How to Solve Range Product Queries of Powers

In the Range Product Queries of Powers problem, you are given a number n and need to derive an array of powers of 2 that sum to n. The goal is to answer multiple range product queries on this array. The challenge is to optimize the solution using bit manipulation and efficient data structures for range queries, considering the size constraints of n and query count.

Problem Statement

Given a positive integer n, construct an array called 'powers', consisting of the minimum number of powers of 2 that sum to n. This array is sorted in non-decreasing order, and there is only one way to form it. The task is to answer multiple range product queries on this array.

You are provided with a 2D integer array queries, where each query defines a range of indices [lefti, righti]. The result of each query is the product of all elements in the 'powers' array within the given range. Since the product may be large, return the result modulo 10^9 + 7 for each query.

Examples

Example 1

Input: n = 15, queries = [[0,1],[2,2],[0,3]]

Output: [2,4,64]

For n = 15, powers = [1,2,4,8]. It can be shown that powers cannot be a smaller size. Answer to 1st query: powers[0] * powers[1] = 1 * 2 = 2. Answer to 2nd query: powers[2] = 4. Answer to 3rd query: powers[0] * powers[1] * powers[2] * powers[3] = 1 * 2 * 4 * 8 = 64. Each answer modulo 109 + 7 yields the same answer, so [2,4,64] is returned.

Example 2

Input: n = 2, queries = [[0,0]]

Output: [2]

For n = 2, powers = [2]. The answer to the only query is powers[0] = 2. The answer modulo 109 + 7 is the same, so [2] is returned.

Constraints

• 1 <= n <= 109
• 1 <= queries.length <= 105
• 0 <= starti <= endi < powers.length

Solution Approach

Array Construction Using Binary Representation

Start by constructing the 'powers' array based on the binary representation of the number n. Each bit in the binary representation represents a power of 2. For example, for n = 15, the binary representation is 1111, which gives the array [1, 2, 4, 8].

Prefix Product Array for Efficient Querying

To efficiently answer the range product queries, construct a prefix product array. This array allows you to compute the product of any subarray in constant time by using the formula: product = prefix[right + 1] / prefix[left].

Modulo Operation for Large Results

Since the result of each query may exceed the modulo limit, always compute the result modulo 10^9 + 7. Perform modular arithmetic on each query to prevent overflow and ensure the solution fits within the problem's constraints.

Complexity Analysis

The time complexity is O(log^2 n + q), where log^2 n arises from constructing the powers array using binary representation and q represents the number of queries. The space complexity is O(log^2 n) due to the storage required for the powers and prefix product arrays.

What Interviewers Usually Probe

• Does the candidate effectively apply bit manipulation to derive the powers array?
• Can the candidate explain the trade-offs between using a brute force solution and an optimized approach with prefix products?
• How well does the candidate handle large query sets and large values of n in terms of both time and space complexity?

Common Pitfalls or Variants

Common pitfalls

• Not using modular arithmetic during product calculations, leading to overflow errors.
• Inefficiently recalculating products for each query instead of utilizing a prefix product array.
• Failing to optimize for the constraints, particularly when handling a large number of queries or large values of n.

Follow-up variants

• The problem can be generalized to other operations like sum or minimum instead of product within a given range.
• Instead of answering multiple queries, solve for a single query at a time in an optimized manner using bit manipulation.
• Extend the problem to handle negative values of n by adjusting the binary representation approach.

How GhostInterview Helps

• GhostInterview helps by guiding you through the efficient construction of the powers array using binary representation.
• It suggests using prefix products to answer range queries efficiently, which drastically reduces the time complexity.
• The solver assists in correctly applying modulo arithmetic to prevent overflow and ensure the results fit within constraints.

Topic Pages

• Array
• Bit Manipulation
• Prefix Sum

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FAQ

What is the primary approach for solving the Range Product Queries of Powers problem?

The primary approach involves using bit manipulation to construct the 'powers' array from the binary representation of n, then using a prefix product array to efficiently answer range product queries.

How do we handle large numbers when solving Range Product Queries of Powers?

To handle large numbers, each product result is computed modulo 10^9 + 7, ensuring that the answers fit within the problem's constraints.

Why is a prefix product array useful in this problem?

A prefix product array allows us to calculate the product of any subarray in constant time, making it much more efficient than recalculating the product for each query.

How do you optimize for large values of n in Range Product Queries of Powers?

Optimizing for large values of n involves efficiently constructing the powers array using the binary representation of n and applying a prefix product array to minimize the number of operations needed for each query.

Can the Range Product Queries of Powers problem be extended to other operations?

Yes, the problem can be adapted to other operations, such as sum or minimum, instead of product within the given range.