【Algorithm】最大连续子数组和

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本文介绍了最大子数组和问题的五种求解方法，包括三种暴力法的不同优化级别及分治法、动态规划法。通过具体实现代码展示了每种算法的特点与效率。

/*
 * MaxSumArray.c
 *
 *  Created on: 2015年5月5日
 *      Author: weike
 */
#include<stdio.h>
int besti,bestj;//返回位置i,位置j
/**
 * algorithm 1 暴力法O(n^3)
 */
int MaxSum1(int n,int *a){
	int sum = 0;
	int i,j,k;
	for(i=1;i<=n;i++){
		for(j=i;j<=n;j++){
			int thissum = 0;
			for(k=i;k<=j;k++){
				thissum += a[k];
			}
			if(thissum >= sum){
				sum = thissum;
				besti = i;
				bestj = j;
			}
		}
	}
	return sum;
}
/**
 * 	algorithm 2 暴力法优化O(n^2)
 */
int MaxSum2(int n,int *a){
	int sum = 0;
	int i,j;
	for(i=1;i<=n;i++){
		int thissum = 0;
		for(j = i;j <= n;j++){
			thissum += a[j];
			if(thissum > sum){
				sum = thissum;
				besti = i;
				bestj = j;
			}
		}
	}
	return sum;
}
/**
 * 	algorithm 暴力法优化
 */
int MaxSum3(int n,int *a){
	int i,j;
	int cumarr[20];
	cumarr[0] = 0;
	for(i=1;i<=n;i++){
		cumarr[i] = cumarr[i-1] + a[i];
	}
	int maxsofar = 0;
	int sum;
	for(i=1;i<=n;i++){
		for(j=0;j<=n;j++){
			sum = cumarr[j] - cumarr[i-1];
			if(sum > maxsofar){
				maxsofar  = sum;
				besti = i;
				bestj = j;
			}

		}
	}
	return maxsofar;
}
/**
 * 	algorithm 4	分治法
 */
int MaxSubSum(int *a,int left,int right){
	int sum = 0;
	int i;
	if(left == right)
		sum = a[left] > 0 ? a[left] : 0;
	else{
		int center = (left + right)/2;
		//(1)
		int leftsum = MaxSubSum(a,left,center);
		//(2)
		int rightsum = MaxSubSum(a,center+1,right);
		//(3)左半部分
		int s1 = 0;
		int lefts = 0;
		for(i=center;i>=left;i--){
			lefts += a[i];
			if(lefts > s1)
				s1 = lefts;
		}
		//(3)右半部分
		int s2 = 0;
		int rights = 0;
		for(i=center+1;i<=right;i++){
			rights += a[i];
			if(rights > s2)
				s2 = rights;
		}
		sum = s1 + s2;
		if(sum < leftsum)
			sum = leftsum;
		if(sum < rightsum)
			sum = rightsum;
	}
	return sum;
}
int MaxSum4(int n,int *a){
	return MaxSubSum(a,1,n);
}
/**
 * 	algorithm 5 动态规划
 */
int MaxSum5(int n,int *a){
	int sum = 0,b = 0;
	int i;
	for(i=1;i<=n;i++){
		if(b>0)
			b += a[i];
		else
			b = a[i];
		if(b > sum)
			sum = b;
	}
	return sum;
}
/**
 * method main
 */
int main(int argc, char **argv) {
	int tmp[] = {0,-2,11,-4,13,-5,-2};
	int a[7];
	int i;
	for(i=1;i<=6;i++){
		a[i] = tmp[i];
	}

	int result1 = MaxSum1(6,a);
	int result2 = MaxSum2(6,a);
	int result3 = MaxSum3(6,a);
	int result4 = MaxSum4(6,a);
	int result5 = MaxSum5(6,a);
	printf("result1 = %d,i = %d,j = %d\n",result1,besti,bestj);
	printf("result2 = %d,i = %d,j = %d\n",result2,besti,bestj);
	printf("result3 = %d,i = %d,j = %d\n",result3,besti,bestj);
	printf("result4 = %d\n",result4);
	printf("result5 = %d\n",result5);
	return 0;
}

参考资料：

《计算机算法设计与分析(第三版)》王晓东著

July博客结构之法算法之道