Smart search: a practical facet of theoretical mathematics Mikhail Volkov Ural Federal University, Ekaterinburg, Russia
Where am I from?
Part I: Smart Search A typical search problem (Data Mining): ◮ A huge data base ◮ Strict time limit ◮ Time depends on the numbers of queries
Part I: Smart Search A typical search problem (Data Mining): ◮ A huge data base ◮ Strict time limit ◮ Time depends on the numbers of queries How can one quickly retrieve data? Theoretical mathematics offers a solution!! [G´ al, Miltersen. ICALP 2003]
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem.
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order.
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards.
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 13 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 76 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 13 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 If during this search every (!) prisoner finds his number in one of the drawers, all prisoners are pardoned.
100 prisoners problem In order to get freedom, 100 prisoners 0 , . . . , 99 have to solve the following problem. In a room there is a cupboard with 100 drawers. Each drawer contains the number of exactly one prisoner in random order. The prisoners enter the room one after another. Each prisoner may open and look into 50 drawers in any order and the drawers are closed again afterwards. 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 If during this search every (!) prisoner finds his number in one of the drawers, all prisoners are pardoned. Before the first prisoner enters the room, the prisoners may discuss their strategy, afterwards no communication is possible.
Naive approach 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 13 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 How big is the chance to get pardoned?
Naive approach 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 How big is the chance to get pardoned? If a prisoner opens 50 drawers at random, he finds his number 100 = 1 50 with the probability 2
Naive approach 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 How big is the chance to get pardoned? If a prisoner opens 50 drawers at random, he finds his number 100 = 1 50 with the probability 2 Then the probability that everyone will find his number is 1 2 100 ≈ 0 . 000000000000000000000000000000 8 � �� � 30 zeroes
Naive approach 0 1 2 42 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 76 34 35 36 37 38 39 40 41 42 43 44 45 46 8 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 How big is the chance to get pardoned? If a prisoner opens 50 drawers at random, he finds his number 100 = 1 50 with the probability 2 Then the probability that everyone will find his number is 1 2 100 ≈ 0 . 000000000000000000000000000000 8 � �� � 30 zeroes Is there a better strategy?
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