PAT 1085 Perfect Sequence

PAT 1085 Perfect Sequence

题目：

Given a sequence of positive integers and another positive integer p. The sequence is said to be a "perfect sequence" if M <= m * p where M and m are the maximum and minimum numbers in the sequence, respectively.

Now given a sequence and a parameter p, you are supposed to find from the sequence as many numbers as possible to form a perfect subsequence.

Input Specification:

Each input file contains one test case. For each case, the first line contains two positive integers N and p, where N (<= 10⁵) is the number of integers in the sequence, and p (<= 10⁹) is the parameter. In the second line there are N positive integers, each is no greater than 10⁹.

Output Specification:

For each test case, print in one line the maximum number of integers that can be chosen to form a perfect subsequence.

Sample Input:

10 8
2 3 20 4 5 1 6 7 8 9

Sample Output:

8

地址：http://pat.zju.edu.cn/contests/pat-a-practise/1085

    注意题意，说的是从数组中取任意多个数字满足完美子序列，所以对数字的要求没有顺序性。我的方法是，先对数组做一个排序，时间为O(nlogn)，然后在用线性的时间找到这个序列，如果该题用O(n^2)的算法则会有一个case超时。现在来说说如何在线性时间找到这个序列：首先，用下标i表示当前找到的最大值，用下标j表示当前找到的最小值。从j=0开始，i可以从0一直遍历到第一不满足data[0]*p >= data[i]，此时，说明序列0到i-1是满足完美子序列，也就是以data[0]为最小值的最大完美子序列，我们把其个数记为count；然后j加一，此时最小值变为data[1]，这是data[1]*p >= data[0]*p>=data[i]，也就是说原来的count个也都满足，但由于data[0]比data[1]要小，所以不满足data[1]为最小值，故count减一，然后在从当前的i开始往后继续找，找到第一个不满足data[1]*p>=data[i]，此时的count为以data[1]为最小值的最大完美子序列的个数，然后j继续加一，count减一，i继续向前遍历，如此循环直到j走到底。由于i,j在遍历时都没有回头，故时间复杂度为线性的。代码：

 1 #include <stdio.h>
 2 #include <algorithm>
 3 using namespace std;  4 
 5 int main()  6 {  7     long long n,p;  8     long long data[100000];  9     while(scanf("%lld%lld",&n,&p) != EOF){ 10         for(int i = 0; i < n; ++i){ 11             scanf("%lld",&data[i]); 12  } 13         sort(data,data+n); 14         int result = 0; 15         int count = 0; 16         int i = 0, j = 0; 17         long long sum; 18         while(i < n){ 19             sum = data[j] * p; 20             while(i < n && data[i] <= sum){ 21                 ++count; 22                 ++i; 23  } 24             if(count > result) 25               result = count; 26             ++j; 27             --count; 28             
29  } 30         printf("%d\n",result); 31  } 32     return 0; 33 }