随机算法 (Spring 2013)/Problem Set 2
- 注意写明自己的姓名与学号。
- 必须包含完整的解题过程，中英文不限。
Problem 1
(Due to Karp)
Consider a bin containing [math]\displaystyle{ d }[/math] balls chosen at random (without replacement) from a collection of [math]\displaystyle{ n }[/math] distinct balls. Without being able to see or count the balls in the bin, we would like to simulate random sampling with replacement from the original set of [math]\displaystyle{ n }[/math] balls. Our only access to the balls in that we can sample without replacement from the bin.
(Spoiler alert: You may want to stop here for a moment and start thinking about the solution before proceeding to read the following part.)
Consider the following strategy. Suppose that [math]\displaystyle{ k\lt d }[/math] balls have been drawn from the bin so far. Flip a coin with probability of HEADS being [math]\displaystyle{ k/n }[/math]. If HEADS appears, then pick one of the [math]\displaystyle{ k }[/math] previously drawn balls uniformly at random; otherwise, draw a random ball from the bin. Show that each choice is independently and uniformly distributed over the space of the [math]\displaystyle{ n }[/math] original balls. How many times can we repeat the sampling?
Problem 2
(Due to Karger and Motwani)
- Let [math]\displaystyle{ S,T }[/math] be two disjoint subsets of a universe [math]\displaystyle{ U }[/math] such that [math]\displaystyle{ |S|=|T|=n }[/math]. Suppose we select a random set [math]\displaystyle{ R\subseteq U }[/math] by independently sampling each element of [math]\displaystyle{ U }[/math] with probability [math]\displaystyle{ p }[/math]. We say that the random sample [math]\displaystyle{ R }[/math] is good if the following two conditions hold: [math]\displaystyle{ R\cap S=\emptyset }[/math] and [math]\displaystyle{ R\cap T\neq\emptyset }[/math]. SHow that for [math]\displaystyle{ p=1/n }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.
- (The hint for the second question was wrong. The question is now optional and use whatever tools you can come up with.)
- Suppose now that the random set [math]\displaystyle{ R }[/math] is chosen by sampling the elements of [math]\displaystyle{ U }[/math] with only pairwise independence. Show that for a suitable choice of the value of [math]\displaystyle{ p }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.
(Hint: Use the second moment.)
Problem 3
- Generalize the LazySelect algorithm for the [math]\displaystyle{ k }[/math]-selection problem: Given as input an array of [math]\displaystyle{ n }[/math] distinct numbers and an integer [math]\displaystyle{ k }[/math], find the [math]\displaystyle{ k }[/math]th smallest number in the array.
- Use the Chernoff bounds instead of Chebyshev's inequality in the analysis of the LazySelect Algorithm and try to use as few random samples as possible.
Problem 4
(Generalization of Chernoff bound to the sum of geometric random variables.)
Let [math]\displaystyle{ X_1,X_2,\ldots,X_n }[/math] be independent geometrically distributed random variables each having expectation 2 (each of the [math]\displaystyle{ X_i }[/math] is an independent experiment counting the number of tosses of an unbiased coin up to and including the first HEADS). Let [math]\displaystyle{ X=\sum_{i=1}^n X_i }[/math] and [math]\displaystyle{ \delta }[/math] be a positive real constant. Use the moment generating functions to derive the best upper bound you can give on [math]\displaystyle{ \Pr[X\gt (1+\delta)(2n)] }[/math].