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| *每道题目的解答都要有<font color="red" size=5>完整的解题过程</font>。中英文不限。
| | == Summary == |
| | | Importing file |
| == Problem 1 ==
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| Recall that in class we show by the probabilistic method how to deduce a <math>\frac{n(n-1)}{2}</math> upper bound on the number of distinct min-cuts in any multigraph <math>G</math> with <math>n</math> vertices from the <math>\frac{2}{n(n-1)}</math> lower bound for success probability of Karger's min-cut algorithm.
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| Also recall that the <math>FastCut</math> algorithm taught in class guarantees to return a min-cut with probability at least <math>\Omega(1/\log n)</math>. Does this imply a much tighter <math>O(\log n)</math> upper bound on the number of distinct min-cuts in any multigraph <math>G</math> with <math>n</math> vertices? Prove your improved upper bound if your answer is "yes", and give a satisfactory explanation if your answer is "no".
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| == Problem 2 ==
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| Consider the function <math>f:\mathbb{R}^n\to\mathbb{R}</math> defined as
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| :<math>f(\vec x)=f(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i)</math>,
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| where <math>\{a_i\}_{1\le i\le n}</math> and <math>\{b_i\}_{1\le i\le n}</math> are '''unknown''' coefficients satisfy that <math>a_i, b_i\in \mathbb{Z}</math> and <math>0\le a_i, b_i \le n</math> for all <math>1\le i\le n</math>.
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| Let <math>p>n</math> be the smallest prime strictly greater than <math>n</math>. The function <math>g:\mathbb{Z}_p^n\to\mathbb{Z}_p</math> is defined as
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| :<math>g(\vec x)=g(x_1,x_2,\dots,x_n)=\prod_{i=1}^{n}(a_ix_i-b_i)</math>,
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| where <math>+</math> and <math>\cdot</math> are defined over the finite field <math>\mathbb{Z}_p</math>.
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| By the properties of finite field, for any value <math>\vec r\in\mathbb{Z}_p^n</math>, it holds that <math>g(\vec r)=f(\vec r)\bmod p</math>.
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| Since the coefficients <math>\{a_i\}_{1\le i\le n}</math> and <math>\{b_i\}_{1\le i\le n}</math> are unknown, you can't calculate <math>f(\vec x)</math> directly. However, there exists an oracle <math>O</math>, each time <math>O</math> gets an input <math>\vec x</math>, it immediately outputs the value of <math>g(\vec x)</math>.
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| 1. Prove that <math>f\not\equiv 0 \Rightarrow g\not\equiv 0</math>.
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| 2. Use the oracle <math>O</math> to design an algorithm to determine whether <math>f\equiv 0</math>, with error probability at most <math>\epsilon</math>, where <math>\epsilon\in (0,1)</math> is a constant.
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