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计算机问题求解 – 论题 1-15 - 问题的难度

计算机问题求解 – 论题 1-15 - 问题的难度. 201 8 年 12 月 25 日. Part I 问题的“固有”难度. 问题 1: 这类拼图问题为什么会很难 ?. 问题 2: 在我们的讨论中, reasonable 这个词是用来修饰什么的,怎么就算 reasonable 呢?. 问题 3 : 在我们的讨论中 tractable 这个词又是修饰什么的呢?它是什么意思?. 问题 3 : 你怎么理解这句话?. Something that slightly annoys or upset you. 问题 4 : 书上举了那些“难”问题的例子?.

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计算机问题求解 – 论题 1-15 - 问题的难度

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  1. 计算机问题求解–论题1-15 - 问题的难度 2018年12月25日

  2. Part I 问题的“固有”难度

  3. 问题1: 这类拼图问题为什么会很难?

  4. 问题2: 在我们的讨论中,reasonable这个词是用来修饰什么的,怎么就算reasonable呢?

  5. 问题3: 在我们的讨论中tractable这个词又是修饰什么的呢?它是什么意思?

  6. 问题3: 你怎么理解这句话?

  7. Something that slightly annoys or upset you

  8. 问题4: 书上举了那些“难”问题的例子?

  9. 问题5: 优化问题与判定问题是什么关系,是否一个问题总可以定义其两种形式?你能说说他们的难度有什么关系吗?

  10. Max Clique: an Example • A maximal complete subgraph of a graph G is called a clique, whose size is the number of vertices in it. • Optimization problem: Find the maximal clique in a given graph G. • Decision problem: Has G a clique of size at least k for some given k?

  11. Max Clique Revisited • The max clique problem can be solved in polynomial time iff. the corresponding decision problem can be solved in polynomial time. • If the size of a max clique can be found in time g(n), the corresponding decision may be settled in that time of course. • If deciClique is algorithm for the decision problem with k in the complexity of f(n), then we apply the algorithm at most n time, for k=n, n-1, ..., 2, 1, and we can solve the optimization problem, and with the complexity no worse than nf(n), which is polynomial only if f(n) is polynomial.

  12. Part II P与NP

  13. 问题6: Magic Coins是什么?你能否以图着色为例,说说它在什么情况下有什么样的作用?

  14. Nondeterministic vs. Deterministic In O(1) Note: (n) for deterministic algorithm void nondetSearch(int k; int[ ] S) inti =genCertif(); if (S[i]=k) Output “yes”; return; void nondetSort(int[ ] S; intn) inti, j; int[ ] out=0; fori =1 tondo j= genCertif(); if out[j ]0 then return; out[j]=S[i ]; fori =1 ton-1 do if out[i ]>out[i+1] thenreturn; S=out; Output(yes); return In O(n) Note: (nlogn) for deterministic algorithm

  15. Max Clique Problem is in NP void nondeteClique(graph G; intn, k) setS=; for inti=1 tokdo intt=genCertif(); iftSthen return; S=S{t }; for all pairs (i,j) with i,j in S and ijdo if (i,j ) is not an edge of G then return; Output(“yes”); In O(n) In O(k2) So, we have an algorithm for the maximal clique problem with the complexity of O(n+k2)=O(n2)

  16. Satisfiability Problem An example of propositional conjunctive normal form (CNF) is like this: Satisfiability Problem Given a CNF formula, is there a truth assignment that satisfies it? In other words, is there a assignment for the set of propositional variable in the CNF, such that the value of the formula is true. void nondetSat(E, n) boolean p[ ]; forint i =1 tondo p[i ]= genCertif(true, false); ifE(p[1], p[2], ..., p[n])=true then Output(“yes”); So, the problem is in NP

  17. Solving a Problem Indirectly T(x) x T Algorithm for Q yes or no answer an input for Q (an input for P ) Algorithm for P The correct answer for P on x is yesif and only if the correct answer for Q on T(x) is yes.

  18. 问题7: 如果问题Q可以多项式时间内规约到问题P,关于这两个问题的难度关系你能说些什么?

  19. 问题8: 你能否证明多项式可规约是传递关系?

  20. NP-complete Problems • A problem Q is NP-hard if every problem P in NP is reducible to Q, that is PPQ. (which means that Q is at least as hard as any problem in NP) • A problem Q is NP-complete if it is in NP and is NP-hard. (which means that Q is at most as hard as to be solved by a polynomially bounded nondeterministic algorithm)

  21. 问题9: 你能理解为什么说这个问题是计算机科学中最重要的问题吗?

  22. 课外作业 • DH pp.186- • 7.2, 7.4, 7.6, 7.10,

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