Which modification to the algorithm is necessary to correctly solve this variant?
hard🎤 Interviewer Follow-up Q15 of Q15
Tree: Depth-First Search - House Robber III (On Tree)
Suppose the problem is modified so that you can rob the same node multiple times (i.e., nodes can be reused any number of times), but still cannot rob directly connected nodes simultaneously. Which modification to the algorithm is necessary to correctly solve this variant?
AUse a bottom-up DP that tracks counts of times each node is robbed to handle reuse.
BUse the same DFS with two-value return; no changes needed since adjacency constraints remain.
CConvert the tree into a graph and run a maximum weighted independent set algorithm with cycle detection.
DModify the DFS to allow revisiting nodes multiple times and use memoization keyed by node and robbing state.
Step-by-Step Solution
Step 1: Understand the reuse constraint
Allowing multiple robberies per node breaks the tree structure assumption and simple one-time visit DFS.
Step 2: Identify necessary algorithmic change
We must track states including how many times a node is robbed and ensure adjacency constraints per robbery. Memoization keyed by node and robbing state prevents exponential recomputation.
Step 3: Why other options fail
Use the same DFS with two-value return; no changes needed since adjacency constraints remain. ignores reuse; Convert the tree into a graph and run a maximum weighted independent set algorithm with cycle detection. is overcomplicated and unnecessary since the structure is still a tree; Use a bottom-up DP that tracks counts of times each node is robbed to handle reuse. is vague and does not address adjacency constraints properly.
Final Answer:
Option D -> Option D
Quick Check:
Memoization with extended state handles reuse and adjacency constraints [OK]
Quick Trick:Reuse requires stateful memoization, not simple DFS [OK]
Common Mistakes:
MISTAKES
Assuming original DFS suffices for reuse
Ignoring adjacency constraints when reusing nodes
Trap Explanation:
PITFALL
Naive DFS fails because it does not track multiple visits or states per node.
Interviewer Note:
CONTEXT
Tests candidate's ability to adapt DP states for problem variants with reuse.
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