def _union_lcs(evaluated_sentences, reference_sentence):
    """
    Returns LCS_u(r_i, C) which is the LCS score of the union longest common subsequence
    between reference sentence ri and candidate summary C. For example, if
    r_i= w1 w2 w3 w4 w5, and C contains two sentences: c1 = w1 w2 w6 w7 w8 and
    c2 = w1 w3 w8 w9 w5, then the longest common subsequence of r_i and c1 is
    “w1 w2” and the longest common subsequence of r_i and c2 is “w1 w3 w5”. The
    union longest common subsequence of r_i, c1, and c2 is “w1 w2 w3 w5” and
    LCS_u(r_i, C) = 4/5.

    :param evaluated_sentences:
        The sentences that have been picked by the summarizer
    :param reference_sentence:
        One of the sentences in the reference summaries
    :returns float: LCS_u(r_i, C)
    :raises ValueError: raises exception if a param has len <= 0
    """
    if len(evaluated_sentences) <= 0:
        raise (ValueError("Collections must contain at least 1 sentence."))

    lcs_union = set()
    reference_words = _split_into_words([reference_sentence])
    combined_lcs_length = 0
    for eval_s in evaluated_sentences:
        evaluated_words = _split_into_words([eval_s])
        lcs = set(_recon_lcs(reference_words, evaluated_words))
        combined_lcs_length += len(lcs)
        lcs_union = lcs_union.union(lcs)

    union_lcs_count = len(lcs_union)
    union_lcs_value = union_lcs_count / combined_lcs_length
    return union_lcs_value