标签：dynamic programming

求最长增加子数列是一道经典算法题，其动态规划解法最简单，但是时间复杂度是O(n²)。stackoverflow上有一个O(nlogn)解法，很是巧妙：

Now the improvement happens at the second loop, basically, you can improve the speed by using binary search. Besides the array dp[], let’s have another array c[], c is pretty special, c[i] means: the minimum value of the last element of the longest increasing sequence whose length is i.

sz = 1;
c[1] = array[0]; /*at this point, the minimum value of the last element of the size 1 increasing sequence must be array[0]*/
dp[0] = 1;
for( int i = 1; i < len; i++ ) {
   if( array[i] < c[1] ) {
  c[1] = array[i]; /*you have to update the minimum value right now*/
  dp[i] = 1;
   }
   else if( array[i] > c[sz] ) {
  c[sz+1] = array[i];
  dp[i] = sz+1;
  sz++;
   }
   else {
  int k = binary_search( c, sz, array[i] ); /*you want to find k so that c[k-1]<array[i]<c[k]*/
  c[k] = array[i];
  dp[i] = k;
   }
}

关键是理解这个c数组，其下标用来表示增加子数列的长度，而值则是这个长度的增加子数列中结尾元素最小的值。比如有两个相同长度的增加子数列：1,2,3和1,2,5，则c[3]的值为3，因为一旦将来有4出现，就可以把1,2,3扩展为1,2,3,4，而无法再把1,2,5扩展。完整的Go程序如下：

package main
import (
    "fmt"
    "os"
)

func bSearch(array []int, start int, end int, value int) int {
    for start <= end {
        mid := start + (end-start)/2
        if array[mid] == value {
            return mid
        } else if array[mid] < value {
            if mid+1 <= end && array[mid+1] > value {
                return mid+1
            } else {
                start = mid+1
            }            
        } else {
            if mid-1 >= start && array[mid-1] < value {
                return mid
            } else {
                end = mid-1
            }
        }
    }
    return start
}
func main() {
 //Enter your code here. Read input from STDIN. Print output to STDOUT
    var num int
    _, err := fmt.Scan(&num)
    if err != nil || num == 0{
        os.Exit(1)
    }
    s := make([]int, num)
    m := make([]int, num)
    e := make([]int, num+1)

    for i := 0; i < num; i++ {
        _, err := fmt.Scan(&s[i])
        if err != nil {
            os.Exit(1)
        }
    }

    m[0] = 1
    max := m[0]
    e[1] = s[0]
    for i := 1; i < num; i++ {
        if s[i] < e[1] {
            e[1] = s[i]
            m[i] = 1
        } else if s[i] > e[max] {
            max++
            e[max] = s[i]
            m[i] = max
        } else {
            k := bSearch(e, 1, max, s[i])
            e[k] = s[i]
            m[i] = k
        }
    }

    fmt.Println(max)
}

求子数组的最大值是一道经典dynamic programming题，解法如下（参考这里）：

public int maxSubArray(int[] A) {
   int newsum=A[0];
   int max=A[0];
   for(int i=1;i<A.length;i++){
       newsum=Math.max(newsum+A[i],A[i]);
       max= Math.max(max, newsum);
   }
   return max;
}

理解这个算法的关键在于：求出数组中以每个元素为子数组的最后一个元素的最大值（上述代码中newsum），这些最大值中的最大者即为解（上述代码中max）。分析如下：从第一个元素A[0]起，newsum和max均为A[0]。而对下一个元素A[1]，以A[1]为子数组的最后一个元素的最大值或者是A[0]+A[1]（A[0]大于0），或是A[1]，取两者最大值。接下来再看A[2]，以A[2]为子数组的最后一个元素的最大值是A[0]+A[1]+A[2]，A[1]+A[2]和A[2]三者之间的最大值。以此类推。。。

这篇笔记摘自Tutorial for Dynamic Programming：

Dynamic programming (usually referred to as DP ) is a very powerful technique to solve a particular class of problems. It demands very elegant formulation of the approach and simple thinking and the coding part is very easy. The idea is very simple, If you have solved a problem with the given input, then save the result for future reference, so as to avoid solving the same problem again.. shortly ‘Remember your Past’ 🙂 . If the given problem can be broken up in to smaller sub-problems and these smaller subproblems are in turn divided in to still-smaller ones, and in this process, if you observe some over-lappping subproblems, then its a big hint for DP. Also, the optimal solutions to the subproblems contribute to the optimal solution of the given problem ( referred to as the Optimal Substructure Property ).

There are two ways of doing this.

1.) Top-Down : Start solving the given problem by breaking it down. If you see that the problem has been solved already, then just return the saved answer. If it has not been solved, solve it and save the answer. This is usually easy to think of and very intuitive. This is referred to as Memoization.

2.) Bottom-Up : Analyze the problem and see the order in which the sub-problems are solved and start solving from the trivial subproblem, up towards the given problem. In this process, it is guaranteed that the subproblems are solved before solving the problem. This is referred to as Dynamic Programming.

Memoization是一种“自顶向下”解决问题的方式，并且顾名思义，它具有保存结果的含义：它把一个问题细化成子问题，如果子问题已经解决，就直接获得结果，反正则解决子问题，并把结果保存起来。
Dynamic Programming则是“自下而上”的解决问题方式，它从处理子问题入手，在解决最终问题之前，保证它的所有子问题都已解决。