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Problem solution · Python

Maximum Partition Factor

Maximum Partition Factor: a Python solution using disjoint set union. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Disjoint set union
Source
Kamyu LeetCode Solutions
Length
206 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Disjoint set union

For Maximum Partition Factor, the implementation maintains connected components and merges them as relationships are processed.

  1. Give each element a component representative.
  2. Merge representatives when a connection is accepted.
  3. Answer connectivity or component queries from the compressed representatives.

Code notes

  • 206 lines of Python from the credited upstream file maximum-partition-factor.py.
  • The implementation visibly relies on sequence storage.
  • No explicit loop blocks detected.

Complexity

Account for every find and union operation; with path compression and ranked merging, the amortized cost is nearly constant per operation.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMaximum Partition Factor · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(n^2 * logn)# Space: O(n^2) # greedy, sort, union find with parityclass Solution(object):    def maxPartitionFactor(self, points):        """        :type points: List[List[int]]        :rtype: int        """        class UnionFind(object):  # Time: O(n * alpha(n)), Space: O(n)            def __init__(self, n):                self.set = range(n)                self.rank = [0]*n                self.parity = [0]*n  # added             def find_set(self, x):                stk = []                while self.set[x] != x:  # path compression                    stk.append(x)                    x = self.set[x]                while stk:                    y = stk.pop()                    self.parity[y] ^= self.parity[self.set[y]]  # added                    self.set[y] = x                return x             def union_set(self, x, y):                ox, oy = x, y  # added                x, y = self.find_set(x), self.find_set(y)                if x == y:                    return self.parity[ox] != self.parity[oy]  # modified                if self.rank[x] > self.rank[y]:  # union by rank                    x, y = y, x                    ox, oy = oy, ox  # added                if self.rank[x] == self.rank[y]:                    self.rank[y] += 1                self.set[x] = self.set[y]                self.parity[x] = self.parity[ox]^self.parity[oy]^1  # added                return True         def dist(u, v):            return abs(points[u][0]-points[v][0])+abs(points[u][1]-points[v][1])         sorted_dists = sorted((dist(u, v), u, v) for u in xrange(len(points)) for v in xrange(u+1, len(points)))        uf = UnionFind(len(points))        return next((d for d, u, v in sorted_dists if not uf.union_set(u, v)), 0)  # Time:  O(n^2 * logn)# Space: O(n^2)# sort, union findclass Solution2(object):    def maxPartitionFactor(self, points):        """        :type points: List[List[int]]        :rtype: int        """        class UnionFind(object):  # Time: O(n * alpha(n)), Space: O(n)            def __init__(self, n):                self.set = range(n)                self.rank = [0]*n             def find_set(self, x):                stk = []                while self.set[x] != x:  # path compression                    stk.append(x)                    x = self.set[x]                while stk:                    y = stk.pop()                    self.set[y] = x                return x             def union_set(self, x, y):                x, y = self.find_set(x), self.find_set(y)                if x == y:                    return False                if self.rank[x] > self.rank[y]:  # union by rank                    x, y = y, x                if self.rank[x] == self.rank[y]:                    self.rank[y] += 1                self.set[x] = self.set[y]                return True         def dist(u, v):            return abs(points[u][0]-points[v][0])+abs(points[u][1]-points[v][1])         sorted_dists = sorted((dist(u, v), u, v) for u in xrange(len(points)) for v in xrange(u+1, len(points)))        uf = UnionFind(len(points))        lookup = [-1]*len(points)        for d, u, v in sorted_dists:            if uf.find_set(u) == uf.find_set(v):                return d            if lookup[u] != -1:                uf.union_set(lookup[u], v)            else:                lookup[u] = v            if lookup[v] != -1:                uf.union_set(lookup[v], u)            else:                lookup[v] = u        return 0  # Time:  O(n^2 * logn)# Space: O(n^2)# binary search, bfs, coordinate compressionclass Solution3(object):    def maxPartitionFactor(self, points):        """        :type points: List[List[int]]        :rtype: int        """        INF = float("inf")        def binary_search_right(left, right, check):            while left <= right:                mid = left+(right-left)//2                if not check(mid):                    right = mid-1                else:                    left = mid+1            return right         def dist(u, v):            return abs(points[u][0]-points[v][0])+abs(points[u][1]-points[v][1])         def is_bipartite(d):            def bfs(u):                if lookup[u] != -1:                    return True                lookup[u] = 0                q = [u]                while q:                    new_q = []                    for u in q:                        for v in xrange(len(points)):                            if not (v != u and dist(v, u) < d):                                continue                            if lookup[v] != -1:                                if lookup[v] != lookup[u]^1:                                    return False                                continue                            lookup[v] = lookup[u]^1                            new_q.append(v)                    q = new_q                return True              lookup = [-1]*len(points)            return all(bfs(u) for u in xrange(len(points)))         sorted_dists = sorted({dist(u, v) for u in xrange(len(points)) for v in xrange(u+1, len(points))}|{INF})        left, right = 0, len(sorted_dists)-1        result = binary_search_right(left, right, lambda i: is_bipartite(sorted_dists[i]))        return sorted_dists[result] if sorted_dists[result] != INF else 0  # Time:  O(n^2 * logr)# Space: O(n)# binary search, bfsclass Solution4(object):    def maxPartitionFactor(self, points):        """        :type points: List[List[int]]        :rtype: int        """        def binary_search_right(left, right, check):            while left <= right:                mid = left+(right-left)//2                if not check(mid):                    right = mid-1                else:                    left = mid+1            return right         def dist(u, v):            return abs(points[u][0]-points[v][0])+abs(points[u][1]-points[v][1])         def is_bipartite(d):            def bfs(u):                if lookup[u] != -1:                    return True                lookup[u] = 0                q = [u]                while q:                    new_q = []                    for u in q:                        for v in xrange(len(points)):                            if not (v != u and dist(v, u) < d):                                continue                            if lookup[v] != -1:                                if lookup[v] != lookup[u]^1:                                    return False                                continue                            lookup[v] = lookup[u]^1                            new_q.append(v)                    q = new_q                return True              lookup = [-1]*len(points)            return all(bfs(u) for u in xrange(len(points)))         mx = max(dist(u, v) for u in xrange(len(points)) for v in xrange(u+1, len(points)))        left, right = 0, mx+1        result = binary_search_right(left, right, is_bipartite)        return result if result != mx+1 else 0

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