SRM-697-DIV2

Div2 Medium: DivisibleSetDiv2

Problem Statement
     You are given a vector <int> b containing a sequence of n positive integers: b[0], ..., b[n-1]. We are now looking for another sequence a[0], ..., a[n-1]. This sequence should have the following properties:

  • Each a[i] should be a number of the form 2^x[i] where x[i] is some positive integer. In other words, each a[i] is one of the numbers 2, 4, 8, 16, ...
  • For each i, the value a[i]^b[i] (that is, a[i] to the power b[i]) should be divisible by P, where P is the product of all a[i].

Determine whether there is at least one sequence with the desired properties. Return "Possible" (quotes for clarity) if such a sequence exists and "Impossible" otherwise.

Definition
    
Class: DivisibleSetDiv2
Method: isPossible
Parameters: vector <int>
Returns: string
Method signature: string isPossible(vector <int> b)
(be sure your method is public)

Limits
    
Time limit (s): 2.000
Memory limit (MB): 256
Stack limit (MB): 256

Constraints
- b will contain between 1 and 50 elements, inclusive.
- Each element in b will be between 1 and 10, inclusive.

Examples
0)  
     
{3,2}
 
Returns: "Possible"
One valid sequence is the sequence {2, 2}. That is, a[0] = a[1] = 2. Clearly, each a[i] is a power of two not smaller than 2. The product of all a[i] is 2*2 = 4. Both a[0]^b[0] = 2^3 = 8 and a[1]^b[1] = 2^2 = 4 are divisible by 4.
1)  
     
{3,3,3}
 
Returns: "Possible"
Here, one valid sequence is {2, 2, 2}.
2)  
     
{1,10}
 
Returns: "Impossible"
Suppose that a[0] = x and a[1] = y. The value a[0]^b[0] = x^1 should be divisible by x*y. This is only possible for y = 1. However, 1 is not a positive power of two, so we cannot have a[1] = 1.
3)  
     
{2, 3, 10}
 
Returns: "Possible"
One valid sequence is {8, 4, 2}.
4)  
     
{7,10,4,6,3}
 
Returns: "Possible"
 
5)  
     
{9,9,9,9,9,9,9,9,9}
 
Returns: "Possible"
 
6)  
     
{3,4,5,6,7}
 
Returns: "Impossible"
 

　　This is a mathematical problem which is very simple to implement but hard to prove. It turns out that the answer is "Possible" if and only if ∑i=0n−11bi≤1∑i=0n-11bi≤1. You can find a detailed proof below.

　　We are asked if there exists a sequence of powers of two (a0,a1,...,an−1)(a0,a1,...,an-1) that for every i:

　　Finally, we should avoid losing accuracy in order to use the perfect formula. Casuing the range of bi is [1, 10], we can multiply a constant C which equal to LCM(1,...,10).

public string isPossible(vector<int> b) {
  int LCM = 2520, sum = 0;
  for (int bi : b)
    sum += LCM / bi; // sum up 1/b multiplied by LCM to avoid floats
  return (sum <= LCM) ? "Possible" : "Impossible";
}

　　Recently, I had met two mathematical problems. The last one is hihoCoder_5 using log to get the maximum value.

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时间： 2024-08-29 23:07:20

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