# Lecture 007

Recur: correct word Recurse: wrong word

If your branching factor is $c\sqrt{n}$, then you need to have $\log (\log n)$ many levels in the tree.

Note that $O(\log (n!)) = n\log n$

• can be asymptotically faster

• often simpler to write (not always)

• break symmetry

• unpredictable

• hard to debug and analyze

## High Probability Bound

We say $W(n) \in O(f(n))$ if, for any constant $k$:

W(n) \in O(k \cdot f(n)) \geq 1 - \left(\frac{1}{n}\right)^k

We want the $k$ to be the same because as the function $W(n)$ violates more and more by constant $k$ compared to $f(n)$, we want it to be higher probability.

## Probability Cheatsheet

Expectation: $E[X] = \sum_{a \in \Omega} Pr\{a\} \cdot X(a)$

Linear Expectation: $X = X_0 + X_1 \implies E[X] = E[X_0] + E[X_1]$

Union Bound: $Pr\{A \cup B\} \leq Pr\{A\} + Pr\{B\}$

Markov Bound: for non-negative random variable $X \geq 0 \land a > 0 \implies Pr\{X \geq a\} \leq \frac{E[X]}{a}$

Conditional Probability: $Pr\{A \cap B\} = Pr\{A\} \cdot Pr\{B | A\} = Pr\{B\} \cdot Pr\{A | B\}$

c^{k \log n} \simeq n^{k \log c}

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