[ENH] Implement Exact Move-Split-Merge (MSM) Barycenter Average

[satwiksps]

Reference Issues/PRs

This enhancement was suggested by @TonyBagnall on Discord. Moreover, it closes #3191 opened to track it.

What does this implement/fix? Explain your changes.

This PR implements the Exact Move-Split-Merge (MSM) Barycenter Average algorithm, filling a gap in aeon's clustering module. This allows centroid-based algorithms (like K-Means) to be used effectively with the MSM metric.

The implementation is a port of the reference Java implementation from JanaHolznigenkemper/msm_mean, optimized for Python using numba and numpy.

Key Implementation Details:

• Solves the N-dimensional shortest path problem to find the mathematically optimal barycenter.
• Uses @njit (Numba) for high-performance JIT compilation.
• Flattens the N-dimensional dynamic programming table into 1D contiguous arrays for memory efficiency and cache locality.
• Implements bitwise predecessor generation to avoid recursion overhead during the wavefront propagation.
• Includes a dynamic queue resizing mechanism to handle potentially large state spaces without crashing.
• Fully integrated into elastic_barycenter_average dispatcher and the aeon clustering API.
• Added comprehensive unit tests covering identity, cost sensitivity, window constraints, and exactness properties.

Does your contribution introduce a new dependency? If yes, which one?

No.

Any other comments?

Important Limitation:
As this is the exact solver, the time and memory complexity scale exponentially with the number of time series (N) and their length (L) (O(L^N))

• It is intended for averaging small batches (typically $N \le 6$) or for use in medoid-based refinements.
• A RuntimeWarning is explicitly raised if the user inputs $N > 6$ to manage performance expectations.
• Multivariate input is currently explicitly blocked (raises ValueError) as the exact solver is defined for univariate series.

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I have added the following labels to this PR based on the title: [ enhancement ].
I have added the following labels to this PR based on the changes made: [ clustering ]. Feel free to change these if they do not properly represent the PR.

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