高级算法 (Fall 2021)/Problem Set 1

• 每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

Recall that in class we show by the probabilistic method how to deduce a [math]\displaystyle{ \frac{n(n-1)}{2} }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices from the [math]\displaystyle{ \frac{2}{n(n-1)} }[/math] lower bound for success probability of Karger's min-cut algorithm.

Also recall that the [math]\displaystyle{ FastCut }[/math] algorithm taught in class guarantees to return a min-cut with probability at least [math]\displaystyle{ \Omega(1/\log n) }[/math]. Does this imply a much tighter [math]\displaystyle{ O(\log n) }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices? Prove your improved upper bound if your answer is "yes", and give a satisfactory explanation if your answer is "no".

Problem 2

Consider the function [math]\displaystyle{ f:\mathbb{R}^n\to\mathbb{R} }[/math] defined as

[math]\displaystyle{ f(\vec x)=f(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i) }[/math],

where [math]\displaystyle{ \{a_i\}_{1\le i\le n} }[/math] and [math]\displaystyle{ \{b_i\}_{1\le i\le n} }[/math] are unknown coefficients satisfy that [math]\displaystyle{ a_i, b_i\in \mathbb{Z} }[/math] and [math]\displaystyle{ 0\le a_i, b_i \le n }[/math] for all [math]\displaystyle{ 1\le i\le n }[/math].

Let [math]\displaystyle{ p\gt n }[/math] be the smallest prime strictly greater than [math]\displaystyle{ n }[/math]. The function [math]\displaystyle{ g:\mathbb{Z}_p^n\to\mathbb{Z}_p }[/math] is defined as

[math]\displaystyle{ g(\vec x)=g(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i) }[/math],

where [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] are defined over the finite field [math]\displaystyle{ \mathbb{Z}_p }[/math].

By the properties of finite field, for any value [math]\displaystyle{ \vec r\in\mathbb{Z}_p^n }[/math], it holds that [math]\displaystyle{ g(\vec r)=f(\vec r)\bmod p }[/math].

Since the coefficients [math]\displaystyle{ \{a_i\}_{1\le i\le n} }[/math] and [math]\displaystyle{ \{b_i\}_{1\le i\le n} }[/math] are unknown, you can't calculate [math]\displaystyle{ f(\vec x) }[/math] directly. However, there exists an oracle [math]\displaystyle{ O }[/math], each time [math]\displaystyle{ O }[/math] gets an input [math]\displaystyle{ \vec x }[/math], it immediately outputs the value of [math]\displaystyle{ g(\vec x) }[/math].

1. Prove that [math]\displaystyle{ f\not\equiv 0 \Rightarrow g\not\equiv 0 }[/math].

2. Use the oracle [math]\displaystyle{ O }[/math] to design an algorithm to determine whether [math]\displaystyle{ f\equiv 0 }[/math], with error probability at most [math]\displaystyle{ \epsilon }[/math], where [math]\displaystyle{ \epsilon\in (0,1) }[/math] is a constant.

Problem 1

Recall that in class we show by the probabilistic method how to deduce a [math]\displaystyle{ \frac{n(n-1)}{2} }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices from the [math]\displaystyle{ \frac{2}{n(n-1)} }[/math] lower bound for success probability of Karger's min-cut algorithm.

Also recall that the [math]\displaystyle{ FastCut }[/math] algorithm taught in class guarantees to return a min-cut with probability at least [math]\displaystyle{ \Omega(1/\log n) }[/math]. Does this imply a much tighter [math]\displaystyle{ O(\log n) }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices? Prove your improved upper bound if your answer is "yes", and give a satisfactory explanation if your answer is "no".

Problem 2

Consider the function [math]\displaystyle{ f:\mathbb{R}^n\to\mathbb{R} }[/math] defined as

[math]\displaystyle{ f(\vec x)=f(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i) }[/math],

where [math]\displaystyle{ \{a_i\}_{1\le i\le n} }[/math] and [math]\displaystyle{ \{b_i\}_{1\le i\le n} }[/math] are unknown coefficients satisfy that [math]\displaystyle{ a_i, b_i\in \mathbb{Z} }[/math] and [math]\displaystyle{ 0\le a_i, b_i \le n }[/math] for all [math]\displaystyle{ 1\le i\le n }[/math].

Let [math]\displaystyle{ p\gt n }[/math] be the smallest prime strictly greater than [math]\displaystyle{ n }[/math]. The function [math]\displaystyle{ g:\mathbb{Z}_p^n\to\mathbb{Z}_p }[/math] is defined as

[math]\displaystyle{ g(\vec x)=g(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i) }[/math],

where [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] are defined over the finite field [math]\displaystyle{ \mathbb{Z}_p }[/math].

By the properties of finite field, for any value [math]\displaystyle{ \vec r\in\mathbb{Z}_p^n }[/math], it holds that [math]\displaystyle{ g(\vec r)=f(\vec r)\bmod p }[/math].

Since the coefficients [math]\displaystyle{ \{a_i\}_{1\le i\le n} }[/math] and [math]\displaystyle{ \{b_i\}_{1\le i\le n} }[/math] are unknown, you can't calculate [math]\displaystyle{ f(\vec x) }[/math] directly. However, there exists an oracle [math]\displaystyle{ O }[/math], each time [math]\displaystyle{ O }[/math] gets an input [math]\displaystyle{ \vec x }[/math], it immediately outputs the value of [math]\displaystyle{ g(\vec x) }[/math].

1. Prove that [math]\displaystyle{ f\not\equiv 0 \Rightarrow g\not\equiv 0 }[/math].

2. Use the oracle [math]\displaystyle{ O }[/math] to design an algorithm to determine whether [math]\displaystyle{ f\equiv 0 }[/math], with error probability at most [math]\displaystyle{ \epsilon }[/math], where [math]\displaystyle{ \epsilon\in (0,1) }[/math] is a constant.

Problem 1

Recall that in class we show by the probabilistic method how to deduce a [math]\displaystyle{ \frac{n(n-1)}{2} }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices from the [math]\displaystyle{ \frac{2}{n(n-1)} }[/math] lower bound for success probability of Karger's min-cut algorithm.

Also recall that the [math]\displaystyle{ FastCut }[/math] algorithm taught in class guarantees to return a min-cut with probability at least [math]\displaystyle{ \Omega(1/\log n) }[/math]. Does this imply a much tighter [math]\displaystyle{ O(\log n) }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices? Prove your improved upper bound if your answer is "yes", and give a satisfactory explanation if your answer is "no".

Problem 2

Consider the function [math]\displaystyle{ f:\mathbb{R}^n\to\mathbb{R} }[/math] defined as

[math]\displaystyle{ f(\vec x)=f(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i) }[/math],

where [math]\displaystyle{ \{a_i\}_{1\le i\le n} }[/math] and [math]\displaystyle{ \{b_i\}_{1\le i\le n} }[/math] are unknown coefficients satisfy that [math]\displaystyle{ a_i, b_i\in \mathbb{Z} }[/math] and [math]\displaystyle{ 0\le a_i, b_i \le n }[/math] for all [math]\displaystyle{ 1\le i\le n }[/math].

Let [math]\displaystyle{ p\gt n }[/math] be the smallest prime strictly greater than [math]\displaystyle{ n }[/math]. The function [math]\displaystyle{ g:\mathbb{Z}_p^n\to\mathbb{Z}_p }[/math] is defined as

[math]\displaystyle{ g(\vec x)=g(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i) }[/math],

where [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] are defined over the finite field [math]\displaystyle{ \mathbb{Z}_p }[/math].

By the properties of finite field, for any value [math]\displaystyle{ \vec r\in\mathbb{Z}_p^n }[/math], it holds that [math]\displaystyle{ g(\vec r)=f(\vec r)\bmod p }[/math].

Since the coefficients [math]\displaystyle{ \{a_i\}_{1\le i\le n} }[/math] and [math]\displaystyle{ \{b_i\}_{1\le i\le n} }[/math] are unknown, you can't calculate [math]\displaystyle{ f(\vec x) }[/math] directly. However, there exists an oracle [math]\displaystyle{ O }[/math], each time [math]\displaystyle{ O }[/math] gets an input [math]\displaystyle{ \vec x }[/math], it immediately outputs the value of [math]\displaystyle{ g(\vec x) }[/math].

1. Prove that [math]\displaystyle{ f\not\equiv 0 \Rightarrow g\not\equiv 0 }[/math].

2. Use the oracle [math]\displaystyle{ O }[/math] to design an algorithm to determine whether [math]\displaystyle{ f\equiv 0 }[/math], with error probability at most [math]\displaystyle{ \epsilon }[/math], where [math]\displaystyle{ \epsilon\in (0,1) }[/math] is a constant.