation

Asymptotic notation, specifically Big O notation, is a mathematical way to describe the limit of a function as the input size approaches infinity. It simplifies the representation of an algorithm's time complexity by focusing on the most significant terms that affect growth rate, ignoring less significant terms and constant factors.

In the context of the exercise, showing that \(T(N) = aN + b\) is of Big O of N, the Big O notation is used to exhibit the upper bound of the running time, regardless of the input size N. As long as there exists a constant 'C' and an input size 'N_0' beyond which the function's growth is at most 'C' times 'N', the function can be described as O(N). This notation is invaluable for comparing algorithms as it distills their behaviour down to the most critical factor - how they scale with increasingly large datasets.