随机算法 (Fall 2011)/Problem set 1

Problem 0

你的姓名、学号。

Problem 1

(Interviewing problem of Google Inc.)

Give a streaming algorithm to maintain a uniform sample from a data stream. The meaning of this sentence is explained as follows:

Suppose that the input is a sequence of items [math]\displaystyle{ A[1], A[2], A[3], \ldots, A[n] }[/math], which is passed to your algorithm by one item at a time in the sequential order. (Equivalently, you can imagine that your algorithm scans over a large array [math]\displaystyle{ A }[/math] in one direction from left to right.)

You algorithm should return an [math]\displaystyle{ A[r] }[/math], where [math]\displaystyle{ r }[/math] is uniformly distributed over [math]\displaystyle{ \{1,2,\ldots, n\} }[/math].

Usually the input "data stream" is from a massive data set (e.g. search engine data), so you cannot afford storing the entire input. Make your algorithm use as small space as possible. We hope for a small space storing only constant number of items.

• Develop an algorithm for the above problem. Give rigorous analysis for the algorithm to justify its correctness and efficiency.
• Develop an algorithm which works even if [math]\displaystyle{ n }[/math] is not known in advance, and also give your analysis for the algorithm. (If your algorithm already satisfies this requirement, it's OK to have one algorithm answer both questions.)

Problem 2

(Interviewing problem of Theory Group in Microsoft Research Asia for student interns.)

We start with [math]\displaystyle{ n }[/math] people, each with 2 hands. None of these hands hold each other.

At each round, we uniformly pick 2 free hands and let them hold together.

• After how many rounds, there are no free hands left?
• What is the expected number of cycles made by people holding hands with each other (one person with left hand holding right hand is also counted as a cycle), when there are no free hands left?

Problem 3

We use the Bloom filters to store playlists of songs, and find people with same taste in music.

Suppose Li Lei has a set [math]\displaystyle{ X }[/math] and Han Meimei has a set [math]\displaystyle{ Y }[/math] (their playlists), both with [math]\displaystyle{ n }[/math] elements. Denote [math]\displaystyle{ \ell=|X\cap Y| }[/math].

We create Bloom filters for both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], using the same number of bits [math]\displaystyle{ m }[/math] and the same [math]\displaystyle{ k }[/math] hash functions (with the uniform hash assumption). Let [math]\displaystyle{ t }[/math] be the expected number of bits where the two Bloom filters differ.

• Represent [math]\displaystyle{ t }[/math] as a function of [math]\displaystyle{ m,n,k,\ell }[/math].
• How can this be used to find people with same taste in music. Is it more efficient than directly comparing playlists? Why?