General comments

• Please read my comments on Canvas.
• Use LaTeX for mathematics, and spell check.
• Academic integrity:
  • You may work with others.
  • If you work with others, you should cite your collaboration on your assignment by listing the names of those you worked with.
  • If you work with others, the work you hand in should represent your own understanding. Don’t hand in solutions you don’t understand.
  • Even if you work with others, the work you hand in should not be identical to the work another student hands in. You should be writing up your solutions on your own, not copying someone else’s solutions.
    • Next time I’m going to have to submit a plagiarism report to the Provost.

Repeat this to yourself until it makes sense

\[ \mbox{} \]

\[ \Large \log_b(x) \mbox{ is the number you raise } b \mbox{ to to get } x. \]

\[ \mbox{} \]

Don’t like the grammar? Put it in your own words.

Basic algebra

• You need to know and apply basic rules of algebra.
  • For example, logarithms and exponents.
  • In the future, if you leave something like \(\log_2 4\) unsimplified, you will lose points (assignment or exam).
• Make a list of rules for exponents and logarithms and memorize it.
  • Don’t just watch videos about it.

Master theorem for solving recurrences

Recursive case of divide and conquer recurrence:

\[ T(n) = rT(n/c) + O(f(n)) \]

1. Root dominates: \(T(n) = O(f(n))\)
   • happens when row sums decrease geometrically.
2. Internal nodes dominate: \(T(n) = O(f(n)\log n)\)
   • happens when row sums are constant.
3. Leaves dominate: \(T(n) = O(n^{\log_c r})\)
   • happens when row sums increase geometrically.

Sample solutions: A, B, C

• \(A(n) = 2A(\frac{n}{4}) + \sqrt{n}\)
  • Row sums: \(\sqrt{n}, \sqrt{n}, \sqrt{n}, \sqrt{n}, \ldots\)
  • Case 2, internal nodes dominate: \(A(n) = O(\sqrt{n} \log n)\)
• \(B(n) = 2B(\frac{n}{4}) + n\)
  • Row sums: \(n, n/2, n/4, n/8, \ldots\)
  • Case 1, root dominates: \(B(n) = O(n)\)
• \(C(n) = 2C(\frac{n}{4}) + n^2\)
  • Row sums: \(n^2, n^2/8, n^2/64, n^2/512, \ldots\)
  • Case 1: root dominates \(C(n) = O(n^2)\)

Sample solutions: D, E, F

• \(D(n) = 3D(\frac{n}{3}) + \sqrt{n}\)
  • Row sums: \(n, n\sqrt{3}, n(\sqrt{3})^2, n(\sqrt{3})^3, \ldots\)
  • Case 3, leaves dominate: \(D(n) = O(n)\)
• \(E(n) = 3E(\frac{n}{3}) + n\)
  • Row sums: \(n, n, n, n, \ldots\)
  • Case 2, internal nodes dominate: \(E(n) = O(n \log n)\)
• \(F(n) = 3F(\frac{n}{3}) + n^2\)
  • Row sums: \(n^2, n^2/3, n^2/9, n^2/27, \ldots\)
  • Case 1, root dominates: \(F(n) = O(n^2)\)

Sample solutions: G, H, I

• \(G(n) = 4G(\frac{n}{2}) + \sqrt{n}\)
  • Row sums: \(\sqrt{n}, \sqrt{8}\sqrt{n}, (\sqrt{8})^2\sqrt{n},(\sqrt{8})^3\sqrt{n}, \ldots\)
  • Case 3, leaves dominate: \(G(n) = O(n^2)\)
• \(H(n) = 4H(\frac{n}{2}) + n\)
  • Row sums: \(n, 2n, 4n, 8n, \ldots\)
  • Case 3, leaves dominate: \(H(n) = O(n^2)\)
• \(I(n) = 4I(\frac{n}{2}) + n^2\)
  • Row sums: \(n^2, n^2, n^2, n^2, \ldots\)
  • Case 2, internal nodes dominate: \(I(n) = O(n^2 \log n)\)

Patterns?

Join your table group and discuss the following. Write any observations you have on the Jamboard.

• What patterns do you notice?
  • For example, can you tell when you are going to be in case \(n\), for \(n = 1,2,3\)?
  • How to \(r\), \(c\), and \(f(n)\) “pull” the asymptotic estimate in different directions?

\[ T(n) = rT(n/c) + O(f(n)) \] - Notice any other patterns or rules-of-thumb?

Recall: Quick Sort

QuickSort(A[1 .. n]):
if (n > 1)
    r <- Partition(A, 1)
    QuickSort(A[1 .. r − 1])
    QuickSort(A[r + 1 .. n])

• Lucky: pivot is the median at each stage.
  • \(T(n) = 2T(n/2) + O(n) \in O(n \log n)\)
• Unlucky: pivot is always at beginning or end at each stage.
  • \(T(n) = T(n-1) + O(n) \in O(n^2)\)

Just a little lucky

With mild assumptions/modifications, we can hope that the pivot will fall in the middle third at each stage:

• Worst case: \(T(n) \leq T(n/3) + T(2n/3) + O(n)\)

This doesn’t exactly match the Master Theorem, but we can still use it.

• See Figure 1.11.

Table Groups

Cavarretta recurrence, revisited.

1. Consider adding a constant term \(K = O(1)\) to the Cavarretta recurrence:

\[ C(n) = \left\{\begin{array}{lr} 1, & \text{for } n=2,\\ 3C\left(\left\lceil\frac{2}{3}n\right\rceil\right) + K, & \text{for } n>2.\\ \end{array}\right. \]

Use a recursion tree (with root \(K\)) to solve this recurrence. What effect did adding the constant term have?

2. What if we added a linear term to the recurrence, instead of a constant term?

Messy recursion tree

3. Solve the recurrence \(K(n) = K(n/2) + 2K(n/3) + 3K(n/4) + n^2\) using a recursion tree.