152. Maximum Product Subarray

Description

Find the contiguous subarray within an array (containing at least one number) which has the largest product.

For example, given the array [2,3,-2,4],
the contiguous subarray [2,3] has the largest product = 6.

Solution

DP, time O(n), space O(1)

This is very similar to the " max cumulative sum subarray" problem. here you keep 2 values: the max cumulative product UP TO current element starting from SOMEWHERE in the past, and the minimum cumuliative product UP TO current element . it would be easier to see the DP structure if we store these 2 values for each index, like maxProduct[i],minProduct[i] .

At each new element, u could either add the new element to the existing product, or start fresh the product from current index (wipe out previous results), hence the 2 Math.max() lines.

If we see a negative number, the “candidate” for max should instead become the previous min product, because a bigger number multiplied by negative becomes smaller, hence the swap().

class Solution {
    public int maxProduct(int[] nums) {
        if (nums == null || nums.length == 0) {
            return 0;
        }
        
        int max = nums[0];
        // imax/imin stores the max/min product of
        // subarray that ends with the current number A[i]
        for (int i = 1, imax = nums[0], imin = nums[0]; i < nums.length; ++i) {
            if (nums[i] < 0) {
                int tmp = imax;
                imax = imin;
                imin = tmp;
            }
            // max/min product for the current number is 
            // either the current number itself
            // or the max/min by the previous number times the current one
            imax = Math.max(nums[i], nums[i] * imax);
            imin = Math.min(nums[i], nums[i] * imin);
            max = Math.max(max, imax);
        }
        
        return max;
    }
}

也可以这样，注意需要用tmp保存中间值：

class Solution {
    public int maxProduct(int[] nums) {
        int maxProduct = Integer.MIN_VALUE;
        
        for (int i = 0, imin = 1, imax = 1; i < nums.length; ++i) {
            int tmp = imax;
            imax = Math.max(nums[i], Math.max(imax * nums[i], imin * nums[i]));
            imin = Math.min(nums[i], Math.min(tmp * nums[i], imin * nums[i]));
            maxProduct = Math.max(maxProduct, Math.max(imax, imin));
        }
        
        return maxProduct;
    }
}