求最长递增子序列的O(NLog(N))方法 让我们维护一个数组，其中第i个元素是一个大小为i的子序列可以结束的最小的数字。

我故意避免进一步的细节，因为投票最多的答案已经解释了它，但这种技术最终导致使用set数据结构的整洁实现(至少在c++中)。

下面是c++中的实现(假设需要严格增加最长子序列的大小)

#include <bits/stdc++.h> // gcc supported header to include (almost) everything
using namespace std;
typedef long long ll;

int main()
{
  ll n;
  cin >> n;
  ll arr[n];
  set<ll> S;

  for(ll i=0; i<n; i++)
  {
    cin >> arr[i];
    auto it = S.lower_bound(arr[i]);
    if(it != S.end())
      S.erase(it);
    S.insert(arr[i]);
  }

  cout << S.size() << endl; // Size of the set is the required answer

  return 0;
}

这是我的Leetcode解决方案使用二进制搜索:->

class Solution:
    def binary_search(self,s,x):
        low=0
        high=len(s)-1
        flag=1
        while low<=high:
              mid=(high+low)//2
              if s[mid]==x:
                 flag=0
                 break
              elif s[mid]<x:
                  low=mid+1
              else:
                 high=mid-1
        if flag:
           s[low]=x
        return s

    def lengthOfLIS(self, nums: List[int]) -> int:
         if not nums:
            return 0
         s=[]
         s.append(nums[0])
         for i in range(1,len(nums)):
             if s[-1]<nums[i]:
                s.append(nums[i])
             else:
                 s=self.binary_search(s,nums[i])
         return len(s)

下面是从动态规划的角度评估问题的三个步骤:

递归定义:maxLength(i) == 1 + maxLength(j) where 0 < j < i and array[i] > array[j] 递归参数边界:可能有0到i - 1个子序列作为参数传递 求值顺序:由于是递增子序列，所以要从0求值到n

如果我们以序列{0,8,2,3,7,9}为例，at index:

我们会得到子序列{0}作为基本情况 [1]有一个新的子序列{0,8} [2]试图评估两个新的序列{0,8,2}和{0,2}通过添加元素在索引2到现有的子序列-只有一个是有效的，所以添加第三个可能的序列{0,2}只到参数列表 .．.

下面是c++ 11的工作代码:

#include <iostream>
#include <vector>

int getLongestIncSub(const std::vector<int> &sequence, size_t index, std::vector<std::vector<int>> &sub) {
    if(index == 0) {
        sub.push_back(std::vector<int>{sequence[0]});
        return 1;
    }

    size_t longestSubSeq = getLongestIncSub(sequence, index - 1, sub);
    std::vector<std::vector<int>> tmpSubSeq;
    for(std::vector<int> &subSeq : sub) {
        if(subSeq[subSeq.size() - 1] < sequence[index]) {
            std::vector<int> newSeq(subSeq);
            newSeq.push_back(sequence[index]);
            longestSubSeq = std::max(longestSubSeq, newSeq.size());
            tmpSubSeq.push_back(newSeq);
        }
    }
    std::copy(tmpSubSeq.begin(), tmpSubSeq.end(),
              std::back_insert_iterator<std::vector<std::vector<int>>>(sub));

    return longestSubSeq;
}

int getLongestIncSub(const std::vector<int> &sequence) {
    std::vector<std::vector<int>> sub;
    return getLongestIncSub(sequence, sequence.size() - 1, sub);
}

int main()
{
    std::vector<int> seq{0, 8, 2, 3, 7, 9};
    std::cout << getLongestIncSub(seq);
    return 0;
}

Petar Minchev的解释帮助我理清了事情，但我很难解析所有内容，所以我做了一个带有过度描述性变量名和大量注释的Python实现。我做了一个简单的递归解，O(n²)解，和O(n log n)解。

我希望它能帮助理清算法!

递归解决方案

def recursive_solution(remaining_sequence, bigger_than=None):
    """Finds the longest increasing subsequence of remaining_sequence that is      
    bigger than bigger_than and returns it.  This solution is O(2^n)."""

    # Base case: nothing is remaining.                                             
    if len(remaining_sequence) == 0:
        return remaining_sequence

    # Recursive case 1: exclude the current element and process the remaining.     
    best_sequence = recursive_solution(remaining_sequence[1:], bigger_than)

    # Recursive case 2: include the current element if it's big enough.            
    first = remaining_sequence[0]

    if (first > bigger_than) or (bigger_than is None):

        sequence_with = [first] + recursive_solution(remaining_sequence[1:], first)

        # Choose whichever of case 1 and case 2 were longer.                         
        if len(sequence_with) >= len(best_sequence):
            best_sequence = sequence_with

    return best_sequence                                                        

O(n²)动态规划解

def dynamic_programming_solution(sequence):
    """Finds the longest increasing subsequence in sequence using dynamic          
    programming.  This solution is O(n^2)."""

    longest_subsequence_ending_with = []
    backreference_for_subsequence_ending_with = []
    current_best_end = 0

    for curr_elem in range(len(sequence)):
        # It's always possible to have a subsequence of length 1.                    
        longest_subsequence_ending_with.append(1)

        # If a subsequence is length 1, it doesn't have a backreference.             
        backreference_for_subsequence_ending_with.append(None)

        for prev_elem in range(curr_elem):
            subsequence_length_through_prev = (longest_subsequence_ending_with[prev_elem] + 1)

            # If the prev_elem is smaller than the current elem (so it's increasing)   
            # And if the longest subsequence from prev_elem would yield a better       
            # subsequence for curr_elem.                                               
            if ((sequence[prev_elem] < sequence[curr_elem]) and
                    (subsequence_length_through_prev >
                         longest_subsequence_ending_with[curr_elem])):

                # Set the candidate best subsequence at curr_elem to go through prev.    
                longest_subsequence_ending_with[curr_elem] = (subsequence_length_through_prev)
                backreference_for_subsequence_ending_with[curr_elem] = prev_elem
                # If the new end is the best, update the best.    

        if (longest_subsequence_ending_with[curr_elem] >
                longest_subsequence_ending_with[current_best_end]):
            current_best_end = curr_elem
            # Output the overall best by following the backreferences.  

    best_subsequence = []
    current_backreference = current_best_end

    while current_backreference is not None:
        best_subsequence.append(sequence[current_backreference])
        current_backreference = (backreference_for_subsequence_ending_with[current_backreference])

    best_subsequence.reverse()

    return best_subsequence                                                   

O(n log n)动态规划解

def find_smallest_elem_as_big_as(sequence, subsequence, elem):
    """Returns the index of the smallest element in subsequence as big as          
    sequence[elem].  sequence[elem] must not be larger than every element in       
    subsequence.  The elements in subsequence are indices in sequence.  Uses       
    binary search."""

    low = 0
    high = len(subsequence) - 1

    while high > low:
        mid = (high + low) / 2
        # If the current element is not as big as elem, throw out the low half of    
        # sequence.                                                                  
        if sequence[subsequence[mid]] < sequence[elem]:
            low = mid + 1
            # If the current element is as big as elem, throw out everything bigger, but 
        # keep the current element.                                                  
        else:
            high = mid

    return high


def optimized_dynamic_programming_solution(sequence):
    """Finds the longest increasing subsequence in sequence using dynamic          
    programming and binary search (per                                             
    http://en.wikipedia.org/wiki/Longest_increasing_subsequence).  This solution   
    is O(n log n)."""

    # Both of these lists hold the indices of elements in sequence and not the        
    # elements themselves.                                                         
    # This list will always be sorted.                                             
    smallest_end_to_subsequence_of_length = []

    # This array goes along with sequence (not                                     
    # smallest_end_to_subsequence_of_length).  Following the corresponding element 
    # in this array repeatedly will generate the desired subsequence.              
    parent = [None for _ in sequence]

    for elem in range(len(sequence)):
        # We're iterating through sequence in order, so if elem is bigger than the   
        # end of longest current subsequence, we have a new longest increasing          
        # subsequence.                                                               
        if (len(smallest_end_to_subsequence_of_length) == 0 or
                    sequence[elem] > sequence[smallest_end_to_subsequence_of_length[-1]]):
            # If we are adding the first element, it has no parent.  Otherwise, we        
            # need to update the parent to be the previous biggest element.            
            if len(smallest_end_to_subsequence_of_length) > 0:
                parent[elem] = smallest_end_to_subsequence_of_length[-1]
            smallest_end_to_subsequence_of_length.append(elem)
        else:
            # If we can't make a longer subsequence, we might be able to make a        
            # subsequence of equal size to one of our earlier subsequences with a         
            # smaller ending number (which makes it easier to find a later number that 
            # is increasing).                                                          
            # Thus, we look for the smallest element in                                
            # smallest_end_to_subsequence_of_length that is at least as big as elem       
            # and replace it with elem.                                                
            # This preserves correctness because if there is a subsequence of length n 
            # that ends with a number smaller than elem, we could add elem on to the   
            # end of that subsequence to get a subsequence of length n+1.              
            location_to_replace = find_smallest_elem_as_big_as(sequence, smallest_end_to_subsequence_of_length, elem)
            smallest_end_to_subsequence_of_length[location_to_replace] = elem
            # If we're replacing the first element, we don't need to update its parent 
            # because a subsequence of length 1 has no parent.  Otherwise, its parent  
            # is the subsequence one shorter, which we just added onto.                
            if location_to_replace != 0:
                parent[elem] = (smallest_end_to_subsequence_of_length[location_to_replace - 1])

    # Generate the longest increasing subsequence by backtracking through parent.  
    curr_parent = smallest_end_to_subsequence_of_length[-1]
    longest_increasing_subsequence = []

    while curr_parent is not None:
        longest_increasing_subsequence.append(sequence[curr_parent])
        curr_parent = parent[curr_parent]

    longest_increasing_subsequence.reverse()

    return longest_increasing_subsequence         

最长递增子序列(Java)

import java.util.*;

class ChainHighestValue implements Comparable<ChainHighestValue>{
    int highestValue;
    int chainLength;
    ChainHighestValue(int highestValue,int chainLength) {
        this.highestValue = highestValue;
        this.chainLength = chainLength;
    }
    @Override
    public int compareTo(ChainHighestValue o) {
       return this.chainLength-o.chainLength;
    }

}


public class LongestIncreasingSubsequenceLinkedList {


    private static LinkedList<Integer> LongestSubsequent(int arr[], int size){
        ArrayList<LinkedList<Integer>> seqList=new ArrayList<>();
        ArrayList<ChainHighestValue> valuePairs=new ArrayList<>();
        for(int i=0;i<size;i++){
            int currValue=arr[i];
            if(valuePairs.size()==0){
                LinkedList<Integer> aList=new LinkedList<>();
                aList.add(arr[i]);
                seqList.add(aList);
                valuePairs.add(new ChainHighestValue(arr[i],1));

            }else{
                try{
                    ChainHighestValue heighestIndex=valuePairs.stream().filter(e->e.highestValue<currValue).max(ChainHighestValue::compareTo).get();
                    int index=valuePairs.indexOf(heighestIndex);
                    seqList.get(index).add(arr[i]);
                    heighestIndex.highestValue=arr[i];
                    heighestIndex.chainLength+=1;

                }catch (Exception e){
                    LinkedList<Integer> aList=new LinkedList<>();
                    aList.add(arr[i]);
                    seqList.add(aList);
                    valuePairs.add(new ChainHighestValue(arr[i],1));
                }
            }
        }
        ChainHighestValue heighestIndex=valuePairs.stream().max(ChainHighestValue::compareTo).get();
        int index=valuePairs.indexOf(heighestIndex);
        return seqList.get(index);
    }

    public static void main(String[] args){
        int arry[]={5,1,3,6,11,30,32,5,3,73,79};
        //int arryB[]={3,1,5,2,6,4,9};
        LinkedList<Integer> LIS=LongestSubsequent(arry, arry.length);
        System.out.println("Longest Incrementing Subsequence:");
        for(Integer a: LIS){
            System.out.print(a+" ");
        }

    }
}

下面的c++实现还包括一些使用名为prev的数组构建实际最长递增子序列的代码。

std::vector<int> longest_increasing_subsequence (const std::vector<int>& s)
{
    int best_end = 0;
    int sz = s.size();

    if (!sz)
        return std::vector<int>();

    std::vector<int> prev(sz,-1);
    std::vector<int> memo(sz, 0);

    int max_length = std::numeric_limits<int>::min();

    memo[0] = 1;

    for ( auto i = 1; i < sz; ++i)
    {
        for ( auto j = 0; j < i; ++j)
        {
            if ( s[j] < s[i] && memo[i] < memo[j] + 1 )
            {
                memo[i] =  memo[j] + 1;
                prev[i] =  j;
            }
        }

        if ( memo[i] > max_length ) 
        {
            best_end = i;
            max_length = memo[i];
        }
    }

    // Code that builds the longest increasing subsequence using "prev"
    std::vector<int> results;
    results.reserve(sz);

    std::stack<int> stk;
    int current = best_end;

    while (current != -1)
    {
        stk.push(s[current]);
        current = prev[current];
    }

    while (!stk.empty())
    {
        results.push_back(stk.top());
        stk.pop();
    }

    return results;
}

没有堆栈的实现只是反转向量

#include <iostream>
#include <vector>
#include <limits>
std::vector<int> LIS( const std::vector<int> &v ) {
  auto sz = v.size();
  if(!sz)
    return v;
  std::vector<int> memo(sz, 0);
  std::vector<int> prev(sz, -1);
  memo[0] = 1;
  int best_end = 0;
  int max_length = std::numeric_limits<int>::min();
  for (auto i = 1; i < sz; ++i) {
    for ( auto j = 0; j < i ; ++j) {
      if (s[j] < s[i] && memo[i] < memo[j] + 1) {
        memo[i] = memo[j] + 1;
        prev[i] = j;
      }
    }
    if(memo[i] > max_length) {
      best_end = i;
      max_length = memo[i];
    }
  }

  // create results
  std::vector<int> results;
  results.reserve(v.size());
  auto current = best_end;
  while (current != -1) {
    results.push_back(s[current]);
    current = prev[current];
  }
  std::reverse(results.begin(), results.end());
  return results;
}