How to go from a recurrence relation to a final complexity

Question

I have an algorithm, shown below, that I need to analyze. Because it's recursive in nature I set up a recurrence relation.

//Input: Adjacency matrix A[1..n, 1..n]) of an undirected graph G  
//Output: 1 (true) if G is complete and 0 (false) otherwise  
GraphComplete(A[1..n, 1..n]) {
  if ( n = 1 )
    return 1 //one-vertex graph is complete by definition  
  else  
    if not GraphComplete(A[0..n − 1, 0..n − 1]) 
      return 0  
    else 
      for ( j ← 1 to n − 1 ) do  
        if ( A[n, j] = 0 ) 
          return 0  
      end
      return 1
}

Here is what I believe is a valid and correct recurrence relation:

$\qquad \begin{align} T(1) &= 0 \\ T(n) &= T(n-1) + n - 1 \quad \text{for } n \geq 2 \end{align}$

The "$n - 1$" is how many times the body of the for loop, specifically the "if A[n,j]=0" check, is executed.

The problem is, where do I go from here? How do I convert the above into something that actually shows what the resulting complexity is?

Answer 1

From what you wrote it seems that for all $k$ you have $T(k)=k-1+T(k-1)$ and $T(1)=0$. Therefore you can get it directly:

$$T(n)=(n-1)+(n-2)+\dots+1+0 = \sum_{k=1}^{n}{(k-1)}=\frac{n(n-1)}{2}$$

So $T(n)$ is in $Θ(n^2)$.

Answer 2

The recursion you have there is this, if we count only matrix accesses:

$\qquad \begin{align} T(1) &= 0 \\ T(n) &= T(n-1) + n - 1 \quad \text{for } n \geq 2 \end{align}$

Note that it is only an upper bound on the runtime function because the loop is not always executed. It is a linear recurrence relation, and there are many ways of solving those. You can probably unroll $T(n)$, spot a pattern and prove it by induction. This does not always work, though, which is why I want to present a general technique.

This is my favorite way, using generating functions. It uses only mechanical calculation (once you get used to the approach). This is the ansatz:

$\qquad \begin{align} \mathcal{T}(z) &= \sum_{n=1}^\infty T(n)z^n \\ &= 0 + \sum_{n=2}^\infty T(n)z^n \\ &= \sum_{n=2}^\infty (T(n-1) + n - 1)z^n \\ &= \sum_{n=2}^\infty T(n-1)z^n + \sum_{n=2}^\infty (n-1)z^n \\ &= z\sum_{n=2}^\infty T(n-1)z^{n-1} + z\sum_{n=2}^\infty (n-1)z^{n-1} \\ &= z\sum_{n=1}^\infty T(n)z^{n} + z\sum_{n=1}^\infty nz^{n} \\ &= z\mathcal{T}(z) + z\cdot\frac{z}{(1-z)^2} \end{align}$

For the identity needed for the last step, see the TCS Cheat Sheet. If we now solve this equation for $\mathcal{T}$, we get:

$\qquad \displaystyle \begin{align} \mathcal{T}(z) &= \frac{z^2}{(1-z)^3} \\ &= z^2\sum_{n=0}^\infty \binom{n+2}{n}z^n \\ &= \sum_{n=0}^\infty \frac{(n+2)(n+1)}{2}z^{n+2} \\ &= \sum_{n=2}^\infty \frac{n(n-1)}{2}z^{n} \\ &= \sum_{n=1}^\infty \frac{n(n-1)}{2}z^{n} \\ \end{align}$

For the first step, see again the Cheat Sheet. Remember that we started with $\mathcal{T}(z) = \sum_{n=1}^\infty T(n)z^n$ so we can read the solution off the last line:

$\qquad \displaystyle T(n) = \frac{n(n-1)}{2} \in \Theta(n^2)$