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| 每道题目的解答都要有<font color="red" >完整的解题过程、分析和证明</font>。中英文不限。
| | Chapter 2. |
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| | Exercise 2.15, 2.16, 2.17, 2.18, 2.29, 2.6 (bonus) |
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| == 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".
| | 请将作业的电子版本(pdf、扫描或拍照)发送到助教处( [mailto:liu.mingmou@smail.nju.edu.cn liu.mingmou@smail.nju.edu.cn] ) |
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| == Problem 2 ==
| | Deadline: 10月11日上午10:09 |
| Two ''rooted'' trees <math>T_1</math> and <math>T_2</math> are said to be '''isomorphic''' if there exists a bijection <math>\phi</math> that maps vertices of <math>T_1</math> to those of <math>T_2</math> satisfying the following condition: for each ''internal'' vertex <math>v</math> of <math>T_1</math> with children <math>u_1,u_2,\ldots, u_k</math>, the set of children of vertex <math>\phi(v)</math> in <math>T_2</math> is precisely <math>\{\phi(u_1), \phi(u_2),\ldots,\phi(u_k)\}</math>, no ordering among children assumed.
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| Give an efficient randomized algorithm with bounded one-sided error (false positive), for testing isomorphism between rooted trees with <math>n</math> vertices. Analyze your algorithm.
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| == Problem 3 ==
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| Fix a universe <math>U</math> and two subset <math>A,B \subseteq U</math>, both with size <math>n</math>. we create both Bloom filters <math>F_A</math>(<math>F_B</math>) for <math>A</math> (<math>B</math>), using the same number of bits <math> m</math> and the same <math>k</math> hash functions.
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| *Let <math>F_C = F_A \and F_B</math> be the Bloom filter formed by computing the bitwise AND of <math>F_A</math> and <math>F_B</math>. Argue that <math>F_C</math> may not always be the same as the Bloom filter that are created for <math>A\cap B </math>.
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| *Bloom filters can be used to estimate set differences. Express the expected number of bits where <math>F_A</math> and <math>F_B</math> differ as a function of <math>m, n, k</math> and <math>|A\cap B|</math>.
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Revision as of 06:25, 21 September 2018
Chapter 2.
Exercise 2.15, 2.16, 2.17, 2.18, 2.29, 2.6 (bonus)
请将作业的电子版本(pdf、扫描或拍照)发送到助教处( liu.mingmou@smail.nju.edu.cn )
Deadline: 10月11日上午10:09