Intro

This lecture will review everything you learned in probabaility

• Basic tools in probability
• Expectations
• High probability events
• Deviations from expectation

Coupon collecting.

• n coupon types. Get a random one each round. How long to get all coupons?
• general example of waiting for combinations of events to happen.
• expected case analysis:
• after get k coupons, each sample has 1 - k/n chance for new coupon
• so wait for (k + 1)^st coupon has geometric distribution.
• expected value of geo dist w/param p is 1/p
• so get harmonic sum
• what standard tools did we use? using conditional expectation to study on phase; used linearity of expectation to add
• expected time for all coupons: n ln n + O(n).

Stable Marriage

Problem:

• complete preference lists
• stable if no two unmarried (to each other) people prefer each other.
• med school
• always exists.

Proof by proposal algorithm:

• rank men arbitrarily
• lowest unmarried man proposes in order of preference
• woman accepts if unmarried or prefers new proposal to current mate.

Time Analysis:

• woman's state only improves with time
• only n improvements per woman
• while unattached man, proposals continue
• (some woman available, since every woman he proposed to is married now)
• must eventually all be attached

Stability Analysis

• suppose X-y are dissatisfied with pairing X-x, Y-y.
• X proposed to y first
• y prefers current Y to X.

Average case analysis

• nonstandard for our course
• random preference lists
• how many proposals?
• principle of deferred decisions
• used intuitively already
• random choices all made in advance
• random choices made algorithm needs them.
• used while discussing autopartition, quicksort
• Proposal algorithm:
• each proposal is random among unchosen women
• still hard
• Each proposal among all women
• stochastic domination: X s.d. Y when Pr[X > z]$\ge$Pr[Y > z] for all z.
• done when all women get a proposal.
• at each step 1/n chance women gts proposal
• This is just coupon collection: O(n log n)

Deviations from Expectation

Sometimes expectation isn't enough. Want to study deviations--probability and magnitude of deviation from expectation.

Example: balls in bins:

• n balls in n bins
• Expected balls per bin: 1 (not very interesting)
• What is max balls we expect to see in a bin?
• Start by bounding probability of many balls

$$\binom{n}{k}$$

$$\le \binom{n}{k}$$

$$\le \left(\vphantom{\frac{ne}{k}}\right. {\frac{ne}{k}} \left.\vphantom{\frac{ne}{k}}\right)^{k}_{}$$

$$\left(\vphantom{\frac{e}{k}}\right. {\frac{e}{k}} \left.\vphantom{\frac{e}{k}}\right)^{k}_{}$$

• 
  
• So prob at least k balls is $\sum_{j \ge k}^{}$(e/j)^j = O((e/k)^k) (geometric series)
• $\le$1/n² if k > (e ln n)/lnln n

¬…       What is probability any bin is over k? 1/n union bound.

¬…       Now can bound expected max:

o      With probability 1 - 1/n, max is O(ln n/lnln n).

o      With probability 1/n, max is bigger, but at most n

o      So, expected max O(ln n/lnln n)

¬…       Typical approach: small expectation as small ``common case'' plus large ``rare case''

Example: coupon collection/stable marriage.

• Probability didn't get k^th coupon after r rounds is (1 - 1/n)^r$\le$e^-r/n
• which is n^{- $\beta$}for r = $\beta$n ln n
• so probability didn't get some coupon is at msot n ^. n^{- $\beta$}= n^{1 - $\beta$}(using union bound)
• we say ``time is O(n ln n) with high probability'' because we can make probability n<sup>- $\beta$</sup>for any desired $\beta$by changing constant that doesn't affect assymptotic claim.
• sometime say ``with high probability'' when prove it for some $\beta$> 1 even if didn't prove it for all.
• Saying ``almost never above O(n ln n)'' is a much stronger statement than saying ``O(n ln n) on average.''

Tail Bounds--Markov Inequality

At other times, don't want to get down and dirty with problem. So have developed set of bounding techniques that are basically problem independent.

• few assumptions, so applicable almost anywhere
• but for same reason, don't give as tight bounds
• the more you require of problem, the tighter bounds you can prove.

Markov inequality.

• Pr[Y$\ge$t]$\le$E[Y]/t
• Pr[Y$\ge$tE[Y]]$\le$1/t.
• Only requires an expectation! So very widely applicable.

Application: ZPP = RP $\cap$coRP.

• If RP $\cap$coRP
• just run both
• if neither affirms, run again
• Each iter has probability 1/2 to affirm
• So expected iterations 2:
• So ZPP.
• If ZPP
• suppose expected time T(n)
• Run for time 2T(n), then stop and give default answer
• Probability of default answer at most 1/2 (Markov)
• So, RP.
• If flip default answer, coRP

On flip side, not very strong: balls in bins Pr[ > ln n]$\le$1/ln n.

Can make much stronger by generalizing: Pr[h(Y) > t]$\le$E[h(Y)]/t for any positive h.