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Problem solution · Python

Total Sum of Interaction Cost in Tree Groups

Total Sum of Interaction Cost in Tree Groups: a Python solution using breadth-first search. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Breadth-first search
Source
Kamyu LeetCode Solutions
Length
133 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

Breadth-first search

For Total Sum of Interaction Cost in Tree Groups, the implementation explores reachable states in layers, which is the standard shape for unweighted shortest paths and minimum-step transitions.

  1. Model each valid configuration as a state and each legal move as an edge.
  2. Seed the queue with the starting state and mark it immediately.
  3. Expand each state once, recording distance or reachability for unseen neighbours.

Code notes

  • 133 lines of Python from the credited upstream file total-sum-of-interaction-cost-in-tree-groups.py.
  • The implementation visibly relies on sequence storage, hash lookup.
  • No explicit loop blocks detected, together with recursive traversal.

Complexity

Verify that each state and transition is processed only a bounded number of times; that determines the traversal cost.

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 codeTotal Sum of Interaction Cost in Tree Groups · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(n * g)# Space: O(n * g) # bfsclass Solution(object):    def interactionCosts(self, n, edges, group):        """        :type n: int        :type edges: List[List[int]]        :type group: List[int]        :rtype: int        """        def bfs():            order, parent = [0], [-1]*len(adj)            for u in order:                for v in adj[u]:                    if v == parent[u]:                        continue                    parent[v] = u                    order.append(v)            return order, parent         adj = [[] for _ in xrange(n)]        for u, v in edges:            adj[u].append(v)            adj[v].append(u)        mx = max(group)        total = [0]*mx        for x in group:            total[x-1] += 1        result = 0        order, parent = bfs()        cnt = [[0]*mx for _ in xrange(n)]        for u in reversed(order):            cnt[u][group[u]-1] += 1            for v in adj[u]:                if u != parent[v]:                    continue                for k in xrange(len(cnt[v])):                    result += cnt[v][k]*(total[k]-cnt[v][k])                    cnt[u][k] += cnt[v][k]        return result  # Time:  O(nlogn)# Space: O(n)import collections  # bfs, small-to-large mergingclass Solution2(object):    def interactionCosts(self, n, edges, group):        """        :type n: int        :type edges: List[List[int]]        :type group: List[int]        :rtype: int        """        def bfs():            order, parent = [0], [-1]*len(adj)            for u in order:                for v in adj[u]:                    if v == parent[u]:                        continue                    parent[v] = u                    order.append(v)            return order, parent         adj = [[] for _ in xrange(n)]        for u, v in edges:            adj[u].append(v)            adj[v].append(u)        total = collections.defaultdict(int)        for x in group:            total[x] += 1        result = 0        order, parent = bfs()        cnt = [collections.defaultdict(int) for _ in xrange(n)]        for u in reversed(order):            cnt[u][group[u]] += 1            for v in adj[u]:                if u != parent[v]:                    continue                for k, c in cnt[v].iteritems():                    result += c*(total[k]-c)                if len(cnt[v]) > len(cnt[u]):                    cnt[u], cnt[v] = cnt[v], cnt[u]                for k, c in cnt[v].iteritems():                    cnt[u][k] += c                cnt[v].clear()        return result  # Time:  O(nlogn)# Space: O(n)import collections  # dfs, small-to-large mergingclass Solution3(object):    def interactionCosts(self, n, edges, group):        """        :type n: int        :type edges: List[List[int]]        :type group: List[int]        :rtype: int        """        def dfs(u, p):            cnt = collections.defaultdict(int)            cnt[group[u]] += 1            for v in adj[u]:                if v == p:                    continue                new_cnt = dfs(v, u)                for k, c in new_cnt.iteritems():                    result[0] += c*(total[k]-c)                if len(new_cnt) > len(cnt):                    cnt, new_cnt = new_cnt, cnt                for k, c in new_cnt.iteritems():                    cnt[k] += c            return cnt         adj = [[] for _ in xrange(n)]        for u, v in edges:            adj[u].append(v)            adj[v].append(u)        total = collections.defaultdict(int)        for x in group:            total[x] += 1        result = [0]        dfs(0, -1)        return result[0]

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