2438. Range Product Queries of Powers
Problem description: Given a positive integer n, an array powers indexed from 0 is constructed, formed by the minimum number of powers of 2 that sum to n. The array powers is sorted in non-decreasing order and there is a unique representation. A 2D array queries is received, where each queries[i] = [lefti, righti] represents a query asking for the product of all elements powers[j] with lefti <= j <= righti. Return an array answers with the answers to each query taken modulo 10^9+7.
Example 1: Input n = 15, queries = [[0,1],[2,2],[0,3]] Output [2,4,64] Explanation: for n = 15 powers = [1,2,4,8]. Product of indices 0 to 1 = 1*2 = 2. Index 2 = 4. Product from 0 to 3 = 1*2*4*8 = 64.
Example 2: Input n = 2, queries = [[0,0]] Output [2] Explanation: for n = 2 powers = [2]. The only query returns 2.
Main idea: The decomposition of n into powers of 2 is obtained directly from its binary representation. Each bit set to 1 at position i corresponds to the power 2^i. The array powers contains these powers in non-decreasing order if we traverse the bits from least significant to most significant.
Efficient solution: Instead of multiplying range by range, we take advantage of the fact that each element is a power of 2. If we store for each position its exponent i, the product of a range corresponds to 2 raised to the sum of those exponents. Steps: 1 Build the list exps with the exponents i such that bit i of n is 1. 2 Calculate a prefix pref of exponent sums where pref[0] = 0 and pref[k+1] = pref[k] + exps[k]. 3 For a query [l,r] calculate sumExp = pref[r+1] - pref[l]. 4 Calculate powmod(2,sumExp,MOD) with MOD = 10^9+7 to obtain the answer. This approach avoids repeated multiplications and is very fast because the size of powers is at most 31 for n <= 10^9.
Complexity: Array construction in O(log n). Preprocessing of pref in O(m) with m number of bits set to 1. Each query is answered in O(log MOD) time due to modular exponentiation, or in O(1) time if optimized binary exponentiation is used. Additional space O(m).
Key implementation points: correctly handle 0-based indices, use modular exponentiation to avoid overflow, and traverse the bits of n from bit 0 to the most significant to obtain sorted powers. Simple alternative: you can also store the array powers with values 1,2,4,... and a prefix of products modulo MOD to answer queries in O(1) per query using modular inverses, although the exponent strategy is usually simpler and numerically stable.
Constraints: 1 <= n <= 10^9, 1 <= queries.length <= 10^5, 0 <= starti <= endi < powers.length.
Hint: The binary representation of n directly generates the array powers and the sum of exponents in the range determines the resulting power of 2.
Conceptual implementation: 1 Create empty vector exps. 2 Traverse i from 0 until n is 0: if n & 1 is 1 add i to exps; n >>= 1; i++. 3 Build pref of accumulated sizes. 4 For each query calculate sumExp and respond with powmod(2,sumExp,MOD).
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