Maximum Product Subarray medium

Description

Given an integer array nums, find a contiguous non-empty subarray within the array that has the largest product, and return the product.

The array can contain negative numbers and zeros.

Examples

> Case 1:
    nums = [2, 3, -2, 4]
    Output: 6
    Explanation: [2, 3] has the largest product 6.
 
> Case 2:
    nums = [-2, 0, -1]
    Output: 0
    Explanation: max product subarray is [0]; subarray [-2, 0, -1] would be 0 too.
 
> Case 3:
    nums = [-2, 3, -4]
    Output: 24
    Explanation: -2 × 3 × -4 = 24.

Constraints

• 1 <= nums.length <= 2 × 10^4
• -10 <= nums[i] <= 10

Why Kadane doesn’t directly work

In Maximum Sub-Array (Kadane), dp[i] = best sum ending at i, and the recurrence is max(nums[i], nums[i] + dp[i - 1]). With products there’s a catch: multiplying by a negative number flips sign. The smallest (most negative) product ending at i - 1 could become the largest product ending at i if nums[i] is negative.

So we have to track both the running max and the running min.

State design

1. What am I computing? max_ending[i] = max product of a subarray ending at i. min_ending[i] = min product of a subarray ending at i.

2. Dimensions? Two scalars per position.

3. Base cases? max_ending[0] = min_ending[0] = nums[0].

4. Transition? At index i, the candidates for max_ending[i] are:

   • nums[i] itself (start fresh),
   • nums[i] × max_ending[i - 1] (extend a positive run),
   • nums[i] × min_ending[i - 1] (extend a negative run that became positive after multiplying by nums[i]).

   Same for min_ending[i], taking the min instead.

5. Order? Increasing i.

6. Answer? max(max_ending[i]) over all i.

We only need the previous (max, min), so use two scalars and run in O(1) space.

Try it yourself

def max_product(nums):
  # track running max AND min (a negative flips them); answer is best max seen
  pass

Code

int maxProduct(vector<int>& nums) {
    int curMax = nums[0], curMin = nums[0], best = nums[0];
    for (int i = 1; i < (int)nums.size(); i++) {
        int x = nums[i];
        int candMax = max({x, x * curMax, x * curMin});
        int candMin = min({x, x * curMax, x * curMin});
        curMax = candMax;
        curMin = candMin;
        best = max(best, curMax);
    }
    return best;
}

def max_product(nums):
    cur_max = cur_min = best = nums[0]
    for x in nums[1:]:
        cand_max = max(x, x * cur_max, x * cur_min)
        cand_min = min(x, x * cur_max, x * cur_min)
        cur_max, cur_min = cand_max, cand_min
        best = max(best, cur_max)
    return best

public int maxProduct(int[] nums) {
    int curMax = nums[0], curMin = nums[0], best = nums[0];
    for (int i = 1; i < nums.length; i++) {
        int x = nums[i];
        int candMax = Math.max(x, Math.max(x * curMax, x * curMin));
        int candMin = Math.min(x, Math.min(x * curMax, x * curMin));
        curMax = candMax;
        curMin = candMin;
        best = Math.max(best, curMax);
    }
    return best;
}

The “split by zeros” alternative

There’s a neat trick: zeros split the array into runs with no zeros. For each run, the answer is either the product of the whole run, or the product up to (but not including) the leftmost negative, or the product from just after the rightmost negative. Compute prefix products from the left and from the right; the max of those two scans (skipping zeros) is the answer.

It’s elegant but trickier to implement; the two-variable Kadane variant above is more interview-friendly.

Analysis

• Time: O(n).
• Space: O(1).

Same skin

• Maximum Subarray Sum — the additive cousin (Kadane).
• Max Sum Circular Subarray — combine Kadane with “total − min subarray sum.”
• Best Sightseeing Pair — two-track DP carrying the best a[i] + i so far.