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:

 #include <stdio.h>
#include <algorithm>
using namespace std; int main()
{
long long n,p;
long long data[];
while(scanf("%lld%lld",&n,&p) != EOF){
for(int i = ; i < n; ++i){
scanf("%lld",&data[i]);
}
sort(data,data+n);
int result = ;
int count = ;
int i = , j = ;
long long sum;
while(i < n){
sum = data[j] * p;
while(i < n && data[i] <= sum){
++count;
++i;
}
if(count > result)
result = count;
++j;
--count; }
printf("%d\n",result);
}
return ;
}

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