• 300. Longest Increasing Subsequence(LIS最长递增子序列 动态规划)


    Given an unsorted array of integers, find the length of longest increasing subsequence.

    For example,
    Given [10, 9, 2, 5, 3, 7, 101, 18],
    The longest increasing subsequence is [2, 3, 7, 101], therefore the length is 4. Note that there may be more than one LIS combination, it is only necessary for you to return the length.

    Your algorithm should run in O(n2) complexity.

    Follow up: Could you improve it to O(n log n) time complexity?

    长度为N的数组记为A={aaa2...an-1}
    A的前i个字符构成的前缀串为Ai= a1a2...ai-1,以ai结尾的最长递增
    子序列记做Li,其长度记为a[i]
    假定已经计算得到了a[0,1…,i-1],如何计算a[i]呢?
    根据定义, Li必须以ai结尾,如果将ai缀到L0 L1…… Li-1的后面,是否
    允许呢?
    如果ajai,则可以将ai缀到Lj的后面,并且使得Lj的长度变长。
     从而:a[i]={max(a(j))+1,  0 j≤i-1a[j]a[i] } 
     需要遍历在i之前的所有位置j,找出满足条件a[j]a[i]a[j]
     计算得到a[0…n-1]后,遍历所有的a[i],找出最大值即为最大递增子序列
    的长度。
     时间复杂度为O(N2)
     思考:如何求最大递增子序列本身?
     记录前驱 


     1 class Solution {
     2     public int lengthOfLIS(int[] a) {
     3         if(a.length==0) return 0;
     4         int[] longs = new int[a.length];
     5         for(int i = 0;i<a.length;i++)
     6             longs[i] = 1;
     7         int max = longs[0];
     8         for(int i = 1;i < a.length;i++){
     9             for(int j = 0;j <= i-1;j++)
    10                 if(a[j]<a[i])
    11                     if(longs[i]<longs[j]+1)
    12                         longs[i] = longs[j]+1;
    13                         //如果求序列本身,在这里记录前驱
    14                 
    15             if(max<longs[i])
    16                 max = longs[i];       
    17         }
    18         return max;
    19     }

     

     1 class Solution {
     2 public:
     3     int lengthOfLIS(vector<int>& a) {
     4         if (a.size()==0) return 0;
     5         int max = 1;
     6         vector<int> dp(a.size(),1);
     7         for (int i = 1 ;i < a.size(); i++) {
     8             for (int j = 0; j < i ;j++) {
     9                 if (a[j] < a[i]) {
    10                 dp[i] = std::max(dp[j]+1,dp[i]);
    11                 }
    12             }
    13             max = std::max(dp[i],max);
    14         }
    15 
    16         return max;
    17     }
    18 };
     1 class Solution(object):
     2     def lengthOfLIS(self, a):
     3         n = len(a)
     4         if n ==0:
     5             return 0
     6         dp = [1] * n
     7         for i in range(n):
     8             for j in range(i):
     9                 if(a[i]>a[j] and dp[i]<dp[j]+1):
    10                     dp[i] = dp[j]+1
    11         
    12         #dp[i]现在存储的即为以a[i]结尾最长递增子序列长度
    13         #求dp数组中最大者即为最长的长度
    14         
    15         return max(dp)
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  • 原文地址:https://www.cnblogs.com/zle1992/p/8532558.html
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