The Minimum Sum of Values by Dividing Array problem gives you an array of values and a list of queries, each specifying a segment and a divisor. For each query, you divide the segment's AND-value by the divisor and collect the resulting values. The goal is to minimize the total sum collected across all queries while maintaining specific AND constraints. This hard Minimum Sum of Values by Dividing Array coding problem requires segment trees and dynamic programming combined with bit manipulation.
Google asks this problem at senior engineer levels because it demands combining three advanced techniques: bit manipulation for AND tracking, segment trees for range queries, and DP for optimal assignment. The array, queue, binary search, bit manipulation, segment tree, and dynamic programming interview pattern are all represented — making it a comprehensive test of algorithmic breadth.
The approach uses DP with segment tree optimization. Process queries by their right endpoints. For each query (l, r, divisor, target AND value), the segment's AND must equal the target. Use a segment tree to maintain current AND values for all prefixes. The DP state captures minimum cost to satisfy all queries up to the current position, and the segment tree enables efficient range AND lookups. Pruning based on AND monotonicity (AND can only decrease as range expands) is critical for efficiency.
Array: [1, 4, 3, 3, 2], queries: [(0, 4, [1,3,3,3,2])]. Partition into segments where AND of each segment = target. DP over partitions: cost[i] = AND-divided value for segment ending at i. Find the partition minimizing total sum.
Hard problems with multiple advanced techniques require a structured approach: first understand each individual component (segment tree for range AND, DP transitions), then integrate them. In an interview, clearly communicate the high-level approach before attempting code. If full implementation isn't possible, demonstrating knowledge of each component and how they connect is valuable. Practice segment trees supporting range AND, OR, and XOR separately before using them inside DP.
