Following this instruction :
1. Strictly adhere to PEP-8 standards, ensuring code is clean, well-structured, and follows Python's style guide for readability.
2. Use appropriate modules like `typing` for type hints, ensuring code clarity, maintainability, and leveraging Python's standard libraries effectively.
3. Focus on time, space, robustness, scalability; write optimized, efficient code that can handle large datasets and scale seamlessly.
4. Implement comprehensive error handling, catching exceptions gracefully, providing meaningful error messages, and ensuring code robustness under unexpected conditions.
5. Ensure the developer agent writes advanced, maintainable Python code, balancing performance and readability while adhering to best practices.



TASK : we have develop five different types of DTW: ( Dynamic Time Warping (DTW) algorithm with backtracking, often used in time series analysis to measure similarity between sequences, even if they differ in length or speed. Here’s a detailed breakdown of each part)

Standard Dynamic Time Warping (DTW)

Measures similarity between sequences by aligning time indices with minimum cumulative distance.
FastDTW (Fast Dynamic Time Warping)

An approximation algorithm for DTW that is faster and works well on large datasets by using multilevel resolution.
Subsequence Dynamic Time Warping (Subsequence DTW)

An adaptation of DTW to find the best matching subsequence within a larger sequence.
Weighted Dynamic Time Warping (WDTW)

A DTW variant that applies weights to penalize larger warp distances, enhancing robustness to noise.
Constrained Dynamic Time Warping (CDTW)

Adds constraints like the Sakoe-Chiba or Itakura bands to reduce the alignment search space and computational cost.




below sample or reference code :

@numba.jit(nopython=True)
def backtrace(trace: np.ndarray):
    i = trace.shape[0] - 1
    j = trace.shape[1] - 1
    trace[0, :] = 2
    trace[:, 0] = 1

    result = []
    while i > 0 or j > 0:
        result.append((i - 1, j - 1))

        if trace[i, j] == 0:
            i -= 1
            j -= 1
        elif trace[i, j] == 1:
            i -= 1
        elif trace[i, j] == 2:
            j -= 1
        else:
            raise ValueError("Unexpected trace[i, j]")

    result = np.array(result)
    return result[::-1, :].T


@numba.jit(nopython=True, parallel=True)
def dtw_cpu(x: np.ndarray):
    N, M = x.shape
    cost = np.ones((N + 1, M + 1), dtype=np.float32) * np.inf
    trace = -np.ones((N + 1, M + 1), dtype=np.float32)

    cost[0, 0] = 0
    for j in range(1, M + 1):
        for i in range(1, N + 1):
            c0 = cost[i - 1, j - 1]
            c1 = cost[i - 1, j]
            c2 = cost[i, j - 1]

            if c0 < c1 and c0 < c2:
                c, t = c0, 0
            elif c1 < c0 and c1 < c2:
                c, t = c1, 1
            else:
                c, t = c2, 2

            cost[i, j] = x[i - 1, j - 1] + c
            trace[i, j] = t

    return backtrace(trace)


def dtw_cuda(x, BLOCK_SIZE=1024):
    from .triton_ops import dtw_kernel

    M, N = x.shape
    assert M < BLOCK_SIZE, f"M should be smaller than {BLOCK_SIZE=}"

    x_skew = (
        F.pad(x, (0, M + 1), value=np.inf).flatten()[: M * (N + M)].reshape(M, N + M)
    )
    x_skew = x_skew.T.contiguous()
    cost = torch.ones(N + M + 2, M + 2) * np.inf
    cost[0, 0] = 0
    cost = cost.cuda()
    trace = torch.zeros_like(cost, dtype=torch.int32)

    dtw_kernel[(1,)](
        cost,
        trace,
        x_skew,
        x_skew.stride(0),
        cost.stride(0),
        trace.stride(0),
        N,
        M,
        BLOCK_SIZE=BLOCK_SIZE,
    )

    trace = trace.T.flatten()[: (M + 1) * (M + N + 3)].reshape(M + 1, M + N + 3)[
        :, : N + 1
    ]
    return backtrace(trace.cpu().numpy())


def dtw(x: torch.Tensor) -> np.ndarray:
    if x.is_cuda:
        try:
            return dtw_cuda(x)
        except (RuntimeError, subprocess.CalledProcessError):
            warnings.warn(
                "Failed to launch Triton kernels, likely due to missing CUDA toolkit; "
                "falling back to a slower DTW implementation..."
            )

    return dtw_cpu(x.double().cpu().numpy())