C#连续区间最大和求解

问题概述 

最大连续区间和是一个经典的问题。给定一个长度为n的序列a[1],a[2]...a[n-1],a[n]，求一个连续的子序列a[i],a[i+1]...a[j-1],a[j]，使得a[i]+a[i+1]...a[j-1]+a[j]最大。


①最简单最容易想到的就是根据定义来枚举。
枚举上下界{i,j | 0<=i<=j<=n}，维护一个max值即可。
其中枚举上下界的时间复杂度为O(n^2)，求区间和的复杂度为O(n)，所以总时间复杂度为O(n^3)。

 算法一

 public int getmax(int[] a, ref int begin, ref int end)
        {
            int max=0;
            for (int i = 1; i < a.Length; i++)
            {
                int sum = 0;
                for (int j = i; j < a.Length; j++)
                {
                    sum += a[j];
                    if (sum > max)
                    {
                        begin = i;
                        end = j;
                        max = sum;
                    }
                }
            }
            return max;
        }
        public int getmin(int[] a,ref int begin,ref int end)
        {
            int min = 0;
            for (int i = 1; i < a.Length; i++)
            {
                int sum = 0;
                for (int j = i; j < a.Length; j++)
                {
                    sum += a[j];
                    if (sum < min)
                    {
                        begin = i;
                        end = j;
                        min = sum;
                    }
                }
            }
            return min;
        }

算法二

第二个时间复杂度为 n 利用了动态规划的思维

 

public int max(int[] a, ref int begin, ref int end)
        {
            int sum = 0;
            int max = 0;
            int[] bit = new int[a.Length];
            for (int i = 1; i < a.Length; i++)
            {
                if (sum <=0)
                {
                    sum = a[i];
                    bit[i] = -1;
                }
                else
                {
                    sum += a[i];
                    if(sum==0)
                        bit[i] = -1;
                }
                if (sum > max)
                {
                    bit[i] = 1;
                    max = sum;
                    end = i;
                }
            }
            bool ff = false;
            for (int i = bit.Length - 1; i >= 0; i--)
            {
                if (ff && bit[i] == -1)
                {
                    begin = i;
                    break;
                }
                if (bit[i] == 1)
                {
                    ff = true;
                }
            }
            return max;
        }
        public int min(int[] a, ref int begin, ref int end)
        {
            int sum = 0;
            int min = 0;
            int[] bit=new int[a.Length];
            for (int i = 1; i < a.Length; i++)
            {
                if (sum >=0)
                {
                    sum = a[i];
                    bit[i] = -1;
                }
                else
                {
                    sum += a[i];
                    if(sum==0)
                        bit[i] = -1;
                }
               
                if (sum < min)
                {
                    bit[i] = 1;
                    min = sum;
                    end = i;
                }
               
            }           bool ff = false;
            for (int i = bit.Length-1; i >= 0; i--)
            {
                if (ff && bit[i] == -1)
                {
                    begin = i;
                    break;
                }
                if (bit[i] == 1)
                {
                    ff = true;
                }
            }           return min;
        }