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Problem solution · Python

Checking Existence of Edge Length Limited Paths II

Checking Existence of Edge Length Limited Paths II: 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
222 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 Checking Existence of Edge Length Limited Paths II, 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

  • 222 lines of Python from the credited upstream file checking-existence-of-edge-length-limited-paths-ii.py.
  • The implementation visibly relies on sequence storage, hash lookup, ordered lookup.
  • 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 codeChecking Existence of Edge Length Limited Paths II · PythonPython
Use this to learn the idea, then write your own version.
# Time:  ctor:  O(mlogm + m * α(n) + nlogn) ~= O(mlogm + nlogn)#        query: O(α(n) + logn) ~= O(logn)# Space: O(nlogn + m) from functools import partial # Template:# https://github.com/kamyu104/GoogleKickStart-2020/blob/main/Round%20D/locked_doors.pyclass TreeInfos(object):  # Time: O(NlogN), Space: O(NlogN), N is the number of nodes    def __init__(self, children):        def preprocess(curr, parent, weight):            if parent != -1:                W[curr].append(weight)                P[curr].append(parent)  # ancestors of the node i            i = 0            while i < len(P[curr]) and i < len(P[P[curr][i]]):                W[curr].append(max(W[curr][i], W[P[curr][i]][i]))                P[curr].append(P[P[curr][i]][i])                i += 1            C[0] += 1            L[curr] = C[0]  # the subtree of the node i is represented by traversal index L[i]..R[i]         def divide(curr, parent, weight):            stk.append(partial(postprocess, curr))            for child, w in reversed(children[curr]):                if child == parent:                    continue                stk.append(partial(divide, child, curr, w))            stk.append(partial(preprocess, curr, parent, weight))         def postprocess(curr):            R[curr] = C[0]  # the subtree of the node i is represented by traversal index L[i]..R[i]         N = len(children)        L, R, P, W, C = [0]*N, [0]*N, [[] for _ in xrange(N)], [[] for _ in xrange(N)], [-1]        for i in xrange(N):            if L[i]:                continue            stk = []            stk.append(partial(divide, i, -1, 0))            while stk:                stk.pop()()        self.L, self.R, self.P, self.W = L, R, P, W        def is_ancestor(self, a, b):  # includes itself        return self.L[a] <= self.L[b] <= self.R[b] <= self.R[a]        def max_weights(self, a, b):        def binary_lift(a, b):            w = 0            for i in reversed(xrange(len(self.P[a]))):  # O(logN)                if i < len(self.P[a]) and not self.is_ancestor(self.P[a][i], b):                    w = max(w, self.W[a][i])                    a = self.P[a][i]            return max(w, self.W[a][0])         w = 0        if not self.is_ancestor(a, b):            w = max(w, binary_lift(a, b))        if not self.is_ancestor(b, a):            w = max(w, binary_lift(b, a))        return w     class UnionFind(object):  # Time: O(n * α(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:            self.set[stk.pop()] = x        return x     def union_set(self, x, y):        x_root, y_root = map(self.find_set, (x, y))        if x_root == y_root:            return False        if self.rank[x_root] < self.rank[y_root]:  # union by rank            self.set[x_root] = y_root        elif self.rank[x_root] > self.rank[y_root]:            self.set[y_root] = x_root        else:            self.set[y_root] = x_root            self.rank[x_root] += 1        return True  class DistanceLimitedPathsExist(object):     def __init__(self, n, edgeList):        """        :type n: int        :type edgeList: List[List[int]]        """        edgeList.sort(key = lambda x:x[2])        self.__uf = UnionFind(n)        self.__adj = [[] for _ in xrange(n)]        for index, (i, j, weight) in enumerate(edgeList):            if not self.__uf.union_set(i, j):                continue            self.__adj[i].append((j, weight))            self.__adj[j].append((i, weight))        self.__tree_infos = TreeInfos(self.__adj)     def query(self, p, q, limit):        """        :type p: int        :type q: int        :type limit: int        :rtype: bool        """        if self.__uf.find_set(p) != self.__uf.find_set(q):            return False        return self.__tree_infos.max_weights(p, q) < limit  # Time:  ctor:  O(mlogm + m * α(n) * logm) ~= O(mlogm)#        query: O(logm + α(n) * logm) ~= O(logm)# Space: O(n + m * α(n) + m) ~= O(n + m)import collectionsimport sortedcontainersimport bisect  class SnapshotArray(object):     def __init__(self, length):        """        :type length: int        """        self.__snaps = collections.defaultdict(lambda:sortedcontainers.SortedList([(0, 0)]))     def set(self, index, val, snap_id):        """        :type index: int        :type val: int        :rtype: None        """        i = self.__snaps[index].bisect_left((snap_id, float("-inf")))        if i != len(self.__snaps[index]) and self.__snaps[index][i][0] == snap_id:            self.__snaps[index].remove(self.__snaps[index][i])        self.__snaps[index].add((snap_id, val))     def get(self, index, snap_id):        """        :type index: int        :type snap_id: int        :rtype: int        """        i = self.__snaps[index].bisect_left((snap_id+1, float("-inf"))) - 1        return self.__snaps[index][i][1]     class VersionedUnionFind(object):  # Time: O(n * α(n)), Space: O(n)     def __init__(self, n):        self.snap_id = 0        self.set = SnapshotArray(n)        for i in xrange(n):            self.set.set(i, i, self.snap_id)        self.rank = SnapshotArray(n)     def find_set(self, x, snap_id):        stk = []        while self.set.get(x, snap_id) != x:  # path compression            stk.append(x)            x = self.set.get(x, snap_id)        while stk:            self.set.set(stk.pop(), x, snap_id)        return x     def union_set(self, x, y):        x_root = self.find_set(x, self.snap_id)        y_root = self.find_set(y, self.snap_id)        if x_root == y_root:            return False        if self.rank.get(x_root, self.snap_id) < self.rank.get(y_root, self.snap_id):  # union by rank            self.set.set(x_root, y_root, self.snap_id)        elif self.rank.get(x_root, self.snap_id) > self.rank.get(y_root, self.snap_id):            self.set.set(y_root, x_root, self.snap_id)        else:            self.set.set(y_root, x_root, self.snap_id)            self.rank.set(x_root, self.rank.get(x_root, self.snap_id)+1, self.snap_id)        return True     def snap(self):        self.snap_id += 1  class DistanceLimitedPathsExist2(object):     def __init__(self, n, edgeList):        """        :type n: int        :type edgeList: List[List[int]]        """        edgeList.sort(key = lambda x:x[2])        self.__uf = VersionedUnionFind(n)        self.__weights = []        for index, (i, j, weight) in enumerate(edgeList):            if not self.__uf.union_set(i, j):                continue            self.__uf.snap()            self.__weights.append(weight)       def query(self, p, q, limit):        """        :type p: int        :type q: int        :type limit: int        :rtype: bool        """        snap_id = bisect.bisect_left(self.__weights, limit)-1        if snap_id == -1:            return False        return self.__uf.find_set(p, snap_id) == self.__uf.find_set(q, snap_id)

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