A report on the problem of sword-pointing offer (Java edition)--Joseph Ring 45

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Introduction

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Common Joseph Ring problem useful loop-linked list do, useful array to do, here to provide a mathematical formula to do, thus, many computer problems if the final use of mathematical knowledge, time complexity will be greatly reduced

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Analyze problems

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First of all we on the 0 to n1 Delete the first number for analysis, the first deleted number must be the number of m-1, because No. 0 number 1, 1th number of 2,m-1 number of M, then should be deleted m-1 number, set the m-1 number is the K number, This is done because the back can be expanded to extend to m-1%n=k

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The remaining numbers are reordered by sequence as follows

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Because the deletion is K, then the first one is K+ 1, this is very good understanding

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Suppose we set the output of the F function to be the original number in n , the number of m per time, and the last left

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And now delete K, only n1 numbers, each time a few m, the last number left is F ' (n-1,m)

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How to establish the relationship between F ' and F. ' ?

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We do a mapping of the numbers after reordering

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The mapping is

P (x)--> Y::: y= (x-k-1)%n

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Then the mapping is the same as the original problem, that is F(n-1,m)

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i.e. P(f ' (n-1,m) =f (n-1,m)

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And now we only have a relationship between F(n,m) and F ' (N-1,M)

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So we need to switch again, and we'll find the inverse of the map.

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and the inverse map of the map p is

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O (Y)-->x x= (y+k+1)%n

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Where K can replace k%n= (m-1)%n

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i.e. o (y)-->x x= (y+m)%n

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Corresponds to the relationship between the previous F (n-1,m) and F ' (n-1,m)

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F ' (n-1,m) =f (n-1,m) +m%n

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and

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F (n,m) =f ' (n-1,m)

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So we have the relationship between F (n,m) and F ' (n-1,m) , and also the relationship between F (n1, M) and F ' (n-1,m)

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The pre-loaded formula is pushed to get

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Solve the problem

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static int Lastremain (int n, int m) {

int last = 0;

if (N < 1 | | m < 1) {

return-1;

}

for (int i = 2; I <= n; i++)

Last = (last + m)% i;

return last;

}

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A report on the problem of sword-pointing offer (Java edition)--Joseph Ring 45