Question

Suppose you are given recurrences of the form \(T(n)=a T(n / b)+g(n)\), with \(T(1)=d>0\) and \(g(n)>0\) for all \(n\), and \(S(n)=a S(n / b)+g(n)\), with \(S(1)=0\) (and the same \(a, b\), and \(g(n))\). Is there any difference in the big \(\Theta\) behavior of the solutions to the two recurrences? What does this say about the influence of the initial condition on the big \(\Theta\) behavior of such recurrences?

Short answer

The big \( \Theta \) behavior is the same for \( T(n) \) and \( S(n) \); initial values do not impact it.

Step-by-step solution

Understand the Recurrences

We have two recurrence relations: \( T(n) = a T(n/b) + g(n) \) with \( T(1) = d > 0 \) and \( S(n) = a S(n/b) + g(n) \) with \( S(1) = 0 \). The goal is to determine if the big \( \Theta \) behavior differs between these two recurrences.

Apply the Master Theorem

The Master Theorem provides a way to determine the asymptotic behavior of recurrences of the form \( T(n) = aT(n/b) + g(n) \). It states that for a recurrence \( T(n) = aT(n/b) + g(n) \), if \( g(n) = \Theta(n^c) \), compare \( c \) with \( \log_b a \):- If \( c < \log_b a \), then \( T(n) = \Theta(n^{\log_b a}) \).- If \( c = \log_b a \), then \( T(n) = \Theta(n^{\log_b a} \log n) \).- If \( c > \log_b a \), then \( T(n) = \Theta(g(n)) \).

Assess \( T(n) \) Using Master Theorem

For \( T(n) = aT(n/b) + g(n) \), apply the Master Theorem. Regardless of \( T(1) = d \), the asymptotic behavior \( \Theta(n^{\log_b a}) \), \( \Theta(n^{\log_b a} \log n) \), or \( \Theta(g(n)) \) is determined provided the conditions on \( g(n) \) hold.

Assess \( S(n) \) Using Master Theorem

Similarly, apply the Master Theorem to \( S(n) = a S(n/b) + g(n) \). The analysis of \( S(n) \) relies solely on the same \( a, b, \) and \( g(n) \) as \( T(n) \), so it will also have the same asymptotic behavior as \( T(n) \).

Conclusion on Initial Condition Influence

For both recurrences, \( T(n) \) and \( S(n) \), the initial conditions \( T(1) = d \) and \( S(1) = 0 \) do not affect the overall big \( \Theta \) behavior. Thus, the initial values do not influence the asymptotic behavior; it is defined entirely by \( a, b, \) and \( g(n) \).

Key concepts

Recurrence Relations

Recurrence relations are equations that recursively define sequences or multi-dimensional arrays. In simpler terms, they express how each term in the sequence relates to its predecessors. In computer science, these often arise when analyzing recursive algorithms, helping us understand their efficiency. For the given exercise, our recurrence relations are defined as

• \( T(n) = aT(n/b) + g(n) \) with an initial condition \( T(1) = d > 0 \)
• \( S(n) = aS(n/b) + g(n) \) with an initial condition \( S(1) = 0 \)

These equations model similar recursive processes, but they begin with different starting values. Understanding these expressions is crucial because solving them can give us insight into the overall runtime and behavior of an algorithm.

Asymptotic Behavior

Asymptotic behavior is crucial when it comes to analyzing algorithms. It helps us understand how functions behave as inputs grow towards infinity, which is critical for predicting the efficiency and limitations of an algorithm. This idea is captured using big \( \Theta \) notation, which provides bounds from above and below by constant factors.
In the context of the problem, we use the Master Theorem, a powerful tool to find asymptotic behavior in recurrence relations of the form \( T(n)=aT(n/b)+g(n) \). It allows us to categorize the growth of

• \( \Theta(n^{\log_b a}) \) if \( c < \log_b a \)
• \( \Theta(n^{\log_b a}\log n) \) if \( c = \log_b a \)
• \( \Theta(g(n)) \) if \( c > \log_b a \)

Where \( g(n) = \Theta(n^c) \), the value of \( c \) compared to \( \log_b a \) determines which asymptotic behavior applies. Thus, it is the function \( g(n) \) and constants \( a \) and \( b \) that set the stage for the result, revealing that the initial conditions don't influence the asymptotic behavior.

Initial Conditions

Initial conditions can seem like a starting point; however, for recurrence relations analyzed with the Master Theorem, they surprisingly don't affect the asymptotic behavior. Initial conditions are the prescribed starting values, like \( T(1) = d \, (> 0) \) and \( S(1) = 0 \), which initially define the base case in recurrences.
While they might influence the first few terms of the sequence, their effect diminishes as \( n \) grows larger due to scaling and the repetitive nature of recurrrences. The overall growth rate, captured by the big \( \Theta \) notation, remains consistent irrespective of these starting values.
The concluding analysis in our problem showed that both \( T(n) \) and \( S(n) \) have identical asymptotic behavior. This clearly demonstrates that while initial conditions formalize the gateway to recursion, they are irrelevant to the long-term growth patterns described by asymptotic analysis.