时间复杂度与求和：递推式T(n,m)=mT(n-1,m)+mc求解咨询

Hey there! Let's break down this recurrence relation step by step, and I'll also show you how to convert it into the summation form you prefer—since you mentioned you're more comfortable working with those.

First, Let's Define the Base Case #

Before we start solving, we need a base case (a starting point for the recurrence). A common base case for algorithmic recurrences like this is:

$T(0, m) = 0$
This makes sense because when $n=0$, there are no subproblems to solve, so the total cost is 0. If your actual base case is different, we can adjust the steps easily!

Step 1: Iterate the Recurrence to Spot a Pattern #

Let's expand the recurrence for a few values of $n$ to see the pattern emerge:

• For $n=1$:
  T(1, m) = m*T(0, m) + m*c = m*0 + mc = mc
• For $n=2$:
  T(2, m) = m*T(1, m) + mc = m*(mc) + mc = m²c + mc
• For $n=3$:
  T(3, m) = m*T(2, m) + mc = m*(m²c + mc) + mc = m³c + m²c + mc
• For $n=k$:
  Looking at the pattern, each term adds another multiple of $c*m^i$, where $i$ goes from 1 to $k$.

Step 2: Convert to Summation Form #

As you requested, we can rewrite this directly as a summation. The recurrence expands to a sum of geometric terms:
T(n, m) = c * Σ (from i=1 to n) mⁱ

This is the summation form you're familiar with! Now we can simplify this to a closed-form expression using geometric series rules, if needed.

Step 3: Simplify the Summation to Closed Form #

The sum of a geometric series $\sum_{i=1}^n m^i$ has a well-known closed-form solution:

• If $m ≠ 1$: $\sum_{i=1}^n m^i = m*(mⁿ - 1)/(m - 1)$
• If $m = 1$: The sum simplifies to $n$ (since every term is 1), so $T(n,1) = c*n$

Substituting back into our recurrence, we get:

• For $m ≠ 1$: T(n, m) = c * m*(mⁿ - 1)/(m - 1)
• For $m = 1$: T(n, 1) = c*n

Step 4: Verify the Result #

Let's test this with a concrete example to make sure it works:

• Let $n=3$, $m=2$, $c=1$:
  • Summation: $1*(2 + 4 + 8) = 14$
  • Closed-form: $12(2³ - 1)/(2-1) = 2*(8-1)/1 = 14$ ✔️
• Let $n=2$, $m=3$, $c=2$:
  • Summation: $2*(3 + 9) = 24$
  • Closed-form: $23(3² -1)/(3-1) = 6*(9-1)/2 = 24$ ✔️

内容的提问来源于stack exchange，提问作者J. Doe