高级算法 (Fall 2019)/Problem Set 1

Under construction

• 每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

Modify the Karger's Contraction algorithm so that it works for the weighted min-cut problem. Prove that the modified algorithm returns a weighted minimum cut with probability at least [math]\displaystyle{ \frac{2}{n(n-1)} }[/math]. The weighted min-cut problem is defined as follows.

• Input: an undirected weighted graph [math]\displaystyle{ G(V, E) }[/math], where every edge [math]\displaystyle{ e \in E }[/math] is associated with a positive real weight [math]\displaystyle{ w_e }[/math];
• Output: a cut [math]\displaystyle{ C }[/math] in [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ \sum_{e \in C} w_e }[/math] is minimized.

Problem 2

Let [math]\displaystyle{ G=(V,E) }[/math] be a graph, where [math]\displaystyle{ n = |V| }[/math] and [math]\displaystyle{ m = |E| }[/math]. In class, we generate a random [math]\displaystyle{ S-T }[/math] cut by sampling [math]\displaystyle{ X_v \in \{0,1\} }[/math] uniformly and independently for each [math]\displaystyle{ v \in V }[/math] and constructing [math]\displaystyle{ S = \{v \in V \mid X_v = 1 \} }[/math] and [math]\displaystyle{ T = \{v \in V \mid X_v = 0 \} }[/math]. Now, consider an alternative way to generate the random [math]\displaystyle{ S-T }[/math] cut. Suppose [math]\displaystyle{ n }[/math] is an even number. Define a collection of subset as

[math]\displaystyle{ \mathcal{F} = \{H \subseteq V: |H| = n /2 \} }[/math]

We sample a random subset [math]\displaystyle{ S \in \mathcal{F} }[/math] uniformly at random and construct [math]\displaystyle{ T = V \setminus S }[/math].

• Find the expected size of such random [math]\displaystyle{ S-T }[/math] cut.
• Let [math]\displaystyle{ \mathcal{R}(\cdot) }[/math] be a random source. Given any [math]\displaystyle{ 0\leq p\leq 1 }[/math], [math]\displaystyle{ \mathcal{R}(p) }[/math] returns an independent random sample [math]\displaystyle{ X \in \{0,1\} }[/math] such that [math]\displaystyle{ \Pr[X= 1] = p }[/math]. Give an algorithm that uses [math]\displaystyle{ \mathcal{R}(\cdot) }[/math] to generate random subset [math]\displaystyle{ S \in \mathcal{F} }[/math] uniformly at random. Analyze the time complexity of your algorithm.

Problem 3

Two rooted trees [math]\displaystyle{ T_1 }[/math] and [math]\displaystyle{ T_2 }[/math] are said to be isomorphic if there exists a bijection [math]\displaystyle{ \phi }[/math] that maps vertices of [math]\displaystyle{ T_1 }[/math] to those of [math]\displaystyle{ T_2 }[/math] satisfying the following condition: for each interval vertex [math]\displaystyle{ v }[/math] of [math]\displaystyle{ T_1 }[/math] with children [math]\displaystyle{ u_1, u_2, ..., u_k }[/math], the set of children of vertex [math]\displaystyle{ \phi(v) }[/math] in [math]\displaystyle{ T_2 }[/math] is precisely [math]\displaystyle{ \{\phi(u_1), \phi(u_2),...,\phi(u_k)\} }[/math], no ordering among children assumed.

Given an efficient randomized algorithm with bounded one-side error (false positive), for testing isomorphism between rooted trees with [math]\displaystyle{ n }[/math] vertices. Analyze your algorithm.

Problem 4

Design a randomized algorithm to decide if an integer sequence [math]\displaystyle{ a_1,...,a_n }[/math] is a permutation of another integer sequence [math]\displaystyle{ b_1,...,b_n }[/math]. Bound the probability of the error and analyze the time complexity.

Problem 5

Let [math]\displaystyle{ X_1,X_2,\ldots,X_n }[/math] be [math]\displaystyle{ n }[/math] random variables, where each [math]\displaystyle{ X_i \in \{0, 1\} }[/math] follows the distribution [math]\displaystyle{ \mu_i }[/math]. For each [math]\displaystyle{ 1\leq i \leq n }[/math], let [math]\displaystyle{ \rho_i = \mathbb{E}[X_i] }[/math] and assume [math]\displaystyle{ \rho_i \geq \frac{1}{2} }[/math]. Consider the problem of estimating the value of

[math]\displaystyle{ Z = \prod_{i = 1}^n \rho_i }[/math]

For each [math]\displaystyle{ 1\leq i \leq n }[/math], the algorithm draws [math]\displaystyle{ s }[/math] random samples [math]\displaystyle{ X_i^{(1)},X_i^{(2)},\ldots,X_i^{(s)} }[/math] independently from the distribution [math]\displaystyle{ \mu_i }[/math], and computes

[math]\displaystyle{ \widehat{Z}_{i}=\frac{1}{s}\sum_{j=1}^s X_i^{(j)} }[/math]

Finally, the algorithm outputs the product of all [math]\displaystyle{ \widehat{Z}_{i} }[/math]:

[math]\displaystyle{ \widehat{Z}=\prod_{i= 1}^n\widehat{Z}_i }[/math]

Express [math]\displaystyle{ s }[/math] as a function of [math]\displaystyle{ n,\varepsilon,\delta }[/math] so that the output [math]\displaystyle{ \widehat{Z} }[/math] satisfies

[math]\displaystyle{ \Pr\left[\mathrm{e}^{-\varepsilon}Z \leq \widehat{Z} \leq \mathrm{e}^{\varepsilon}Z\right] \geq 1- \delta }[/math]

Try to make [math]\displaystyle{ s }[/math] as small as possible.