Laura Bassi was the first female professor at a European university.  
Despite her immense intellect, she was not always allowed to lecture publicly.

One day a professor with very strong beliefs against women in academia sent some `agents` to find Bassi and end her career.

Help her escape by telling her the safest places in town!

Task

```if:python,javascript
Implement the function `advice(agents, n)` where
```
```if:haskell
Implement the function `advice agents n :: [(Int,Int)] -> Int -> [(Int,Int)]` where
```
```if:swift
Implement the function `advice(agents: [[Int]], n: Int) -> [[Int]]` where
```

- `agents` is an array of agent coordinates.
- `n` defines the size of the city that Bassi needs to hide in, in other words the side length of the square grid. 

The function should return a list of coordinates that are the furthest away (by Manhattan distance) from all agents.

As an example, say you have a 6x6 map, and agents at locations

```python
[(0, 0), (1, 5), (5, 1)]
```

The distances to the nearest agent look like this.



The safest spaces are the ones with distance `4`, marked in bright red. So the function should return

```python
[(2, 2), (3, 3), (4, 4), (5, 5)]
```

in any order.

Edge cases:
- If there is an agent on every grid cell, there is no safe space, so return an empty list.
- If there are no agents, then every cell is a safe spaces, so return all coordinates.
- if `n` is `0`, return an empty list.
- If agent coordinates are outside of the map, they are simply not considered.
- There are no duplicate agents on the same square.

Performance

All reference solutions run in around 6 seconds.
You might not pass the tests if you use a brute-force solution.

```if-not:swift,javascript
There are `200` random tests with `n <= 50`. Inefficient solutions might time out.
```
```if:javascript
There are `300` random tests with `n <= 70`. Inefficient solutions might time out.
```
```if:swift
There are `200` random tests with `n <= 70`. Inefficient solutions might time out.
```

This kata is inspired by ThoughtWorks' coding challenge

Difficulty: EASY
Starter code: def advice(agents, n):
	

Solutions:
def advice(agents, n):
	frontier = {(x, y) for (x, y) in agents if 0 <= x < n and 0 <= y < n}
	bag = {(x, y) for x in range(n) for y in range(n)}
	if frontier == bag:
		return []
	while frontier and bag > frontier:
		bag -= frontier
		frontier = {pos for (x, y) in frontier for pos in ((x + 1, y), (x - 1, y), (x, y + 1), (x, y - 1)) if pos in bag}
	return sorted(bag)


def advice(agents, n):
	agents = [(a[0], a[1]) for a in agents if 0 <= a[0] < n and 0 <= a[1] < n]
	if len(agents) == n * n or n == 0:
		return []
	grid = [[0] * n for _ in range(n)]
	for (ax, ay) in agents:
		grid[ax][ay] = 1
	for (ax, ay) in agents:
		for (dx, dy) in [[1, 0], [0, 1], [-1, 0], [0, -1]]:
			(nx, ny) = (ax + dx, ay + dy)
			if 0 <= nx < n and 0 <= ny < n and (grid[nx][ny] == 0):
				grid[nx][ny] = grid[ax][ay] + 1
				agents.append((nx, ny))
	max_dist = max((d for row in grid for d in row))
	return [(x, y) for x in range(n) for y in range(n) if grid[x][y] == max_dist]


def advice(agents, n):
	agents = [(r, c) for (r, c) in agents if 0 <= r < n and 0 <= c < n]
	if n == 0 or len(set(agents)) == n * n:
		return []
	field = [[0 for _ in range(n)] for _ in range(n)]
	edges = []
	for (r, c) in agents:
		field[r][c] = 1
		edges.append((r, c))
	previous_edges = [(r, c) for r in range(n) for c in range(n)]
	while edges:
		new_edges = []
		for (r, c) in edges:
			for (dr, dc) in [(0, 1), (1, 0), (0, -1), (-1, 0)]:
				(rr, cc) = (r + dr, c + dc)
				if 0 <= rr < n and 0 <= cc < n and (field[rr][cc] == 0):
					field[rr][cc] = 1
					new_edges.append((rr, cc))
		(previous_edges, edges) = (edges, new_edges)
	return previous_edges


from itertools import starmap, product, count

def _complex_to_point(c):
	return (int(c.real), int(c.imag))

def _neighbors(point):
	return (point + 1j ** i for i in range(4))

def advice(agents, n):
	if not n:
		return []
	city = {complex(i, j): None for (i, j) in product(list(range(n)), repeat=2)}
	last = set((agent for agent in starmap(complex, agents) if agent in city))
	if not last:
		return sorted((_complex_to_point(cell) for cell in city))
	for dist in count():
		city.update({cell: dist for cell in last})
		next_cells = set((neighbor for cell in last for neighbor in _neighbors(cell) if neighbor in city and city[neighbor] is None))
		if not next_cells:
			return [] if not dist else sorted((_complex_to_point(cell) for cell in last))
		last = next_cells


from collections import deque
INF = 1000000

class Field:

	def __init__(self, n, agents):
		self.n = n
		self.agents = agents
		self.table = [[INF] * n for _ in range(n)]
		self._queue = deque()
		self._calculate_safety()

	def _calculate_safety(self):
		for (i, j) in self.agents:
			if 0 <= i < self.n and 0 <= j < self.n:
				self._add_to_queue(i, j, 0)
		self._process_queue()

	def _add_to_queue(self, i, j, value):
		self._queue.append((i, j, value))

	def _process_queue(self):
		while self._queue:
			(i, j, value) = self._queue.popleft()
			if self.table[i][j] != INF:
				continue
			self.table[i][j] = value
			if i < self.n - 1:
				self._add_to_queue(i + 1, j, value + 1)
			if i > 0:
				self._add_to_queue(i - 1, j, value + 1)
			if j < self.n - 1:
				self._add_to_queue(i, j + 1, value + 1)
			if j > 0:
				self._add_to_queue(i, j - 1, value + 1)

	@property
	def safest_places(self):
		locations = []
		max_el = 1
		for i in range(self.n):
			for j in range(self.n):
				if self.table[i][j] > max_el:
					max_el = self.table[i][j]
					locations = [(i, j)]
				elif self.table[i][j] == max_el:
					locations.append((i, j))
		return locations

def advice(agents, n):
	field = Field(n, agents)
	return field.safest_places


def advice(agents, n):
	if n <= 0:
		return []
	city = [[-1] * n for _ in range(n)]
	agents = [(i, j) for (i, j) in agents if i >= 0 and j >= 0 and (i < n) and (j < n)]
	d = 0
	while agents:
		t = []
		for (i, j) in agents:
			city[i][j] = d
		d += 1
		for (i, j) in agents:
			for (di, dj) in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
				(x, y) = (i + di, j + dj)
				if x < 0 or y < 0 or x >= n or (y >= n) or (city[x][y] >= 0):
					continue
				city[x][y] = d
				t.append((x, y))
		agents = t
	m = max((max(r) for r in city))
	return [(i, j) for i in range(n) for j in range(n) if city[i][j] == m] if m != 0 else []


def advice(agents, n):
	A = {(i, j) for (i, j) in agents if 0 <= i < n and 0 <= j < n}
	B = {(i, j) for i in range(n) for j in range(n)}
	if A == B:
		return []
	while len(A) > 0 and A != B:
		B = B - A
		A = {(i, j) for (i, j) in B if (i + 1, j) in A or (i - 1, j) in A or (i, j + 1) in A or ((i, j - 1) in A)}
	return sorted(list(B))