Find Closest Node to Given Two Nodes

class Solution {
public:
    void exploreGraph(int current, vector<int>& connections, vector<int>& distance, vector<bool>& visited) {
        visited[current] = true;
        int next = connections[current];
        if (next != -1 && !visited[next]) {
            distance[next] = 1 + distance[current];
            exploreGraph(next, connections, distance, visited);
        }
    }

    int findClosestNode(vector<int>& connections, int start1, int start2) {
        int sz = connections.size();
        vector<int> distance1(sz, numeric_limits<int>::max()), distance2(sz, numeric_limits<int>::max());
        vector<bool> seen1(sz), seen2(sz);
        distance1[start1] = 0, distance2[start2] = 0;

        exploreGraph(start1, connections, distance1, seen1);
        exploreGraph(start2, connections, distance2, seen2);

        int closestNode = -1, minimumDistance = numeric_limits<int>::max();
        for (int i = 0; i < sz; i++) {
            if (minimumDistance > max(distance1[i], distance2[i])) {
                closestNode = i;
                minimumDistance = max(distance1[i], distance2[i]);
            }
        }

        return closestNode;
    }
};

This C++ solution efficiently identifies the closest node to two given nodes in a directed graph represented as a list of connections, where each index and its value portray a directional connection from index to value.

• Initiate by defining the exploreGraph function to explore the graph and compute the distance from the starting node to all other nodes using simple DFS. This function updates the distance array with the number of steps taken to reach each node and uses a visited array to keep track of explored nodes.
• Proceed to define the findClosestNode function:
  1. Prepare two distance vectors distance1 and distance2, initializing them with maximum integer values. These arrays will store the minimal distances from the two start nodes defined by start1 and start2.
  2. Two boolean vectors seen1 and seen2 are used to record if a node has been visited starting from start1 and start2.
  3. Call exploreGraph for both start nodes to fill out the distance1 and distance2 arrays.
  4. Initialize closestNode to -1 to indicate that no node has been found initially. Iterate over all nodes. For each node, determine the greater of the two distances (max(distance1[i], distance2[i])). Update the closestNode with the node index if this maximum is less than the previously recorded minimum distance.
  5. Eventually, return the closestNode which represents the node that has the shortest largest distance to both start nodes.

This implementation excels in clarity and scalability, effectively capturing both single-path distances and identifying optimal node accessibility in the context of varying graph topologies.

class GraphTraversal {
    public void depthFirstSearch(int vertex, int[] connections, int[] pathLength, Boolean[] visited) {
        visited[vertex] = true;
        int next = connections[vertex];
        if (next != -1 && !visited[next]) {
            pathLength[next] = pathLength[vertex] + 1;
            depthFirstSearch(next, connections, pathLength, visited);
        }
    }

    public int findClosestNode(int[] connections, int start1, int start2) {
        int totalVertices = connections.length;
        int[] path1 = new int[totalVertices], path2 = new int[totalVertices];
        Arrays.fill(path1, Integer.MAX_VALUE);
        Arrays.fill(path2, Integer.MAX_VALUE);
        path1[start1] = 0;
        path2[start2] = 0;

        Boolean[] visited1 = new Boolean[totalVertices], visited2 = new Boolean[totalVertices];
        Arrays.fill(visited1, Boolean.FALSE);
        Arrays.fill(visited2, Boolean.FALSE);

        depthFirstSearch(start1, connections, path1, visited1);
        depthFirstSearch(start2, connections, path2, visited2);

        int minIndex = -1, minDistance = Integer.MAX_VALUE;
        for (int i = 0; i < totalVertices; i++) {
            if (minDistance > Math.max(path1[i], path2[i])) {
                minIndex = i;
                minDistance = Math.max(path1[i], path2[i]);
            }
        }

        return minIndex;
    }
}

This Java program helps find the closest node between two given nodes within a graph represented by an adjacency list. The solution utilizes a Depth-First Search (DFS) approach to determine the shortest path lengths from each starting node to all other nodes in the graph. Here’s how the provided solution operates:

• DepthFirstSearch Method:

  • Initialize each traversal with the start vertex marked as visited.
  • Explore all connected vertices recursively.
  • Keep tracking the path length from the start vertex to the current vertex, updating the shortest path found.

• FindClosestNode Method:

  • Set up initial conditions, marking path lengths from the start nodes to infinite (except to themselves where it's 0) and marking all nodes as unvisited.
  • Conduct individual depth-first searches starting from each of the two given nodes.
  • Post DFS, evaluate each vertex in the graph to find the vertex that has the smallest maximum distance from either start node.

The outcome is an integer representing the index of the closest node to the two given start nodes, minimizing the longest path from either start node. This solution efficiently maps out paths to determine the most centrally located vertex relative to the two specified starting nodes, leveraging DFS to handle potentially complex connection scenarios in the graph.