Product of Array except self

Difficulty: 🟡 Medium

Techniques: PRECOMPUTE

Links: LeetCode

Problem Statement

Given an integer array nums, return an array answer such that answer[i] is equal to the product of all the elements of nums except nums[i].

The product of any prefix or suffix of nums is guaranteed to fit in a 32-bit integer. Try solving it without using the division operation /

1. Using Division operation (violating instructions)

Cases to observe

Two or more zero elements: All products will be zero
One zero element: All products will be zero except at the zero element’s position
No zero elements: The product at each postition will be combined product of all elements divided by current element (we’ll use the division operator)

C++ code

vector<int> productExceptSelf (vector<int> &nums) {
    int n = nums.size();
    // Find the zero element (stop when zero count reaches 2)
    vector<int> zeroValueIndices;
    for (int i = 0; i < n; i++) {
        if (zeroValueIndices.size() >= 2) break;
        if (nums[i] == 0) zeroValueIndices.push_back(i);
    }
    // Case 1: Two or more zero elements present
    if (zeroValueIndices.size() > 1) {
        // All products will be zero
        return vector<int>(n, 0);
    }
    // Case 2: One zero element present
    if (zeroValueIndices.size() == 1) {
        int zeroIndex = zeroValueIndices.front();
        // All products will be zero except for the one at zero element
        int productAtZeroElement = 1;
        // Multiplying elements present on the left
        for (int i = 0; i < zeroIndex; i++) {
            productAtZeroElement *= nums[i];
        }
        // Multiplying elements present on the right
        for (int i = zeroIndex + 1; i < n; i++) {
            productAtZeroElement *= nums[i];
        }
        vector<int> temp(n, 0);
        // Set the product at zero element
        temp[zeroIndex] = productAtZeroElement;
        return temp;
    }
    // Case 3: No zero element present
    int productOfAll = 1;
    // Get the product of all elements
    for (int curr : nums) {
        productOfAll *= curr;
    }
    vector<int> ans(n, 1);
    // The product at each element will be
    // the overall product divided by current element
    for (int i = 0; i < n; i++) {
        // NOTE: We're using division operator here 🙅‍♂️
        ans[i] = productOfAll / nums[i];
    }
    return ans;
}

Metric	Complexity
Time	\( O(N) \) … 3 traversals at worst
Space	\( O(1) \)

2. Maintain Prefix and Suffix product arrays

Product of all elements except self means product of elements to it’s left multiplied by the product of elements to it’s right

Create 2 arrays of size 'n'

Traverse array left to right (skipping one element) and calculate PREFIX products

Traverse array right to left (skipping one element) and calculate SUFFIX products

Create answer array
Each element in answer array will be the product of corresponding values in prefix and suffix product arrays

The first element of prefix array and last element of suffix array will be 1 (as their prefix and suffix are nothing i.e. empty arrays). So, prefix[0] = 1 = suffix[n-1].
The remaining values are calucated during traversals as:
- prefix[i] = prefix[i-1] * nums[i-1]
- suffix[i] = suffix[i+1] * nums[i+1]

Metric	Complexity
Time	\( O(N) \) … 3 traversals
Space	\( O(N) \) … Extra space for `prefix[]` , `suffix[]`

C++ code

vector<int> productExceptSelf (vector<int> &nums) {
    int n = nums.size();

    vector<int> prefix(n, 1);
    // Calculate prefix product values starting from second element
    for (int i = 1; i < n; i++) {
        prefix[i] = prefix[i - 1] * nums[i - 1];
    }

    vector<int> suffix(n, 1);
    // Calculate prefix product values starting from second-last element
    for (int i = n - 2; i >= 0; i--) {
        suffix[i] = suffix[i + 1] * nums[i + 1];
    }

    vector<int> ans(n, 1);
    // Construct answer value as product of prefix and suffix values
    for (int i = 0; i < n; i++) {
        ans[i] = prefix[i] * suffix[i];
    }
    return ans;
}

Optimization: The multiplications can be done in-place right within the answer array

Create answer array of size n

Traverse input array from left:
    Fill answer values as prefix product values

Traverse input array from right:
    Calculate suffix product value
    Multiply the existing prefix value in answer with the suffix product value

Metric	Complexity
Time	\( O(N) \) … 2 traversals
Space	\( O(1) \) … Only `ans[]` , no extra space

C++ code

vector<int> productExceptSelf (vector<int> &nums) {
    int n = nums.size();

    vector<int> ans(n, 1);
    for (int i = 1; i < n; i++) {
        // Set prefix product values in answer
        ans[i] = ans[i - 1] * nums[i - 1];
    }

    int suffix = 1;
    for (int i = n - 2; i >= 0; i--) {
        // Calculate suffix product
        suffix *= nums[i + 1];
        // Multiply existing prefix product with calculated suffix product
        ans[i] *= suffix;
    }

    return ans;
}