semi-dynamic range minimum query

The semi-dynamic range minimum query algorithm that maintains an array A and supports the following two operations:

• append(x):
  append x to the end of A in amortized constant time.
• rmq(i, j):
  return the (rightmost) position of the minimum value in the subarray A[i], ..., A[j] in constant time.

Example

#include "RMQ.h"

[...]
    {
        RMQ r(8); // number of elements to be appended.
        vector<int> X = { 4, 6, 5, 7, 3, 4, 5, 3 };
        for (auto x : X) r.append(x);

        Assert::AreEqual(4, r.rmq(0, 7)); // argmin(X) = 4 because X[4] == 3. 
        Assert::AreEqual(7, r.rmq(4, 7)); // X[4] == X[7] == 3, but return the rightmost one.
    }

Implementations

• semi-dynamicRMQ/RMQ.*
  The proposed algorithm in [2].
• semi-dynamicRMQ/pm1RMQ.*
  The ±1RMQ algorithm in [1] based on the sparse table algorithm and table lookup algorithm.
• semi-dynamicRMQ/SparseTable.*
  The sparse table algorithm proposed in [1].
• semi-dynamicRMQ/BitTableLookup.*
  The table lookup algorithm proposed in [1] using some bitwise operations.
• UnitTest/unittest1.cpp
  Unittest. To run this code, Microsoft Visual Studio is required.

Remark

To compile with gcc or clang, the C++11 option -std=c++11 is required.
I have not yet implemented the semi-dynamic RMQ algorithm which increases the array size exponentially, i.e., the number of elements to be appended is required beforehand in order to achieve the amortized constant time complexity.

References

[1] M. A. Bender & M. Farach-Colton: The LCA problem revisited. In LATIN 2000, pp.88-94, 2000.
[2] Y. Ueki, Diptarama, M. Kurihara, Y. Matsuoka, K. Narisawa, R. Yoshinaka, H. Bannai, S. Inenaga, A. Shinohara: Longest common subsequence in at least k length order-isomorphic substrings. In SOFSEM 2017 https://arxiv.org/abs/1609.03668