<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def best_case(self, matrix, m_list, indices_left):
 """ Computes a best case given a matrix and manipulation list. Args: matrix: the current matrix (with some permutations already performed) m_list: [(multiplication fraction, number_of_indices, indices, species)] describing the manipulation indices: Set of indices which haven't had a permutation performed on them. """

        m_indices = []
        fraction_list = []
        for m in m_list:
            m_indices.extend(m[2])
            fraction_list.extend([m[0]] * m[1])

        indices = list(indices_left.intersection(m_indices))

        interaction_matrix = matrix[indices, :][:, indices]

        fractions = np.zeros(len(interaction_matrix)) + 1
        fractions[:len(fraction_list)] = fraction_list
        fractions = np.sort(fractions)

        # Sum associated with each index (disregarding interactions between
        # indices)
        sums = 2 * np.sum(matrix[indices], axis=1)
        sums = np.sort(sums)

        # Interaction corrections. Can be reduced to (1-x)(1-y) for x,y in
        # fractions each element in a column gets multiplied by (1-x), and then
        # the sum of the columns gets multiplied by (1-y) since fractions are
        # less than 1, there is no effect of one choice on the other
        step1 = np.sort(interaction_matrix) * (1 - fractions)
        step2 = np.sort(np.sum(step1, axis=1))
        step3 = step2 * (1 - fractions)
        interaction_correction = np.sum(step3)

        if self._algo == self.ALGO_TIME_LIMIT:
            elapsed_time = datetime.utcnow() - self._start_time
            speedup_parameter = elapsed_time.total_seconds() / 1800
            avg_int = np.sum(interaction_matrix, axis=None)
            avg_frac = np.average(np.outer(1 - fractions, 1 - fractions))
            average_correction = avg_int * avg_frac

            interaction_correction = average_correction * speedup_parameter \
                + interaction_correction * (1 - speedup_parameter)

        best_case = np.sum(matrix) + np.inner(sums[::-1], fractions - 1) \
            + interaction_correction

        return best_case