Block partition a string

Question

Inspiration.

Consider a list l, consisting of numbers. Define a block operation at index i on the list l to be the act of moving 3 consecutive elements starting from i in l to the end.

Example:

l, i (1-indexing) -> l (after applying block operation at index i)
[1,2,3,4,5], 1 -> [4,5,1,2,3]
[1,2,3,4,5,6,7], 3 -> [1,2,6,7,3,4,5]

Given a list consisting of only 0 and 1, your challenge is to partition it so that zeros are at the front, and ones are at the back, using only block operations. Output should be the indices in the order they are applied on the list.

Because this is impossible for the list [1,0,1,0], the list length is guaranteed to be at least 5.

Test cases (1-indexing)

(there are other valid outputs)

[1,1,1,0,0] -> [1]
[0,1,0,1,0] -> [1,2,1,1]
[0,0,0,1,1,1,0,0,0] -> [4]

Use this script to generate more test cases. (only input. The rplc ' ';',' part is used to replace spaces with commas in the output)

Winning criteria

code-challenge is the main winning criteria, and fastest-code is the tie-breaker. In particular:

• The solution with the shortest output length (least number of block operations) with the test case (n_elem = 500, random_seed = {secret value}) wins. You should be able to run your solution to completion with the test case (n_elem = 500, random_seed = 123456).
• In case of ties, the solution that can handles the largest power-of-2 value of n_elem with random_seed = {secret value} within 10 seconds (for me) wins.
• In case of ties, the solution that takes less time on that test case wins.

Answer 1

Python 3, (0.397 n + 3.58) steps

1000-point polynomial regression by numpy.polyfit.

---

• Number of steps for version 1: 0.0546 n² + 2.80 n - 221
• Number of steps for version 2: 0.0235 n² + 0.965 n - 74
• Number of steps for version 3: 0.00965 n² + 2.35 n - 111
• Number of steps for version 4: 1.08 n - 36.3
• Number of steps for version 5: 0.397 n + 3.58

---

• Secret test case score for version 1: 14468
• Secret test case score for version 2: 5349
• Secret test case score for version 3: 4143
• Secret test case score for version 4: 450
• Secret test case score for version 5: 205

---

def partite(L):
	endgame5 = [9,9,1,9,0,0,1,9,
		0,1,0,1,0,1,1,9,
		0,0,1,0,0,0,1,0,
		0,0,0,1,0,0,0,9]
	endgame6 = [9,9,2,9,1,1,2,9,0,2,0,0,1,2,2,9,
		0,1,2,1,0,1,2,1,0,1,0,2,1,1,0,9,
		0,0,1,0,1,0,1,1,0,0,0,0,1,1,1,1,
		0,0,2,2,0,0,2,2,0,0,0,0,0,0,0,9]
	endgame = [9,9,3,9,2,2,3,9,1,0,3,0,2,0,3,9,0,1,3,3,2,2,3,0,1,0,1,0,2,1,0,9,
		0,0,2,1,0,0,2,2,1,0,1,2,0,0,0,2,0,1,3,3,3,3,3,0,1,1,1,1,1,3,0,9,
		0,0,0,0,1,0,1,1,1,0,3,0,1,0,1,0,0,1,0,0,1,1,0,0,1,0,0,0,2,0,1,0,
		0,0,2,0,0,0,2,0,0,0,2,0,0,0,2,0,0,0,3,0,3,0,3,0,3,0,2,3,3,0,0,9]
	offset = 1
	steps = []
	def update(L,steps,ind):
		steps.append(offset + ind)
		if 0 <= ind and ind+3 < len(L):
			return (steps,L[:ind]+L[ind+3:]+L[ind:ind+3])
		else:
			print(offset,ind,L)
			raise
	if len(L) == 5:
		while endgame5[L[0]*16+L[1]*8+L[2]*4+L[3]*2+L[4]] != 9:
			steps, L = update(L,steps,endgame5[L[0]*16+L[1]*8+L[2]*4+L[3]*2+L[4]])
		return steps
	if len(L) == 6:
		while endgame6[L[0]*32+L[1]*16+L[2]*8+L[3]*4+L[4]*2+L[5]] != 9:
			steps, L = update(L,steps,endgame6[L[0]*32+L[1]*16+L[2]*8+L[3]*4+L[4]*2+L[5]])
		return steps
	if 1 not in L:
		return []
	while len(L) > 7 and 0 in L:
		wf_check = len(L)
		while L[0] != 0:
			pos = [-1]
			wf_check2 = -1
			while True:
				i = pos[-1]+1
				while i < len(L):
					if L[i] == 0:
						pos.append(i)
						i += 1
					else:
						i += 3
				assert len(pos) > wf_check2
				wf_check2 = len(pos)
				space = (pos[-1]-len(L)+1)%3
				ind = -1
				tail = pos.pop()
				i = len(L)-1
				while i >= 0:
					if tail == i:
						while tail == i:
							i -= 1
							tail = pos.pop() if pos else -1
						i -= 2
					elif i < len(L)-3 and L[i+space] == 0:
						ind = i
						break
					else:
						i -= 1
				if ind == -1:
					break
				steps, L = update(L, steps, ind)
				pos = pos or [-1]
			if L[0] == 0:
				break
			pos = [-1]
			while L[0] != 0:
				pos = [-1]
				found = False
				for i in range(1,len(L)):
					if L[i] == 0:
						if i%3 == (pos[-1]+1)%3:
							pos.append(i)
						else:
							found = found or i
				if found > len(L)-4:
					found = False
				if not found:
					break
				triple = []
				for i in range(1,len(L)-1):
					if L[i-1] == 1 and L[i] == 1 and L[i+1] == 1:
						triple.append(i)
					if len(triple) > 3:
						break
				space = (pos[-1]-len(L)+1)%3
				if space == 0:
					if found >= 2 and found-2 not in pos and found-1 not in pos:
						# ... _ 1 _ [0] 0 ...
						if found-2 in triple:
							triple.remove(found-2)
						if found-3 in triple:
							triple.remove(found-3)
						if L[-1] == 1:
							steps, L = update(L, steps, found-2)
							steps, L = update(L, steps, len(L)-4)
							steps, L = update(L, steps, len(L)-4)
						elif triple:
							steps, L = update(L, steps, found-2)
							if found < triple[0]:
								triple[0] -= 3
							steps, L = update(L, steps, triple[0]-1)
							steps, L = update(L, steps, len(L)-4)
						else:
							break
						assert L[-3] == 0
					elif found >= 1 and found-1 not in pos and found+1 not in pos:
						# ... _ 1 [0] _ 0 ...
						if found-2 in triple:
							triple.remove(found-2)
						if L[-2] == 1 and L[-1] == 1:
							steps, L = update(L, steps, found-1)
							steps, L = update(L, steps, len(L)-5)
							steps, L = update(L, steps, len(L)-5)
						elif triple:
							steps, L = update(L, steps, found-1)
							if found < triple[0]:
								triple[0] -= 3
							steps, L = update(L, steps, triple[0]-1)
							steps, L = update(L, steps, len(L)-5)
						elif L[-1] == 1:
							steps, L = update(L, steps, found-1)
							steps, L = update(L, steps, len(L)-4)
							steps, L = update(L, steps, len(L)-4)
							steps, L = update(L, steps, len(L)-4)
						else:
							break
						assert L[-3] == 0
					else:
						break
				elif space == 1:
					# ... 1 1 [0] 0 ...
					if found >= 2 and found-2 not in pos and found-1 not in pos:
						if found-2 in triple:
							triple.remove(found-2)
						if found-3 in triple:
							triple.remove(found-3)
						if triple:
							steps, L = update(L, steps, found-2)
							if found < triple[0]:
								triple[0] -= 3
							steps, L = update(L, steps, triple[0]-1)
							steps, L = update(L, steps, len(L)-5)
						elif L[-2] == 1 and L[-1] == 1:
							steps, L = update(L, steps, found-2)
							steps, L = update(L, steps, len(L)-4)
							steps, L = update(L, steps, len(L)-5)
						else:
							break
						assert L[-2] == 0
					else:
						break
				else:
					if found+1 not in pos and found+2 not in pos:
						# ... 0 [0] _ 1 _ ...
						if found+2 in triple:
							triple.remove(found+2)
						if found+3 in triple:
							triple.remove(found+3)
						if L[-2] == 1 and L[-1] == 1:
							steps, L = update(L, steps, found)
							steps, L = update(L, steps, len(L)-5)
						elif L[-1] == 1:
							steps, L = update(L, steps, found)
							steps, L = update(L, steps, len(L)-4)
							steps, L = update(L, steps, len(L)-4)
						elif triple:
							steps, L = update(L, steps, triple[0]-1)
							if triple[0] < found:
								found -= 3
							steps, L = update(L, steps, found)
							steps, L = update(L, steps, len(L)-5)
						else:
							break
						assert L[-1] == 0
					elif found >= 1 and found-1 not in pos and found+1 not in pos:
						# ... 0 _ [0] 1 _ ...
						if found+2 in triple:
							triple.remove(found+2)
						if L[-1] == 1:
							steps, L = update(L, steps, found-1)
							steps, L = update(L, steps, len(L)-4)
						elif triple:
							steps, L = update(L, steps, triple[0]-1)
							if triple[0] < found:
								found -= 3
							steps, L = update(L, steps, found-1)
							steps, L = update(L, steps, len(L)-4)
						else:
							break
						assert L[-1] == 0
					else:
						break
			if L[0] == 0:
				break
			if 0 in L[::3]:
				assert L[::3].index(0) < wf_check
				wf_check = L[::3].index(0)
			steps, L = update(L, steps, 0)
		assert L[0] == 0
		offset += L.index(1)
		del L[:L.index(1)]
		continue
	if 0 in L:
		offset -= 7-len(L)
		L = [0]*(7-len(L))+L
		assert(len(L) == 7)
		while endgame[L[0]*64+L[1]*32+L[2]*16+L[3]*8+L[4]*4+L[5]*2+L[6]] != 9:
			steps, L = update(L,steps,endgame[L[0]*64+L[1]*32+L[2]*16+L[3]*8+L[4]*4+L[5]*2+L[6]])
	return steps

Try it online!

Answer 2

Python 3, ~179 steps for n=500 (on average)

A heuristic greedy approach. Kinda slow but still works. Uses an optimal solver for small sizes.

def incomplete_groups(l):
    r = 0
    ones = 0
    for x in l:
        if x == "1":
            ones += 1
        else:
            if ones % 3:
                r += 1
            ones = 0
    # Ones at the end don't count as an incomplete group.

    return r

def move(l, i):
    return l[:i] + l[i+3:] + l[i:i+3]

def best_pos(l, hist):
    r = []
    cleanup = incomplete_groups(l) == 0

    candidates = []
    for i in range(len(l) - 3):
        block = l[i:i+3]
        if block == "111" and cleanup:
            return i
        elif block == "111":
            continue

        new = move(l, i)
        bad_start = i < 3 and "10" in l[:3]
        candidates.append((new not in hist, -incomplete_groups(new), bad_start, block != "000", i))

    candidates.sort(reverse=True)
    return candidates[0][-1]

def done(l):
    return list(l) == sorted(l)



class IDAStar:
    def __init__(self, h, neighbours):
        """ Iterative-deepening A* search.

        h(n) is the heuristic that gives the cost between node n and the goal node. It must be admissable, meaning that h(n) MUST NEVER OVERSTIMATE the true cost. Underestimating is fine.

        neighbours(n) is an iterable giving a pair (cost, node, descr) for each node neighbouring n
        IN ASCENDING ORDER OF COST. descr is not used in the computation but can be used to
        efficiently store information about the path edges (e.g. up/left/right/down for grids).
        """

        self.h = h
        self.neighbours = neighbours
        self.FOUND = object()


    def solve(self, root, is_goal, max_cost=None):
        """ Returns the shortest path between the root and a given goal, as well as the total cost.
        If the cost exceeds a given max_cost, the function returns None. If you do not give a
        maximum cost the solver will never return for unsolvable instances."""

        self.is_goal = is_goal
        self.path = [root]
        self.is_in_path = {root}
        self.path_descrs = []
        self.nodes_evaluated = 0

        bound = self.h(root)

        while True:
            t = self._search(0, bound)
            if t is self.FOUND: return self.path, self.path_descrs, bound, self.nodes_evaluated
            if t is None: return None
            bound = t

    def _search(self, g, bound):
        self.nodes_evaluated += 1

        node = self.path[-1]
        f = g + self.h(node)
        if f > bound: return f
        if self.is_goal(node): return self.FOUND

        m = None # Lower bound on cost.
        for cost, n, descr in self.neighbours(node):
            if n in self.is_in_path: continue

            self.path.append(n)
            self.is_in_path.add(n)
            self.path_descrs.append(descr)
            t = self._search(g + cost, bound)

            if t == self.FOUND: return self.FOUND
            if m is None or (t is not None and t < m): m = t

            self.path.pop()
            self.path_descrs.pop()
            self.is_in_path.remove(n)

        return m

def h(l):
    """Number of groups of 1 with length <= 3 that come before a zero."""
    h = 0
    num_ones = 0
    complete_groups = 0
    incomplete_groups = 0
    for x in l:
        if x == "1":
            num_ones += 1
        else:
            while num_ones > 3:
                num_ones -= 3
                h += 1
                complete_groups += 1
            if num_ones > 0:
                h += 1
                incomplete_groups += 1
            num_ones = 0

    return complete_groups + incomplete_groups

def neighbours(l):
    inc_groups = incomplete_groups(l)
    final = inc_groups == 0

    candidates = []
    for i in range(len(l) - 3):
        left = l[:i]
        block = l[i:i+3]
        right = l[i+3:]
        cand = (1, left + right + block, i)

        # Optimal choice.
        if final and (block != "111" or i >= len(l.rstrip("1"))):
            continue

        candidates.append(cand)

    candidates.sort(key=lambda c: c[2], reverse=True)

    return candidates


def is_goal(l):
    return all(l[i] <= l[i+1] for i in range(len(l)-1))

opt_solver = IDAStar(h, neighbours)

def partite(l):
    if isinstance(l, list):
        l = "".join(map(str, l))
    if len(l) < 10:
        return [i + 1 for i in opt_solver.solve(l, is_goal)[1]]
    moves = []
    hist = [l]
    while not done(l):
        i = best_pos(l, hist)
        l = move(l, i)
        moves.append(i+1)
        hist.append(l)
    return moves