#3367

Hard

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Maximize Sum of Weights after Edge Removals

Maximize the sum of edge weights in a tree after removals, using dynamic programming and tree traversal techniques.

dynamic programming tree depth first search

Problem Statement

You are given an undirected tree with n nodes numbered from 0 to n - 1. A 2D array edges contains n - 1 elements, where each element [ui, vi, wi] indicates an edge between nodes ui and vi with weight wi.

Your task is to remove zero or more edges to maximize the sum of the remaining edge weights after the removals. The problem requires you to determine the optimal set of removals to achieve this maximum sum.

Examples

Example 1

Input: edges = [[0,1,4],[0,2,2],[2,3,12],[2,4,6]], k = 2

Output: 22

Example 2

Input: edges = [[0,1,5],[1,2,10],[0,3,15],[3,4,20],[3,5,5],[0,6,10]], k = 3

Output: 65

Constraints

2 <= n <= 105
1 <= k <= n - 1
edges.length == n - 1
edges[i].length == 3
0 <= edges[i][0] <= n - 1
0 <= edges[i][1] <= n - 1
1 <= edges[i][2] <= 106
The input is generated such that edges form a valid tree.

Solution Approach

Depth-First Search (DFS) Traversal

Start by using DFS to traverse the tree, calculating the maximum possible weight sums in subtrees after considering potential edge removals. This will allow you to process each node while maintaining the sum of its subtree's edge weights.

Dynamic Programming State Tracking

Use dynamic programming to track the optimal solution at each node. For each edge, decide whether it should be removed based on its contribution to the total weight, considering the subtree's edge sums and potential removals.

Greedy Edge Removal Strategy

While traversing the tree, prioritize removing edges that provide the largest reduction in weight, and use dynamic programming to adjust the remaining edge sums accordingly.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the final approach, but generally, it will be O(n) due to the DFS traversal of the tree. Space complexity is also O(n), considering the storage required for dynamic programming states and the tree structure.

What Interviewers Usually Probe

Assesses problem-solving ability using tree traversal and dynamic programming.
Evaluates candidate's understanding of DFS-based approaches and state tracking in optimization problems.
Tests knowledge of tree structure manipulation for performance optimization.

Common Pitfalls or Variants

Common pitfalls

Incorrectly tracking subtree sums during DFS, leading to suboptimal edge removal decisions.
Failing to consider all possible edges for removal and missing the maximum possible sum.
Overcomplicating the solution, resulting in a slower-than-necessary approach.

Follow-up variants

Consider different tree shapes (e.g., skewed trees) and how they might affect the traversal strategy.
Varying the number of edge removals (k) to test scalability of the solution.
Adding constraints like weighted nodes instead of edges to explore tree manipulation complexities.

FAQ

What is the core idea behind maximizing sum of weights in this problem?

The core idea is to traverse the tree using DFS, dynamically tracking the weight sums while making optimal edge removal decisions.

How does dynamic programming help in this problem?

Dynamic programming is used to track the maximum sum achievable at each node, considering the best edge removal strategies for each subtree.

Can DFS be used to solve this problem?

Yes, DFS is the most natural traversal method for this problem, as it allows for efficient computation of subtree sums and edge removal decisions.

What is a common mistake in solving this problem?

A common mistake is failing to track the maximum sum correctly during DFS traversal, which leads to incorrect edge removal decisions.

How can GhostInterview help with this problem?

GhostInterview helps break down the DFS traversal and dynamic programming transitions, guiding you through the correct edge removal strategy to maximize the sum.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def maximizeSumOfWeights(self, edges: List[List[int]], k: int) -> int:
        def dfs(u: int, fa: int) -> Tuple[int, int]:
            s = 0
            t = []
            for v, w in g[u]:
                if v == fa:
                    continue
                a, b = dfs(v, u)
                s += a
                if (d := (w + b - a)) > 0:
                    t.append(d)
            t.sort(reverse=True)
            return s + sum(t[:k]), s + sum(t[: k - 1])

        n = len(edges) + 1
        g: List[List[Tuple[int, int]]] = [[] for _ in range(n)]
        for u, v, w in edges:
            g[u].append((v, w))
            g[v].append((u, w))
        x, y = dfs(0, -1)
        return max(x, y)

Continue Practicing

#3241 time taken to mark all nodes #3544 subtree inversion sum #3559 number of ways to assign edge weights ii