How do you solve Find the Maximum Sum of Node Values on LeetCode?

Find the Maximum Sum of Node Values (LeetCode #3068) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(n) space complexity. Key insight: XOR operation can change the values significantly, and careful selection of edges can maximize the sum.

What is the brute force solution for Find the Maximum Sum of Node Values?

The brute force approach for Find the Maximum Sum of Node Values is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves trying every possible combination of edges and calculating the resulting sums. This is straightforward but inefficient as it explores all combinations, leading to high computational costs. The optimal Optimal Solution approach improves this to O(n).

What are the interview tips for Find the Maximum Sum of Node Values?

Understand the properties of XOR and how it affects numbers. Be prepared to discuss the implications of tree traversal methods like DFS. Practice explaining your thought process clearly, especially when discussing dynamic programming.

What are common mistakes when solving Find the Maximum Sum of Node Values?

Not considering the effect of multiple XOR operations on the same node. Failing to account for the tree structure when calculating sums.

#3068

Find the Maximum Sum of Node Values

Hard

ArrayDynamic ProgrammingGreedyBit ManipulationTreeSortingDynamic ProgrammingTree Traversal

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(n)

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Intuition

Time O(n)Space O(n)

The optimal solution uses dynamic programming to efficiently compute the maximum sum by considering the effect of each edge on the subtree sums. By leveraging the tree structure, we can avoid redundant calculations.

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Algorithm

3 steps

1Step 1: Construct the tree from the edges.
2Step 2: Use DFS to calculate the maximum sum for each subtree, considering the XOR operation with k.
3Step 3: For each node, calculate the maximum sum with and without the XOR operation, and propagate the results up the tree.

solution.py21 lines

1# Full working Python code
2from collections import defaultdict
3
4def max_sum_optimal(nums, k, edges):
5    tree = defaultdict(list)
6    for u, v in edges:
7        tree[u].append(v)
8        tree[v].append(u)
9
10    def dfs(node, parent):
11        total = nums[node]
12        total_with_xor = nums[node] ^ k
13        subtree_sum = 0
14        for neighbor in tree[node]:
15            if neighbor == parent:
16                continue
17            child_sum = dfs(neighbor, node)
18            subtree_sum += child_sum
19        return max(total + subtree_sum, total_with_xor + subtree_sum)
20
21    return dfs(0, -1)

ℹ

Complexity note: The complexity is linear because we traverse each node once in the DFS, and the space complexity is also linear due to the storage of the tree structure.

1XOR operation can change the values significantly, and careful selection of edges can maximize the sum.
2The tree structure allows for efficient traversal and computation of subtree sums.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.