标签：first ati ddl sum different ref possibly diff prefix
We have a lot of ways to solve the maximum subsequence sum problem, but different ways take different time.
1、Brute-force algorithm
int maxSubSum1(const vector<int> &a)
{
int maxSum=0;
for(int i=0;i<a.size();i++)
for(int j=i;j<a.size();j++)
{
int sum=0;
for(int k=i;k<=j;k++)
sum+=a[k];
if(sum>maxSum)
maxSum=sum;
}
return maxSum;
}
/*The running time is O(n^3)
It takes too much time.
*/
2、a little imporvement
int maxSubSum2(const vector<int>& a )
{
int maxSum=0;
for(int i=0;i<a.size();i++)
{
int sum=0;
for(int j=i;j<a.size();j++)
{
sum+=a[j];
if(maxSum<sum)
{
maxSum=sum;
}
}
}
return maxSum;
}
3. Divide-conquer algorithm
We can divide this problem into three parts:
(1) First half;
(2) cross the middle parts;
(3) second part;
What we need to do is to find the max sum of the three part.
int max3(int a, int b, int c)
{
if(a>b)
{
if(a>c)return a;
else return c;
}
else
{
if(c>b)return c;
else return b;
}
}
int maxSubSum3(cosnt vector<int >& a, int left, int right)
{
if(left==right)
if(a[left]>0) return a[left];
else return 0;
int center= (left+right)/2;
int maxLeftSum=maxSumRec(a, left, center);
int maxRightSum=maxSumRec(a, center+1, right);
int maxLeftBoderSum=0, leftBoderSum=0;
for(int i=center;i>=left;i--)
{
leftBoderSum+=a[i];
if(leftBoderSum>maxLeftBoderSum)
maxLeftBoderSum=leftBoderSum;
}
int maxRightBoderSum=0, leftBoderSum=0;
for(int i=center+1;i<=right;i++)
{
rightBoderSum+=a[i];
if(rightBoderSum>maxRightBoderSum)
maxRightBoderSum=rightBoderSum;
}
return max3(maxLeftSum, maxLeftBoderSum+maxRightBoderSum,maxRightSum);
}
4. The best algorithm
If the start is negative, the sum of the subsequence can not be the max. Hence, any negative subsequence cannot possibly be a prefix of the optimal subsequence.
int maxSubSum4(const vector<int> & a)
{
int maxSum=0, sum=0;
for(int i=0;i<a.size();i++)
{
sum+=a[i];
if(sum>maxSum)
maxSum=sum;
else if(sum<0)
sum=0;
}
return maxSum;
}
Maxmum subsequence sum problem
标签：first ati ddl sum different ref possibly diff prefix
原文地址：http://www.cnblogs.com/KennyRom/p/5994914.html
