Range Minimum Query

Range Minimum Query (RMQ) is a fundamental algorithmic problem that asks for the index or value of the minimum element within a specified contiguous subsegment of a linear array. Formally, given an array A of n elements, a query Q(i, j) returns the index k such that A[k] = min(A[i…j]) for 0 <= i <= j < n. In high-performance systems, RMQ serves as the foundational reduction target for the Lowest Common Ancestor (LCA) problem in trees, string suffix tree indexing, and computational geometry spatial partitioning optimizations.

#include <vector>
#include <cmath>
#include <algorithm>
#include <stdexcept>

class RangeMinimumQuery {
private:
    std::vector<std::vector<int>> st;
    std::vector<int> log_table;
public:
    explicit RangeMinimumQuery(const std::vector<int>& arr) {
        size_t n = arr.size();
        if (n == 0) return;
        int K = static_cast<int>(std::log2(n));
        st.assign(n, std::vector<int>(K + 1));
        log_table.assign(n + 1, 0);
        
        for (size_t i = 2; i <= n; ++i) {
            log_table[i] = log_table[i / 2] + 1;
        }
    }
};

    void build(const std::vector<int>& arr) {
        size_t n = arr.size();
        int K = static_cast<int>(std::log2(n));
        
        for (size_t i = 0; i < n; ++i) {
            st[i][0] = arr[i];
        }
        
        for (int j = 1; j <= K; ++j) {
            for (size_t i = 0; i + (1 << j) <= n; ++i) {
                st[i][j] = std::min(st[i][j - 1], st[i + (1 << (j - 1))][j - 1]);
            }
        }
    }

    int query(int L, int R) const {
        if (L > R || L < 0 || R >= static_cast<int>(st.size())) {
            throw std::out_of_range("Invalid query bounds");
        }
        int j = log_table[R - L + 1];
        return std::min(st[L][j], st[R - (1 << j) + 1][j]);
    };

#cxx -O3 -std=c++20 rmq.cpp -o rmq
#include <cassert>

int main() {
    std::vector<int> data = {4, 6, 1, 5, 7, 3, 2, 8};
    RangeMinimumQuery rmq(data);
    rmq.build(data);
    
    assert(rmq.query(0, 3) == 1);
    assert(rmq.query(2, 2) == 1);
    assert(rmq.query(4, 7) == 2);
    return 0;
}

// Optimized Cache-Friendly Layout
class CacheOptimizedRMQ {
    // Transposed storage: st[j][i] instead of st[i][j]
    std::vector<std::vector<int>> st;
    std::vector<int> log_table;
public:
    // Implementation trades minor indexing complexity for L1/L2 cache line hits
};

Mathematical Formulation & Data Structure Variants

The standard approach to Range Minimum Query is the Sparse Table, which exploits the idempotency of the minimum operator (min(x, x) = x). Because overlapping identical intervals do not alter the outcome of a minimum query, an arbitrary range [L, R] can be covered by exactly two precomputed power-of-two intervals of length k = floor(log2(R – L + 1)). The query time complexity collapses from O(n) down to O(1). For dynamic scenarios where array elements are mutated frequently, alternative architectures like Segment Trees or Fenwick Trees (Binary Indexed Trees) are preferred, trading the O(1) query time for O(log n) updates and queries.

Advanced O(n) Construction via Cartesian Trees

While the standard Sparse Table requires O(n log n) space and construction time, theoretical limits established by Harel, Tarjan, Bender, and Farach-Colton achieve O(n) preprocessing time and O(1) query time. This is accomplished by converting the input array into a Cartesian Tree—where an in-order traversal yields the original array and heap-property invariants hold universally. By mapping sub-tree structures to localized bitmasks (via Farach-Colton and Bender's algorithm), repetitive tree blocks are indexed using precalculated lookup tables, optimizing both memory footprints and construction bottlenecks.

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Editorial Engineering Team — Senior Engineering Contributor

Specializing in enterprise infrastructure, low-latency caching, and full-stack performance optimization.