最大子列和的四种算法比较

本文将会介绍最大子列和的四种不同复杂度算法。

算法一：暴力枚举法，算法复杂度为O（N三次方）

int maxsequence1(int A[], int N)
{
int ThisSum , MaxSum=0;
int i, j, k;
for (i = 0; i <= N; i++) {

for (j = i; j <= N; j++) {
ThisSum = 0;
for (k = i; k <= j; k++)
ThisSum += A[k];
if (ThisSum > MaxSum)
MaxSum = ThisSum;

}
}

return MaxSum;
}

算法二：算法一的改进，减少一个for循环，复杂度为O（N平方）

int maxsequence2(int A[], int N)
{
int ThisSum, MaxSum = 0;
int i, j;
for (i = 0; i <= N; i++) {
ThisSum = 0;
for (j = i; j <= N; j++) {
ThisSum += A[j];
if (ThisSum > MaxSum)
MaxSum = ThisSum;
}
}
return MaxSum;
}

算法三：分而治之。算法复杂度为O（NlogN）

int Max(int A, int B, int C)
{
/*return A > B ? A > C ? A :C: B > C ? B : C;*/
if (A > B) {
if (A > C)
return A;
else
return C;
}
else if (B > C)
return B;
else return C;

}
int DivideAndConquer(int List[], int left, int right) {
int MaxLeftSum, MaxRightSum;
int MaxLeftBoardSum, MaxRightBoardSum;
int LeftBoardSum, RightBoardSum;
int center,i;
/*递归终止条件*/
if (left == right) {
if (List[left] > 0)
return List[left];
else
return 0;
}

center = (right + left) / 2;
MaxLeftSum = DivideAndConquer(List, left, center);
MaxRightSum= DivideAndConquer(List, center+1, right);

MaxLeftBoardSum = 0; LeftBoardSum = 0;
for (i = center;i >= left; i--)
LeftBoardSum += List[i];
if (LeftBoardSum > MaxLeftBoardSum)
MaxLeftBoardSum = LeftBoardSum;
MaxRightBoardSum = 0; RightBoardSum = 0;

for (i = center + 1; i <= right; i++)
RightBoardSum += List[i];
if (RightBoardSum > MaxRightBoardSum)
MaxRightBoardSum = RightBoardSum;

return Max(MaxLeftSum, MaxRightSum, MaxRightBoardSum + MaxLeftBoardSum);

}
int maxsequence3(int A[], int N)
{
return DivideAndConquer(A, 0, N-1);
}

算法四：在线处理算法，代码最简单，且复杂度最低，为O（N）

int maxsequence4(int A[], int N)
{
int ThisSum,MaxSum ;
int i;
ThisSum = MaxSum = 0;
for (i = 0; i <= N; i++) {
ThisSum += A[i];
if (ThisSum > MaxSum)
MaxSum = ThisSum;
else if (ThisSum < 0)
ThisSum = 0;
}
return MaxSum;

}

主程序如下

#include <stdio.h>
int maxsequence1(int A[], int N);
int maxsequence2(int A[], int N);
int maxsequence3(int A[], int N);
int maxsequence4(int A[], int N);
int main(void)
{
int K,j=0;
int a,max;
int num[100000];
printf("请输入数字个数：\n");
scanf("%d", &K);
printf("请输入%d个数字\n", K);
for (a = 0; a < K; a++) {
scanf("%d", &num[a]);
if (num[a] < 0)
j++;
}
if (j == K)
printf("0\n");
else {
/*用哪个选哪个*/
max = maxsequence1(num, K);
max = maxsequence2(num, K);
max = maxsequence3(num, K);
max = maxsequence4(num, K);
printf("最大子列和为：%d\n", max);
}

}

以上即为求最大子列和的四种算法。