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台大資工系 呂學一 1.

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Presentation on theme: "台大資工系 呂學一 1."— Presentation transcript:

1 台大資工系 呂學一 1

2 2  計算  難題  人生

3  Input:  A graph G  Output:  A smallest vertex subset of G that covers all edges of G. 3

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5 Clever ways to solve computational problems 5

6 Being clever in solving a computation problem could make a big difference. 6

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8 It makes a big difference 8

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11 Coffee and Milk 11

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17  The Factorization Problem  Input: a number N  Output:  “yes” if N is a prime number;  A factorization of N if N is not a prime number.  For example,  N =  Output = *

18 Is there an efficient recipe for the Factorization Problem? 18

19 The security of many encryption schemes is based upon the assumption that the factorization problem is difficult. 19

20 20 Rivest ShamirAdleman

21 21 Challenge Number Prize ($US) Challenge Number Prize ($US) RSA-576$10,000RSA-896$75,000 RSA-640$20,000RSA-1024$100,000 RSA-704$30,000RSA-1536$150,000 RSA-768$50,000RSA-2048$200,000

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23    At the same time, Adi Shamir gave two talks at NTU (Dec. 4, 2003 ) 23

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25  F. Bahr, M. Boehm, J. Franke, T. Kleinjung    Efforts:  GHz-Opteron-CPU years  over five months of calendar time. 25

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31  Input:  A graph with edge lengths  Output:  A shortest tour visiting each node of the input graph exactly once. 31

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33  Clay Mathematics Institute (Cambridge, MA, USA) offered US$1,000,000 for each of seven open problems on May 24, 2000 at Paris.  | Birch and Swinnerton-Dyer Conjecture | Hodge Conjecture | Navier-Stokes Equations | P vs NP | Poincare Conjecture | Riemann Hypothesis | Yang- Mills Theory |Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincare Conjecture Riemann Hypothesis Yang- Mills Theory 33

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39  Input:  A graph G  Output:  A smallest vertex subset of G that covers all edges of G. 39

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41  Clay Mathematics Institute (Cambridge, MA, USA) offered US$1,000,000 for each of seven open problems on May 24, 2000 at Paris.  | Birch and Swinnerton-Dyer Conjecture | Hodge Conjecture | Navier-Stokes Equations | P vs NP | Poincare Conjecture | Riemann Hypothesis | Yang- Mills Theory |Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincare Conjecture Riemann Hypothesis Yang- Mills Theory 41

42 Being clever in solving computational problems could make a big difference. 42

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44 Approximation Algorithms ( 近似演算法 ) 44

45 放下對完美的堅持, 往往就可以找到新的出路。 知所進退, 則近道矣 ! 45

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47 康熙大學士兼禮部尚書張英 一紙書來只為牆, 讓他三尺又何妨。 長城萬里今猶在, 不見當年秦始皇。 47

48 路燈問題 (Vertex Cover) 48

49  Input:  A graph G  Output:  A smallest vertex subset of G that covers all edges of G. 49

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52  Input:  A graph G  Output:  A near smallest vertex subset of G that covers all edges of G. 52

53  Initially, let S be an empty set.  Repeat until G has no edges:  Arbitrarily choose an edge (u, v) of G.  Insert u and v into S.  Delete all edges of G incident to u or v.  Output S. 53

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55  Q1: Is the output vertex set indeed a vertex cover of the input graph?  Q2: Does the algorithm run in polynomial time?  Q3: Is the quality of the output solution close to optimal? 55

56  Yes. 56

57  It is not difficult to see that this algorithm runs in linear time.  O(n + m) time, where n is the number of vertices and m is the number of edges in G. 57

58  The output vertex cover has size at most 2 times that of any optimal vertex cover. 58

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60  Criterion 1: feasibility  Always output a feasible solution.  Criterion 2: tractability  Always runs in polynomial time.  Criterion 3: quality  The solution’s quality is always provably not too far from that of an optimal solution. 60

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62  That 2-approximation was known for 30 years.  It remains the best known approximation algorithm for the vertex cover problem!  Finding a approximation is known to be NP- hard.  Even a 1.9-approximation would be a significant breakthrough.  向公園路燈管理局致敬 62

63 掃黑的藝術 (maximum cut) 63

64  Input:  A graph G  Output:  A partition of G’s nodes into A and B that maximizes the number of edges between A and B. 64

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66  Repeat the following randomized subroutine for, say, 100 times, and then output the best cut among them.  For each node v of G,  put v into A with probability ½. 66

67  Any partition is a feasible solution. 67

68  Throwing a fair coin is “easy” to simulate by computers.  As a matter of fact, the existence of pseudo-random generator implies NP≠P. 68

69  One can prove that this simple algorithm is a 2- approximation with very high probability (something like 1-1/2 100 ). 69

70  For about 20 years, the above 2-approximation was the best known result for maximum cut. 70

71 Goemans and Williamson, ACM STOC

72  1/0.878-approximation for MAXCUT  Initiate a series of research in Operations Research, Scientific Computing, and Approximation Algorithms. 72

73 73 Integer Linear Program Integral solution Linear Program Fractional solution relaxation Rounding Approximation Solver

74 74 Quadratic Program Scalar solution Semi-definite Program Vector solution relaxation Rounding Approximation Solver

75  Being clever in solving problem can make a big difference  Most of the real life problems cannot be solved perfectly in reasonable amount of time.  Approximate solutions might be a nice alternative. 75

76  God, grant us the serenity to accept the things we cannot change, the courage to change the things we can, and the wisdom to know the difference. 76


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