Recurrence Relation T(n) = n*T(n-1) | Substitution Method | Algorithm
Understand the Problem
The question is asking about the recurrence relation T(n) = n*T(n-1) and how to solve it using the substitution method in the context of algorithms. It requires an understanding of recursive functions and mathematical induction to derive a solution.
Answer
$T(n) = n! * c$
Answer for screen readers
The solution to the recurrence relation is $T(n) = n! * c$.
Steps to Solve
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Identify the Base Case To solve the recurrence relation, we need a base case. Let's assume for $n = 1$, $T(1) = c$, where $c$ is a constant.
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Unroll the Recurrence Unroll the recurrence relation for a few values to identify a pattern.
- For $n = 2$: $$ T(2) = 2 * T(1) = 2 * c $$
- For $n = 3$: $$ T(3) = 3 * T(2) = 3 * (2 * c) = 3! * c $$
- For $n = 4$: $$ T(4) = 4 * T(3) = 4 * (3! * c) = 4! * c $$
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Formulate a General Expression From the unrolling above, we can see a pattern. The expression suggests that: $$ T(n) = n! * c $$
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Verify the Pattern To confirm the pattern, we can use mathematical induction:
- Base case: For $n = 1$, $T(1) = 1! * c$, which holds true.
- Inductive step: Assume it holds for $n = k$, i.e., $T(k) = k! * c$. Now check for $n = k + 1$: $$ T(k + 1) = (k + 1) * T(k) = (k + 1) * (k! * c) = (k + 1)! * c $$
This shows the formula holds for $n = k + 1$.
- State the Final Result Based on the above deduction, the solution to the recurrence relation is: $$ T(n) = n! * c $$
The solution to the recurrence relation is $T(n) = n! * c$.
More Information
The factorial function grows very quickly, and the use of recurrence relations like this is common in algorithm analysis, especially in contexts such as dynamic programming or recursive algorithms.
Tips
- Incorrectly identifying the base case can lead to errors in the entire solution. Always verify your base case before proceeding.
- Misapplying the inductive step by failing to correctly assume or prove the previous case can also distort the result.
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