随机算法 (Fall 2015)/Problem Set 2: Difference between revisions
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* Use the above bound to estimate the number of distinct <math>\alpha</math>-min cuts in <math>G</math>. | * Use the above bound to estimate the number of distinct <math>\alpha</math>-min cuts in <math>G</math>. | ||
==Problem | ==Problem 2 == | ||
Suppose that we flip a fair coin <math>n</math> times to obtain <math>n</math> random bits. Consider all <math>m={n\choose 2}</math> pairs of these bits in some order. Let <math>Y_i</math> be the exclusive-or of the <math>i</math>th pair of bits, and let <math>Y=\sum_{i=1}^m Y_i</math> be the number of <math>Y_i</math> that equal 1. | Suppose that we flip a fair coin <math>n</math> times to obtain <math>n</math> random bits. Consider all <math>m={n\choose 2}</math> pairs of these bits in some order. Let <math>Y_i</math> be the exclusive-or of the <math>i</math>th pair of bits, and let <math>Y=\sum_{i=1}^m Y_i</math> be the number of <math>Y_i</math> that equal 1. | ||
# Show that the <math>Y_i</math> are '''NOT''' mutually independent. | # Show that the <math>Y_i</math> are '''NOT''' mutually independent. | ||
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# Using Chebyshev's inequality, prove a bound on <math>\Pr[|Y-\mathbf{E}[Y]|\ge n]</math>. | # Using Chebyshev's inequality, prove a bound on <math>\Pr[|Y-\mathbf{E}[Y]|\ge n]</math>. | ||
==Problem | ==Problem 3== | ||
Show that the maximum load when <math>n</math> balls are hashed into <math>n</math> bins using a hash function chosen from a 2-universal family of hash functions is at most <math>O(\sqrt{n})</math> with probability at least 0.99. Generalize this argument to <math>k</math>-universal hash functions. | Show that the maximum load when <math>n</math> balls are hashed into <math>n</math> bins using a hash function chosen from a 2-universal family of hash functions is at most <math>O(\sqrt{n})</math> with probability at least 0.99. Generalize this argument to <math>k</math>-universal hash functions. | ||
Hint: Perhaps the only information we can use about a 2-universal hash function is the number of collisions. What does it become for <math>k</math>-universal hashing? | Hint: Perhaps the only information we can use about a 2-universal hash function is the number of collisions. What does it become for <math>k</math>-universal hashing? | ||
Revision as of 03:33, 13 November 2015
Problem 1
For any [math]\displaystyle{ \alpha\ge 1 }[/math], a cut [math]\displaystyle{ C }[/math] in an undirected (multi)graph [math]\displaystyle{ G(V,E) }[/math]is called an [math]\displaystyle{ \alpha }[/math]-min-cut if [math]\displaystyle{ |C|\le\alpha|C^*| }[/math] where [math]\displaystyle{ C^* }[/math] is a min-cut in [math]\displaystyle{ G }[/math].
- Give a lower bound to the probability that a single iteration of Karger's Random Contraction algorithm returns an [math]\displaystyle{ \alpha }[/math]-min-cut in a graph [math]\displaystyle{ G(V,E) }[/math] of [math]\displaystyle{ n }[/math] vertices.
- Use the above bound to estimate the number of distinct [math]\displaystyle{ \alpha }[/math]-min cuts in [math]\displaystyle{ G }[/math].
Problem 2
Suppose that we flip a fair coin [math]\displaystyle{ n }[/math] times to obtain [math]\displaystyle{ n }[/math] random bits. Consider all [math]\displaystyle{ m={n\choose 2} }[/math] pairs of these bits in some order. Let [math]\displaystyle{ Y_i }[/math] be the exclusive-or of the [math]\displaystyle{ i }[/math]th pair of bits, and let [math]\displaystyle{ Y=\sum_{i=1}^m Y_i }[/math] be the number of [math]\displaystyle{ Y_i }[/math] that equal 1.
- Show that the [math]\displaystyle{ Y_i }[/math] are NOT mutually independent.
- Show that the [math]\displaystyle{ Y_i }[/math] satisfy the property [math]\displaystyle{ \mathbf{E}[Y_iY_j]=\mathbf{E}[Y_i]\mathbf{E}[Y_j] }[/math].
- Compute [math]\displaystyle{ \mathbf{Var}[Y] }[/math].
- Using Chebyshev's inequality, prove a bound on [math]\displaystyle{ \Pr[|Y-\mathbf{E}[Y]|\ge n] }[/math].
Problem 3
Show that the maximum load when [math]\displaystyle{ n }[/math] balls are hashed into [math]\displaystyle{ n }[/math] bins using a hash function chosen from a 2-universal family of hash functions is at most [math]\displaystyle{ O(\sqrt{n}) }[/math] with probability at least 0.99. Generalize this argument to [math]\displaystyle{ k }[/math]-universal hash functions.
Hint: Perhaps the only information we can use about a 2-universal hash function is the number of collisions. What does it become for [math]\displaystyle{ k }[/math]-universal hashing?