Question

If \(S(n)=a S(n-1)+g(n)\) and \(g(n)=c^{n}\) with \(0

Short answer

\(S(n)\) grows as \(\Theta(c^n)\).

Step-by-step solution

Understanding the Recurrence

We are given a recurrence relation for a sequence: \(S(n) = a S(n-1) + c^n\). Here, \(g(n) = c^n\) represents the non-homogeneous part of the recurrence. The constants \(a\) and \(c\) satisfy the condition \(0 < a < c\).

Characterizing the Homogeneous Solution

The homogeneous solution for the recurrence \(S(n) = aS(n-1)\) is described by the general geometric series solution: \(S_h(n) = C_1 a^n\), where \(C_1\) is a constant determined by initial conditions.

Finding a Particular Solution

A particular solution can be assumed in a form that resembles the non-homogeneous part. Assuming \(S_p(n) = C_2 c^n\) and substituting into the recurrence gives \(C_2 c^n = a C_2 c^{n-1} + c^n\). Solving for \(C_2\), we find \(C_2 = \frac{1}{1 - \frac{a}{c}}\).

General Solution of the Recurrence

The general solution \(S(n)\) is the sum of the homogeneous and particular solutions: \(S(n) = C_1 a^n + C_2 c^n\). As \(n\) becomes large, the term \(c^n\) will dominate because \(a < c\), making \(S(n) = \Theta(c^n)\).

Concluding Big Theta Notation

Since \(c^n\) grows faster than \(a^n\), the dominant term in the general solution is \(C_2 c^n\). Therefore, \(S(n)\) grows at the rate of \(\Theta(c^n)\).

Key concepts

Big Theta Notation

When analyzing algorithms, it's essential to understand how to describe their growth rates. This is where Big Theta notation (\( \Theta \)) comes into play. Big Theta notation describes a function's growth rate using tight bounds. It indicates that a function grows asymptotically as fast as another function within constant factors.
To be precise, we say that a function \( f(n) \) is \( \Theta(g(n)) \) if there exist positive constants \( c_1, c_2, ext{and} n_0 \) such that:

• \( c_1g(n) \leq f(n) \leq c_2g(n) \) for all\( n \ge n_0 \).

This means that \( g(n) \) serves as both an upper and a lower bound on \( f(n) \). It's a very powerful way to simplify complex expressions and focus on the most significant terms, especially as \( n \) becomes large.
In the context of our problem, understanding Big Theta helps us predict that as \( n \) increases, the growth rate of \( S(n) \) will mimic \( \Theta(c^n) \) because the term \( c^n \) eventually dominates the expression \( C_1a^n + C_2c^n \).

Homogeneous Solution

The homogeneous solution of a recurrence involves finding a formula that fulfills the equation without considering the non-homogeneous part. For the recurrence \( S(n) = aS(n-1) \), the homogeneous solution comes from solving it as if there were no additional terms. This forms a simple geometric series.
When we solve it, \( S_h(n) = C_1a^n \), where \( C_1 \) is a constant determined based on initial conditions. This form is derived from the standard solution approach for linear homogeneous recurrences. The term \( a^n \) illustrates that this solution is exponentially based on \( n \), but due to the \( a < c \) condition, \( a^n \) grows slower than \( c^n \).The homogeneous solution is important because it serves as a base for constructing the complete solution of the recurrence, combining with the particular solution for an overall understanding of the behavior of the sequence.

Particular Solution

To find a particular solution of a non-homogeneous recurrence, you seek a specific instance that satisfies the entire equation. In our given recurrence \( S(n) = a S(n-1) + c^n \), the particular solution addresses the presence of \( c^n \).
We assume the particular solution to have a form related closely to \( c^n \), since this aligns with the non-homogeneous part. Thus, let's suggest \( S_p(n) = C_2 c^n \). If we substitute this into the recurrence, we end up with the condition \( C_2 c^n = ac^{n-1}C_2 + c^n \). By comparing coefficients, we solve for \( C_2 \) and get \( C_2 = \frac{1}{1 - \frac{a}{c}} \).
This clever insight gives us a component of the general solution. By itself, a particular solution doesn't give the full picture, but it's crucial for constructing the total solution.

Geometric Series

A geometric series is a sequence of terms where each next term is a constant multiple of the previous one. It's central to many areas of mathematics and computer science, particularly when dealing with recurrence relations.
For the homogeneous part of our recurrence, \( S_h(n) = C_1a^n\), this sequence represents a geometric series with the first term \( C_1 \) and a common ratio of \( a \). The behavior of a geometric series is dictated by its common ratio. If \( |a| < 1 \), the series converges as \( n \rightarrow \infty \); otherwise, it diverges.
Our context, where \( a < c \), signifies a series growing at a moderate pace compared to the component \( c^n \), allowing us to largely focus on the particular solution's growth, \( c^n \), to determine the overall asymptotic growth, which is \( \Theta(c^n) \). Understanding geometric series aids in predicting and characterizing the behavior of such recurrences.