NOI 2009 Solutions to Contest Tasks
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1 NOI 2009 Solutions to Contest Tasks Martin Henz March 14, 2009 Martin Henz NOI 2009 Solutions to Contest Tasks 1
2 The Scientific Committee of NOI 2009 Ben Leong Chang Ee-Chien Colin Tan Frank Stephan Martin Henz Sung Wing Kin Martin Henz NOI 2009 Solutions to Contest Tasks 2
3 1 XMAS Martin Henz NOI 2009 Solutions to Contest Tasks 3
4 Program in Pascal 1 XMAS Program in Pascal Martin Henz NOI 2009 Solutions to Contest Tasks 4
5 XMAS Program in Pascal At a Secret Santa Christmas party, each guest brings a gift and places it under the Christmas tree. Each guest gets exactly one gift from under the tree. Martin Henz NOI 2009 Solutions to Contest Tasks 5
6 XMAS Program in Pascal At a Secret Santa Christmas party, each guest brings a gift and places it under the Christmas tree. Each guest gets exactly one gift from under the tree. You are given a list that says whose gift each person receives, and need to compile a list of guests that pick each person s gift. Martin Henz NOI 2009 Solutions to Contest Tasks 6
7 XMAS Program in Pascal Alice brings an apple, Bob brings a balloon, and Carol brings a cake. Given: Alice gets the balloon, Bob gets the cake, and Carol gets the apple. Required list: Apple goes to Carol, balloon goes to Alice, cake goes to Bob. Martin Henz NOI 2009 Solutions to Contest Tasks 7
8 XMAS Program in Pascal Picking a gift corresponds to a permutation of the list of guests. Martin Henz NOI 2009 Solutions to Contest Tasks 8
9 XMAS Program in Pascal Picking a gift corresponds to a permutation of the list of guests. Task compute the inverse of the permutation. Martin Henz NOI 2009 Solutions to Contest Tasks 9
10 XMAS Program in Pascal Idea For each gift, remember its receiver. Then output the receivers in a loop. Martin Henz NOI 2009 Solutions to Contest Tasks 10
11 Program in Pascal XMAS Program in Pascal program xmas ; var R e c e i v e r : array [ ] of i n t e g e r ; g u e s t s, g i f t s, guest, g i f t : i n t e g e r ; begin readln ( g u e s t s ) ; g i f t s := g u e s t s ; f o r g u e s t := 1 to g u e s t s do begin readln ( g i f t ) ; R e c e i v e r [ g i f t ] := g u e s t ; end ; f o r g i f t := 1 to g i f t s do w r i t e l n ( R e c e i v e r [ g i f t ] ) ; end. Martin Henz NOI 2009 Solutions to Contest Tasks 11
12 Complexity XMAS Program in Pascal There is a fixed amount of work done in each loop body. Martin Henz NOI 2009 Solutions to Contest Tasks 12
13 Complexity XMAS Program in Pascal There is a fixed amount of work done in each loop body. Each loop runs through all guests (gifts). Martin Henz NOI 2009 Solutions to Contest Tasks 13
14 Complexity XMAS Program in Pascal There is a fixed amount of work done in each loop body. Each loop runs through all guests (gifts). Overall: the runtime grows proportional to the number of guests. Martin Henz NOI 2009 Solutions to Contest Tasks 14
15 1 XMAS Martin Henz NOI 2009 Solutions to Contest Tasks 15
16 Investment options: oil, shares, steel, silver and gold. Investor has to commit to an investment plan over m months. Investor has to commit to a sum of $2520 which is invested each month. Each month, the investor has to buy one product for $2520. Investor has to buy the same product in the first 15 months of the plan (or in all months if it is shorter than 15 months). Two consecutive changes of the product bought must be separated by at least 14 months, in which the product remains unchanged. Martin Henz NOI 2009 Solutions to Contest Tasks 16
17 XMAS For a 36 months plan it is possible to buy gold the first 18 months, then silver the next 15 months and then gold again the remaining three months. Martin Henz NOI 2009 Solutions to Contest Tasks 17
18 Input XMAS First six rows contain buy price in given month; last row contains sell price after last month. Martin Henz NOI 2009 Solutions to Contest Tasks 18
19 Desired Output XMAS Best returns: The amount of money that can be earned from selling all investments at the end of the period, after investing each month in the best possible way. Martin Henz NOI 2009 Solutions to Contest Tasks 19
20 A Naive Approach Consider all possible combinations of investments. : G-G-O-O-O-S For each combination, check that it meets the rules. If it meets the rules, compute the returns. Output the highest return. Martin Henz NOI 2009 Solutions to Contest Tasks 20
21 XMAS Complexity! We have to go through 5 months combinations! Martin Henz NOI 2009 Solutions to Contest Tasks 21
22 First Idea XMAS Observation We are allowed to change the investment product only after at least 15 months Martin Henz NOI 2009 Solutions to Contest Tasks 22
23 First Idea XMAS Observation We are allowed to change the investment product only after at least 15 months Reduce combinations Only consider months in which a change of product happens. Martin Henz NOI 2009 Solutions to Contest Tasks 23
24 First Idea XMAS Observation We are allowed to change the investment product only after at least 15 months Reduce combinations Only consider months in which a change of product happens. Complexity reduces to 5 months/15 combinations! Martin Henz NOI 2009 Solutions to Contest Tasks 24
25 Second Idea XMAS Memoization Once we have computed the best return when starting with a certain product on a given month, we don t need to compute it any more; we simply remember the computed value in a table. Martin Henz NOI 2009 Solutions to Contest Tasks 25
26 Second Idea Memoization Once we have computed the best return when starting with a certain product on a given month, we don t need to compute it any more; we simply remember the computed value in a table. Complexity The overall runtime reduces to numberofproducts numberofmonths Martin Henz NOI 2009 Solutions to Contest Tasks 26
27 1 XMAS Martin Henz NOI 2009 Solutions to Contest Tasks 27
28 XMAS The map of a house is given by a grid like this one: B F X X F F X X F S A lazy cat at a S tarting point wants to pick up all F ood items and then go to B ed. Martin Henz NOI 2009 Solutions to Contest Tasks 28
29 Rules B F X X F F X X F S Only step left, right, up or down Do not step on walls marked with X Martin Henz NOI 2009 Solutions to Contest Tasks 29
30 Traveling Salesman (TSP) The problem Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. Martin Henz NOI 2009 Solutions to Contest Tasks 30
31 Traveling Salesman (TSP) The problem Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. Martin Henz NOI 2009 Solutions to Contest Tasks 31
32 Traveling Salesman (TSP) The problem Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. Optimal tour visiting Germany s 15 largest cities Martin Henz NOI 2009 Solutions to Contest Tasks 32
33 Traveling Salesman (TSP) The problem Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. Optimal tour visiting Germany s 15 largest cities Shortest among 43,589,145,600 possible tours Martin Henz NOI 2009 Solutions to Contest Tasks 33
34 Traveling Salesman (TSP) The problem Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. Optimal tour visiting Germany s 15 largest cities Shortest among 43,589,145,600 possible tours There is no known algorithm that can solve this problem efficiently Martin Henz NOI 2009 Solutions to Contest Tasks 34
35 Traveling Salesman (TSP) The problem Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. Optimal tour visiting Germany s 15 largest cities Shortest among 43,589,145,600 possible tours There is no known algorithm that can solve this problem efficiently Runtime of best algorithm grows exponentially with number of cities Martin Henz NOI 2009 Solutions to Contest Tasks 35
36 TSP With A Twist XMAS B F X X F F X X F S Martin Henz NOI 2009 Solutions to Contest Tasks 36
37 TSP With A Twist XMAS B F X X F F X X F S What is the distance between F ood items? We need to compute the shortest path between two food items, considering the walls marked with X Martin Henz NOI 2009 Solutions to Contest Tasks 37
38 Overall Strategy XMAS Compute the distances between each pair of food items and between S and B and the food items using a shortest path algorithm Martin Henz NOI 2009 Solutions to Contest Tasks 38
39 Overall Strategy Compute the distances between each pair of food items and between S and B and the food items using a shortest path algorithm Try all possible combinations starting with S, going through all F, and ending in S and compute the travel distance Martin Henz NOI 2009 Solutions to Contest Tasks 39
40 Overall Strategy Compute the distances between each pair of food items and between S and B and the food items using a shortest path algorithm Try all possible combinations starting with S, going through all F, and ending in S and compute the travel distance Remember the shortest distance obtained so far, and at the end, output that number Martin Henz NOI 2009 Solutions to Contest Tasks 40
41 : Cryptography Some History Breaking a Symmetric Key Cipher Alice sends a message m to Bob. The message may be intercepted by Eve along the way. Martin Henz NOI 2009 Solutions to Contest Tasks 41
42 : Cryptography Some History Breaking a Symmetric Key Cipher Alice sends a message m to Bob. The message may be intercepted by Eve along the way. Solution Alice encrypts m using a key k that is only known to Bob. Martin Henz NOI 2009 Solutions to Contest Tasks 42
43 Some History Breaking a Symmetric Key Cipher : Cryptography Alice sends a message m to Bob. The message may be intercepted by Eve along the way. Solution Alice encrypts m using a key k that is only known to Bob. Symmetric key cipher c = E(m, k) m = D(c, k) Martin Henz NOI 2009 Solutions to Contest Tasks 43
44 Some History Breaking a Symmetric Key Cipher Engima Cryptography in WW-I and WW-II German encryption machine The most common encryption technique in WW-II employed by the German military was based on an encryption machine called Enigma. Martin Henz NOI 2009 Solutions to Contest Tasks 44
45 Some History Breaking a Symmetric Key Cipher Breaking Enigma Weaknesses Enigma had cryptographic weaknesses, but ultimately the following factors led to the breaking of codes: procedural flaws, operator mistakes, occasional captured machines and key tables Martin Henz NOI 2009 Solutions to Contest Tasks 45
46 Some History Breaking a Symmetric Key Cipher Breaking Enigma Weaknesses Enigma had cryptographic weaknesses, but ultimately the following factors led to the breaking of codes: procedural flaws, operator mistakes, occasional captured machines and key tables Guessing messages Keine besonderen Vorkommnisse (in English: No particular events ) Martin Henz NOI 2009 Solutions to Contest Tasks 46
47 Some History Breaking a Symmetric Key Cipher Alan Turing Turing s role in breaking German ciphers Alan Turing worked at Bletchley Park, Britain s codebreaking centre. Martin Henz NOI 2009 Solutions to Contest Tasks 47
48 Some History Breaking a Symmetric Key Cipher Alan Turing Turing s role in breaking German ciphers Alan Turing worked at Bletchley Park, Britain s codebreaking centre. Bombe Turing devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine that could find settings for the Enigma machine. Martin Henz NOI 2009 Solutions to Contest Tasks 48
49 Some History Breaking a Symmetric Key Cipher Alan Turing Turing s role in breaking German ciphers Alan Turing worked at Bletchley Park, Britain s codebreaking centre. Bombe Turing devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine that could find settings for the Enigma machine. Father of computer science Turing provided an influential formalisation of the concept of the algorithm and computation with the Turing machine. Martin Henz NOI 2009 Solutions to Contest Tasks 49
50 Some History Breaking a Symmetric Key Cipher Breaking a Symmetric Key Cipher Symmetric key cipher c = E(m, k) m = D(c, k) Martin Henz NOI 2009 Solutions to Contest Tasks 50
51 Some History Breaking a Symmetric Key Cipher Breaking a Symmetric Key Cipher Symmetric key cipher c = E(m, k) m = D(c, k) Attacker stituation Eve gets hold of c, m, E and D, and wants to find k. Martin Henz NOI 2009 Solutions to Contest Tasks 51
52 Some History Breaking a Symmetric Key Cipher Breaking a Symmetric Key Cipher Symmetric key cipher c = E(m, k) m = D(c, k) Attacker stituation Eve gets hold of c, m, E and D, and wants to find k. Strategy Try all possible keys k i and test: c = E(m, k i )? Martin Henz NOI 2009 Solutions to Contest Tasks 52
53 Some History Breaking a Symmetric Key Cipher Double Encoding Idea c = E(E(m, k 1 ), k 2 ) m = D(D(c, k 2 ), k 1 ) Martin Henz NOI 2009 Solutions to Contest Tasks 53
54 Some History Breaking a Symmetric Key Cipher Double Encoding Idea c = E(E(m, k 1 ), k 2 ) m = D(D(c, k 2 ), k 1 ) Attacker stituation Eve gets hold of c, m, E and D, and wants to find k 1 and k 2. Martin Henz NOI 2009 Solutions to Contest Tasks 54
55 Some History Breaking a Symmetric Key Cipher Double Encoding Idea c = E(E(m, k 1 ), k 2 ) m = D(D(c, k 2 ), k 1 ) Attacker stituation Eve gets hold of c, m, E and D, and wants to find k 1 and k 2. Naive strategy Try all possible keys k1 i and kj 2, an compute: c = E(E(m, k1 i ), kj 2 ) Martin Henz NOI 2009 Solutions to Contest Tasks 55
56 Better Approach XMAS Some History Breaking a Symmetric Key Cipher Idea Work forward and backward, and check if results match Martin Henz NOI 2009 Solutions to Contest Tasks 56
57 Better Approach XMAS Some History Breaking a Symmetric Key Cipher Idea Work forward and backward, and check if results match Implementation Try all possible keys k1 i x = E(m, k1 i ) and x = D(c, k2 i ) and compute: Martin Henz NOI 2009 Solutions to Contest Tasks 57
58 Better Approach XMAS Some History Breaking a Symmetric Key Cipher Idea Work forward and backward, and check if results match Implementation Try all possible keys k1 i x = E(m, k1 i ) and x = D(c, k2 i ) and compute: Hash table Use a hash table to remember all x values, and for each x value, look in the hash table for a match. Martin Henz NOI 2009 Solutions to Contest Tasks 58
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