To solve the problem of finding the product of all integers in an array without using the multiplication operator, we can use a helper function that employs the Russian Peasant Multiplication algorithm. This algorithm efficiently multiplies two numbers using bit shifting and addition, avoiding direct multiplication. The solution handles edge cases such as empty arrays, zeros, and negative numbers appropriately.

1. Problem Analysis: The task is to compute the product of all elements in an array without using the multiplication operator. The solution must handle:

   • Empty arrays (return 1 by convention).
   • Arrays containing zeros (product becomes zero).
   • Negative numbers (product sign is handled correctly).

2. Key Insight: The Russian Peasant Multiplication algorithm uses bit shifting and addition to multiply two numbers. This method efficiently computes the product by:

   • Handling the sign of the result separately.
   • Using bit shifting to double one number and halve the other iteratively.
   • Adding the doubled number to the result when the halved number is odd.

3. Algorithm Selection:

   • Helper Function (multiply):
     • Converts both numbers to positive and tracks the sign.
     • Uses bit shifting to multiply without the operator.
     • Returns the result with the correct sign.
   • Main Function (product_array):
     • Iterates through each number in the array.
     • Uses the helper function to compute the cumulative product.
     • Early termination if the product becomes zero (optimization).

4. Complexity Analysis:

   • Time Complexity: O(n log m), where n is the number of elements in the array and m is the maximum absolute value in the array. The Russian Peasant method runs in O(log m) per multiplication.
   • Space Complexity: O(1), as only a few variables are used for computation.

Solution Code

def multiply(a, b):
    if a == 0 or b == 0:
        return 0
    sign = 1
    if a < 0:
        a = -a
        sign = -sign
    if b < 0:
        b = -b
        sign = -sign
    res = 0
    while b:
        if b & 1:
            res += a
        a <<= 1
        b >>= 1
    return res if sign == 1 else -res
def product_array(arr):
    if not arr:
        return 1
    result = 1
    for num in arr:
        if result == 0:
            break
        result = multiply(result, num)
    return result

Explanation

1. Helper Function (multiply):

   • Zero Handling: If either number is zero, returns zero immediately.
   • Sign Handling: Converts both numbers to positive and tracks the sign of the result.
   • Russian Peasant Algorithm:
     • Loop: While the second number (b) is non-zero:
       • If b is odd (using bitwise AND), adds the first number (a) to the result.
       • Doubles a (left shift) and halves b (right shift) in each iteration.
     • Sign Adjustment: Returns the result with the correct sign based on the tracked sign.

2. Main Function (product_array):

   • Edge Case: Returns 1 for an empty array.
   • Product Calculation:
     • Initializes the result as 1.
     • Iterates through each number in the array, multiplying cumulatively using the helper function.
     • Early Termination: If the result becomes zero, breaks the loop early since further multiplications will not change the result.
   • Result: Returns the computed product of all elements in the array.

This approach efficiently computes the product without using the multiplication operator, handling all edge cases and optimizing for performance.