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Post-order DFS was the right call and I got there, but explaining why I needed to track the downward gain separately from the global max took me longer than it should have.
Use a post-order DFS to compute the maximum path sum. At each node, calculate the maximum gain from its left and right subtrees, update the global maximum with the sum of the node's value plus both gains, and return the node's value plus the larger gain to its parent.
Pro tip: Emphasize that the global maximum must be updated at every node, not just at the root, and discuss how to handle negative values by taking max(0, gain) to avoid including negative paths.
Confirm that a path can start and end at any node, must follow parent-child connections, and each node is used at most once. Ask if the tree can be empty or contain negative values.
Define a helper function that returns the maximum gain (sum) from a subtree rooted at a given node, considering only paths that go straight down. This gain is the node's value plus the maximum of 0, left gain, and right gain.
At each node, compute the sum of the node's value plus the left gain and right gain (if positive). Update the global maximum if this sum is larger.
If the node is null, return 0. When computing gains, use max(0, gain) to ignore negative contributions, ensuring the path sum is maximized.
Explain that each node is visited once, so time complexity is O(n). Space complexity is O(h) for recursion stack, where h is tree height.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Explain that you would augment the DP state to store the predecessor (or the action taken) that led to the optimal value at each state, then backtrack from the final state to reconstruct the path. Emphasize that this adds O(1) extra space per state and O(path length) time for reconstruction, without changing the overall time complexity.
Pro tip: Mention that you can often avoid storing full paths in each state by storing only the decision (e.g., which neighbor gave the best value) and then reconstructing the path by following those decisions backward. This keeps memory usage linear in the number of states, which is crucial for large inputs.
Restate the DP recurrence and the meaning of each state, ensuring that the optimal value is computed correctly. Confirm that the DP table stores the best sum up to each state.
For each state, store either the predecessor state or the action (e.g., direction) that yielded the optimal value. This can be a separate array or integrated into the DP table.
Starting from the final optimal state, follow the stored predecessor pointers backward until reaching the start state. Collect the node values along the way.
Since backtracking yields the path in reverse order, reverse it to get the correct sequence from start to end. Print or return the sequence of node values.
Discuss the additional space (O(number of states)) and time (O(path length)) for reconstruction, and compare with alternative approaches like storing full paths in each state.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.