Find Products of Elements of Big Array

Find Products of Elements of Big Array

What is this problem about?

The Find Products of Elements of Big Array interview question is a high-level bitwise math problem. A "Big Array" is constructed by taking every positive integer $i$ and appending the powers of 2 that sum to $i$ (in ascending order). For example, $3 = 2^0 + 2^1$, so the parts for 3 are $\{0, 1\}$. You are given queries $[from, to]$ and need to find the product of all elements in that range of the Big Array, modulo $m$.

Why is this asked in interviews?

IBM and other technical firms ask this to test a candidate's mastery of Bit Manipulation and Binary Search. It is a "Hard" problem because the Big Array is virtually infinite. You must use combinatorial counting to find which integers contribute to a specific range of the array without actually building the array. It evaluates your ability to handle large-scale numeric patterns.

Algorithmic pattern used

This follows the Digit/Bit Counting and Binary Search pattern.

1. Find Integer Range: Use binary search to find the integer $N$ such that the total number of set bits in all integers from $1 \dots N$ is approximately equal to the query index.
2. Count Bits: For each bit position $j$, calculate how many times the $j^{th}$ bit is set in the range $[1, N]$.
3. Power Calculation: The product of $2^a imes 2^b \dots$ is $2^{a+b+\dots}$. Calculate the total sum of all exponents in the query range.
4. Modulo Math: Use binary exponentiation (pow(2, exponent_sum, m)) to find the final product.

Example explanation

Integers: 1 (bits: {0}), 2 (bits: {1}), 3 (bits: {0, 1}). Big Array: [0, 1, 0, 1, ...]

• index 0: $2^0$
• index 1: $2^1$
• index 2: $2^0$
• index 3: $2^1$ Query $[0, 1]$: $2^0 imes 2^1 = 2^{0+1} = 2$.

Common mistakes candidates make

• Building the array: Trying to actually append elements to a list. The indices can be up to $10^{15}$, which is impossible to store.
• Counting bits: Inefficiently counting bits for each number instead of using the mathematical pattern for bit frequency in a range.
• Modulo: Forgetting that you cannot take the modulo of the exponent directly unless you use Euler's Totient Theorem (the exponent sum must be used in a power function).

Interview preparation tip

Learn the formula for counting set bits in the range $[1, N]$. For the $j^{th}$ bit, the pattern $0 \dots 0, 1 \dots 1$ repeats every $2^{j+1}$ numbers. This is a foundational Bit Manipulation interview pattern for large-scale problems.