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imported>TCSseminar |
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| == Problem 1==
| | 截止2018.10.18 21:00，作业3已提交名单如下 |
| 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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| | | |171250623 || 姜勇刚 |
| 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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| | | |DZ1733021 || 夏瑞 |
| == Problem 2 ==
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| 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.
| | |DZ1833028 || 徐闽泽 |
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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.
| | |151220130 || 伍昱名 |
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| == Problem 3 ==
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| Consider the following problem:
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| *Fixed a Universe <math>U</math> of size <math>m</math>, given a set <math>S \subseteq U</math> of size <math>n</math> and for each <math>x \in S</math> a <math>r</math>-bit variable <math>v_x</math>, we want to store <math>S</math> and <math>\{v_x\}_{x\in S}</math> into a table, so that the query "Is <math>x\in S</math>, and if so, what's <math>v_x</math>?" can be answer quickly.
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| The above problem, which is often refered to as '''dictionary''', is a generalization of the set membership problem that we have introduced in class, and there is another hash table called '''bloomier filter''' that solves this problem in linear (<math>O(nr)</math>) space and constant query time with tiny one-sided error probability.
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| Now if we are allowed to answer arbitrarily when <math>x\notin S</math>, then without resorting to bloomier filter, show that we can store any instance(<math>S</math> and <math>\{v_x\}_{x\in S}</math>) into a table using at most <math>O(nr+\log\log m)</math> bits space, and then correctly answer this relaxed query based on table contents only.
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| ''Hint: You don't have to actually design an algorithm, just prove its existence is fine. You might want to use probabilistic method and [https://en.wikipedia.org/wiki/Perfect_hash_function perfect hashing].''
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