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long_comps_topcoder  Problem Statement
Contest: Marathon Match 44
Problem: LaggedGenerator

Problem Statement

    In this problem your task is to break a pseudorandom number generator given its last K outputs. Each `random' number will be generated as a function of the previous M `random' numbers, for some M. After the function is evaluated, the result is taken modulo a prime P, and that is the output. For instance, a simple generator function would be Sn = Sn-1+Sn-2 (mod P). A more complicated one is Sn = Sn-11*Sn-2 + Sn-1 + 1 (mod P). In general we will consider polynomials functions of the last M numbers. Your task is to write a method crack that takes a int[] S, a int, M, an int P, and returns your best guess for the next M numbers output by the generator.



The prime, P will be between 101 and 40000, inclusive. M will be chosen as floor(rand()*rand()*996)+5, where rand() is a uniform random number in [0,1]. The polynomial will be generated by first picking D, the maximum degree as 1, 2, or 3. The number of terms will then be chosen in [2,min(200,1+M+(D>1?M(M+1)/2)+(D>2?M(M+1)(M+2)/6)], and each term will be generated with a degree in [0,D]. After the degree d is chosen, d (not necessarily distinct) previous random numbers (offsets) are selected as the components of the term. If this generates a term which has already been generated, the process is repeated (so, degree 0 will only occur once, for instance). After the variables are chosen, the coefficient is chosen in [0,P-1]. You will be given the K sequential random numbers for K in [M,min(1000000,3*MD)]. The random number generator will be initialized with M random values, and then run for K+100*M iterations, of which you will be given the final K. All random choices are uniform and independent.



Your score will be the number of the M entries that are correct. Your overall score will simply be the sum of your individual scores.



You can download the source code and class files to generate a sequence of random numbers. To run, simply enter 'java LaggedGenerator <seed>'. The output will be the M initial values, followed by an asterisk, followed by 100M random values, followed by an asterisk, followed by K random values, followed by an asterisk, and finally the M values to predict. The seeds for the examples are 1-9 and 18, in order.
 

Definition

    
Class:LaggedGenerator
Method:crack
Parameters:int[], int, int
Returns:int[]
Method signature:int[] crack(int[] S, int P, int M)
(be sure your method is public)
    
 

Notes

-The time limit is 30 seconds.
-The memory limit is 1024M.
-In the examples, N is the number of terms in the polynomial.
-There are 100 non-example test cases.
 

Examples

0)
    
P = 101

N = 4

D = 2

M = 5

K = 50

1)
    
P = 36299

N = 7

D = 1

M = 16

K = 44

2)
    
P = 23789

N = 177

D = 3

M = 102

K = 668069

3)
    
P = 13687

N = 62

D = 2

M = 155

K = 4147

4)
    
P = 28403

N = 26

D = 1

M = 25

K = 40

5)
    
P = 17257

N = 19

D = 3

M = 611

K = 682883

6)
    
P = 21481

N = 51

D = 1

M = 115

K = 193

7)
    
P = 24281

N = 180

D = 2

M = 211

K = 125395

8)
    
P = 27847

N = 127

D = 1

M = 565

K = 928

9)
    
P = 39317

N = 169

D = 2

M = 26

K = 1485


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