def split(lower_bound, upper_bound, middle):
    """
    Helper function to split the indices for BFS.
    """
    if lower_bound == middle:
        L = None
        R = [middle + 1, upper_bound]
    elif upper_bound == middle:
        L = [lower_bound, middle - 1]
        R = None
    else:
        L = [lower_bound, middle - 1]
        R = [middle + 1, upper_bound]
    
    return (L, R)