1.0 - Recurrences

1.1 - Divide and Conquer Algorithms

1.1.1 - Merge Sort

• merge_sort(A, p, r) sorts the sub-array $A[p..r]$ which uses the merge(A, p, q, r) subroutine
• merge(A, p, q, r) takes the sorted sub-arrays $A[p,q]$ and $A[q+1,r]$ and merges them to produce the sorted array $A[p, r]$.
• This assumes merge(A, p, q, r) is $\Theta(n)$ where $n=r-p+1$
  • i.e. we’re measuring the input in terms of the number of items that need to be merged

if p < r                    // If there are two or more elements to merge, we divide:
		q = ⌊(p + r) / 2⌋        // Divide
		merge_sort(A, p, q)     // solve sub-problem recursively T(n/2) work
    merge_sort(A, q+1, r)   // solve sub-problem recursively T(n/2) work
    merge(A, p, q, r)       // combine

• Let $n=r-p+1$

  $$\begin{aligned} T(n)&=\Theta(1)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if } n\le 1\\ T(n)&=\Theta(1)+2T(\frac n2)+\Theta(n) \ \ \ \ \ \ \ \text{if } n > 0\\ &=2 T(\frac n2)+\Theta(n) \end{aligned}$$

• If we have hit the base case, (we assume) that the method is going to take a constant amount of time to execute an return otherwise.

• Otherwise, we have to perform $2T(\frac n2)$ work to sort the two sub-problems, as well as the $\Theta(n)$ work required to merge the two sub-problems together.

• $T(n)=2T(\frac n2) +\Theta(n)$ is shorthand for $T(n)=2T(\frac n2)+f(n)$ for some function $f(n)\in\Theta(n)$

1.2 - Recurrences

• To be well-defined, a recurrence needs:

  • base case, and
  • recursive case(s) that converge on the base case

• Running time is usually bounded by a constant for constant-sized inputs, and so we often omit the base case, and assume that

  $$T(n)=\Theta(1) \ \ \ \ \ \ \text{for } n\le c$$

  where $c$ is a constant.

1.1 - Divide and Conquer Algorithms

• Given a divide and conquer algorithm that:
  • Takes $\color{lightblue} D(n)$ time to divide the problem, and
  • Takes a problem of size $\color{lightblue} n$, and
  • Breaks it into $\color{lightblue} a$ parts, each of size $\color{lightblue} \frac nb$
  • Takes $C(n)$ time to combine solutions to sub problems, we get the following recurrence:

1.1.1 - Merge Sort

• merge_sort(A, p, r) sorts the sub-array $A[p..r]$
• merge(A, p, q, r) takes the sorted sub-arrays $A[p,q]$ and $A[q+1,r]$ and merges them to produce the sorted array $A[p, r]$.
• This assumes merge(A, p, q, r) is $\Theta(n)$ where $n=r-p+1$

if p < r
		q = ⌊(p + r) / 2⌋        // Divide
		merge_sort(A, p, q)     // solve sub-problem recursively
    merge_sort(A, q+1, r)   // solve sub-problem recursively.
    merge(A, p, q, r)       // combine

• Let $n=r-p+1$

  $$\begin{aligned} T(n)&=\Theta(1)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if } n\le 1\\ T(n)&=\Theta(1)+2T(\frac n2)+\Theta(n) \ \ \ \ \ \ \ \text{if } n > 0\\ &=2 T(\frac n2)+\Theta(n) \end{aligned}$$

• $T(n)=2T(\frac n2) +\Theta(n)$ is shorthand for $T(n)=2T(\frac n2)+f(n)$ for some function $f(n)\in\Theta(n)$

  • That is, $f(n)$ is a yet-unknown function that is $\Theta(n)$