Tree of Coprimes

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from collections import defaultdict
from math import gcd

class Solution:
    def getCoprimes(self, nums: List[int], edges: List[List[int]]) -> List[int]:
        answer = [-1] * len(nums)
        path = [[] for _ in range(51)] # Keeps track of ancestors for each number
        graph = defaultdict(list)
        seen = set() # Keeps track of visited nodes

        for u, v in edges:
            graph[u].append(v)
            graph[v].append(u)

        def dfs(node, depth):
            if node in seen:
                return
            seen.add(node)
            largestDepth = -1
            for x in range(1, 51):
                if gcd(nums[node], x) == 1: # Check if co-prime
                    if len(path[x]) > 0:
                        topNode, topDepth = path[x][-1]
                        if largestDepth < topDepth:  # Pick the node with the largest depth and co-prime with the current node as the ancestor
                            largestDepth = topDepth
                            answer[node] = topNode
            path[nums[node]].append((node, depth)) # Add the current node to the path
            for neighbor in graph[node]:
                dfs(neighbor, depth + 1)
            path[nums[node]].pop() # Backtrack

        dfs(0, 0)
        return answer

Identifying Problem Isomorphism

An approximate isomorphism: “Closest Leaf in a Binary Tree”.

In “Tree of Coprimes”, you’re given a tree (represented as an array of nodes, where the index is the node value, and the value at each index represents its parent), and an array of numbers representing the value at each node. The objective is to find for each node the closest node in its subtree that is coprime with the node’s value.

In “Closest Leaf in a Binary Tree”, you’re given a binary tree (represented as a set of edges) and a specific node K. The goal is to find the closest leaf node to the given node K.

The similarity is that both problems involve traversing a tree to find a node that satisfies a certain condition. In the “Tree of Coprimes”, it’s finding a node that’s coprime with the current node, while in “Closest Leaf in a Binary Tree”, it’s finding the nearest leaf node.

“Tree of Coprimes” is more complex due to its need to calculate coprimality and handle the added complexity of the tree being a generic tree (with any number of children) rather than a binary tree.

While not the same, the traversal methods used in these problems could be similar, involving depth-first search or breadth-first search to explore the tree. Both problems can be approached using depth-first search from each node to find the node that meets the specific condition. The difference is that, in the “Tree of Coprimes”, you will need to keep track of the ancestors for each node during the traversal due to the coprime condition.

10 Prerequisite LeetCode Problems

“1766. Tree of Coprimes” involves graph theory, depth-first search (DFS), and number theory (coprime concept). Here are 10 problems to prepare for this:

  1. 133. Clone Graph: This problem helps in understanding the basic concept of graph traversal using DFS.

  2. 207. Course Schedule: This problem is a classic application of the concept of detecting a cycle in a directed graph.

  3. 684. Redundant Connection: This problem helps in understanding the union-find algorithm which is quite helpful in solving graph-related problems.

  4. 743. Network Delay Time: This problem involves finding the shortest path in a graph which is an essential concept in graph theory.

  5. 872. Leaf-Similar Trees: This problem helps to practice tree traversal techniques, particularly DFS.

  6. 199. Binary Tree Right Side View: This problem can improve your understanding of tree traversal and breadth-first search (BFS).

  7. 863. All Nodes Distance K in Binary Tree: This problem is a good exercise in DFS and BFS in the context of a tree.

  8. 204. Count Primes: This problem helps in understanding number theory which is required to understand the concept of coprime numbers.

  9. 959. Regions Cut By Slashes: This is an advanced graph problem that involves manipulating the graph in a specific way before performing a standard graph traversal or connectivity check.

  10. 1245. Tree Diameter: This problem can give you practice in finding the longest path in a tree.