Returns an integer that is equal to the margin of victory of the election profile, that is,
    the smallest number k such that changing k votes can change the winners.

    :ivar Profile profile: A Profile object that represents an election profile.

def AppMoVCopeland(profile, alpha=0.5):
    """
    Returns an integer that is equal to the margin of victory of the election profile, that is,
    the smallest number k such that changing k votes can change the winners.

    :ivar Profile profile: A Profile object that represents an election profile.
    """

    # Currently, we expect the profile to contain complete ordering over candidates.
    elecType = profile.getElecType()
    if elecType != "soc" and elecType != "toc":
        print("ERROR: unsupported profile type")
        exit()

    # Initialization
    n = profile.numVoters
    m = profile.numCands

    # Compute the original winner d
    # Initialize each Copeland score as 0.0.
    copelandscores = {}
    for cand in profile.candMap.keys():
        copelandscores[cand] = 0.0

    # For each pair of candidates, calculate the number of votes in which one beat the other.
    wmgMap = profile.getWmg()
    for cand1, cand2 in itertools.combinations(wmgMap.keys(), 2):
        if cand2 in wmgMap[cand1].keys():
            if wmgMap[cand1][cand2] > 0:
                copelandscores[cand1] += 1.0
            elif wmgMap[cand1][cand2] < 0:
                copelandscores[cand2] += 1.0

            # If a pair of candidates is tied, we add alpha to their score for each vote.
            else:
                copelandscores[cand1] += alpha
                copelandscores[cand2] += alpha
    d = max(copelandscores.items(), key=lambda x: x[1])[0]

    #Compute c* = argmin_c RM(d,c)
    relative_margin = {}
    alter_without_d = delete(range(1, m + 1), d - 1)
    for c in alter_without_d:
        relative_margin[c] = RM(wmgMap, n, m, d, c, alpha)
    c_star = min(relative_margin.items(), key=lambda x: x[1])[0]

    return relative_margin[c_star]*(math.ceil(log(m)) + 1)