Solving Recurrence Relations: A Comprehensive Guide Using the Given Example

Recurrence relations are a fundamental concept in computer science and mathematics, often encountered in the analysis of algorithms. This article delves into solving a specific recurrence relation: T_n 3T_{sqrt{n}} log n. We will follow a systematic approach to break down and solve this problem, enhancing our understanding of recurrence relations and the methods to solve them.

Step-by-Step Solution

The first step involves making a change of variables to simplify the given recurrence relation.

Step 1: Change of Variables

Let n 2^k. This change of variables is particularly useful because it simplifies the square root term. Let's derive the new form of the recurrence relation with this substitution:

Expression for sqrt{n}

sqrt{n} sqrt{2^k} 2^{k/2}

Substitution into the Original Recurrence Relation

Substituting n 2^k into the original recurrence relation, we get:

T_{2^k} 3T_{2^{k/2}} log 2^k

Using the property of logarithms, we have:

log 2^k k log 2

The constant log 2 can be ignored as it does not affect the growth rate. Thus, we rewrite the recurrence relation as:

T_{2^k} 3T_{2^{k/2}} k

Step 2: Define a New Function

We define a new function:

S_k T_{2^k}

Substituting this into the recurrence relation, we get:

S_k 3S_{k/2} k

Step 3: Solve the New Recurrence Relation

To solve this new recurrence relation, we can either use the Master Theorem or expand the relation.

Expansion of the Recurrence Relation

Let's expand the relation step-by-step:

S_k 3S_{k/2} k

S_k 3(3S_{k/4} frac{k}{2}) k 9S_{k/4} frac{3k}{2} k 9S_{k/4} frac{5k}{2}

S_k 9(3S_{k/8} frac{k}{4}) frac{5k}{2} 27S_{k/8} frac{9k}{4} frac{5k}{2}

frac{9k}{4} frac{5k}{4} frac{19k}{4}

Continuing this process, you can observe that each time you expand the coefficient of S grows by a factor of 3 while the added k terms accumulate.

Formulating a General Expression

After m expansions, the general expression is:

S_k 3^m S_{k/2^m} k sum_{i0}^{m-1} frac{3^i}{2^i}

The series sum_{i0}^{m-1} left(frac{3}{2}right)^i is a geometric series that can be summed up to:

sum_{i0}^{m-1} left(frac{3}{2}right)^i frac{1 - left(frac{3}{2}right)^m}{1 - frac{3}{2}} 2left(1 - left(frac{3}{2}right)^{-m}right)

Base Case

The base case is when k is small, say k 0, which implies S_0 T_1 c.

Step 4: Asymptotic Behavior

For large values of k, the term 3^m dominates, leading to:

S_k approx 3^m S_c k cdot 2left(1 - left(frac{3}{2}right)^{-m}right)

As k becomes large, the term 3^m dominates the behavior, giving us:

S_k O(3^m) O(n^{log_2 3})

Given that log n grows slower than n^{log_2 3}, we conclude that:

T_n Theta(n^{log_2 3}) which simplifies to:

T_n Theta(n^{1.585})

Conclusion

This solution provides a comprehensive guide to solving a recurrence relation of the form T_n 3T_{sqrt{n}} log n. The key steps include a change of variables, defining a new function, expanding the recurrence relation, and identifying the asymptotic behavior. Understanding these steps is crucial for tackling more complex recurrence relations encountered in algorithm analysis and computer science.