def cluster_distance(cluster1: Cluster,
                    cluster2: Cluster,
                    distance_agg: Callable = min) -> float:
    """Compute all the pairwise distances between cluster1 and cluster2 and apply the aggregation function
    _distance_agg_ to the resulting list"""
    
    return distance_agg([distance(v1,v2)]
                       for v1 in get_values(cluster1)
                       for v2 in get_values(cluster2))
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def knn_classify(k: int, labeled_points: List[Vector],
                 new_point: Vector) -> str:
    # Order the labeled points from nearest to farthest
    by_distance = sorted(labeled_points,
                         key=lambda lp: distance(lp.point, new_point))

    # Find the labels for the k closest
    k_nearest_labels = [lp.label for lp in by_distance[:k]]

    # and let them vote
    return majority_vote(k_nearest_labels)
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def find_eigenvector(A, tolerance=0.00001):
    guess = [random.random() for _ in A]

    while True:
        result = matrix_operate(A, guess)
        length = magnitude(result)
        next_guess = scalar_multiply(1 / length, result)

        if distance(guess, next_guess) < tolerance:
            return next_guess, length  # eigenvector, eigenvalue

        guess = next_guess
def find_eigenvector(A, tolerance=0.00001):
    guess = [random.random() for _ in A]

    while True:
        result = matrix_operate(A, guess)
        length = magnitude(result)
        next_guess = scalar_multiply(1/length, result)

        if distance(guess, next_guess) < tolerance:
            return next_guess, length               # eigenvector, eigenvalue

        guess = next_guess
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def knn_classify(k, labeled_points, new_point):
    """each labeled point should be a pair (point, label)"""

    # order the labeled points from nearest to farthest
    by_distance = sorted(labeled_points,
                         key=lambda (point, _): distance(point, new_point))

    # find the labels for the k closest
    k_nearest_labels = [label for _, label in by_distance[:k]]

    # and let them vote
    return majority_vote(k_nearest_labels)
def knn_classify(k, labeled_points, new_point):
    """each labeled point should be a pair (point, label)"""
    
    # order the labeled points from nearest to farthest
    by_distance = sorted(labeled_points,
                         key=lambda (point, _): distance(point, new_point))
    
    # find the labels for the k closest
    k_nearest_labels = [label for _, label in by_distance[:k]]
    
    # and let them vote
    return majority_vote(k_nearest_labels)
def find_eigenvector(m: Matrix, tolerance: float = 0.00001) -> Tuple[Vector, float]:
    guess = [random.random() for _ in m]
    
    while True:
        result = matrix_times_vector(m, guess) # transform the guess
        norm = magnitude(result) # compute the norm
        next_guess = [x/norm for x in result] # rescale
        
        if distance(guess, next_guess) < tolerance:
            # convergence so return (eigenvector, eigenvalue)
            return next_guess, norm
        
        guess = next_guess

        
    return 1 if (i,j) in friend_pairs or (j,i) in friend_pairs else 0
import random
import sys

sys.path.insert(0, "../linear_algebra")
from vector_operations import distance

def step(v, direction, step_size):
    """move step_size in the direction from v"""
    return [v_i + step_size * direction_i
            for v_i, direction_i in zip(v, direction)]

def sum_of_squares_gradient(v):
    return [2 * v_i for v_i in v]

# pick a random starting point
v = [random.randint(-10, 10) for i in range(3)]

tolerance = 0.0000001

while True:
    gradient = sum_of_squares_gradient(v)       # compute the gradient at v
    next_v = step(v, gradient, -0.01)           # take a negative gradient step
    if distance(next_v, v) < tolerance:         # stop if we're converging
        break
    v = next_v

print v
    return [2 * v_i for v_i in v]


# pick a random starting point
v = [random.uniform(-10, 10) for i in range(3)]

for epoch in range(1000):
    # compute gradient
    grad = sum_of_squares_gradient(v)
    # update vector in the gradient direction
    v = gradient_step(
        v, grad,
        step_size=-0.01)  # take a step in the negative gradient direction
    print(epoch, v)

print("Distance = ", distance(v, [0, 0, 0]))

assert distance(v, [0, 0, 0]) < 0.001  # v should be close to zero
"""Using the gradient descent algorith to find the slope and intercept of a linear equation"""
# Create a linear equation with know parameters (slope = 20, intercept = 5)
input = [(x, 20 * x + 5) for x in range(-50, 50)]


def linear_gradient(x: float, y: float, theta: Vector) -> Vector:
    slope, intercept = theta
    predicted = slope * x + intercept  # prediction of a linear model
    error = (predicted - y)
    squared_error = error**2  # minimize squared error
    grad = [2 * error * x, 2 * error]  # using its gradient
    return grad
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def random_distance(dim: int, num_pairs: int) -> List[float]:
    return [
        distance(random_point(dim), random_point(dim))
        for _ in range(num_pairs)
    ]