Shortest Distance After Road Addition Queries I

class Solution {
public:
    int calculateMinDistance(vector<vector<int>> &graph, int nodes) {
        vector<int> dist(nodes);
        dist[nodes - 1] = 0;  // Initialize the end node distance to zero
    
        for (int i = nodes - 2; i >= 0; i--) {
            int smallestDist = nodes;
            for (auto neighbor : graph[i]) {
                smallestDist = min(smallestDist, dist[neighbor] + 1);
            }
            dist[i] = smallestDist;
        }
        return dist[0];
    }
    
    vector<int> processDistanceQueries(int nodes, vector<vector<int>> &edges) {
        vector<int> results;
        vector<vector<int>> graph(nodes);
    
        for (int i = 0; i < nodes - 1; i++) {
            graph[i].push_back(i + 1);
        }
    
        for (auto &edge : edges) {
            int from = edge[0], to = edge[1];
            graph[from].push_back(to);
    
            results.push_back(calculateMinDistance(graph, nodes));
        }
        return results;
    }
};

The given C++ solution addresses the problem of finding the shortest distance after road addition queries on a graph represented through its adjacency list.

The program contains two main functions:

1. calculateMinDistance: This function computes the minimum distance from the starting node to the last node of the graph. It employs a backward traversal technique, starting from the last node, and uses dynamic programming to calculate the shortest paths. Initialized with the end node set to zero distance, the function iterates through the nodes, updating the shortest distance using the minimal distance of its neighbors plus one.

2. processDistanceQueries: This function prepares a graph from a given number of nodes and edges, and initially connects consecutive nodes. It then processes each edge addition and immediately invokes calculateMinDistance to compute the shortest path after each road (edge) addition. The results from each query are stored in a vector that is returned at the end of the function.

The solution involves populating an adjacency list for the graph, updating it with each query, and calculating the shortest path efficiently with each update, providing a dynamic way to see the impact of adding each road in real-time on the distance calculations.

class Solution {
    public int calculateMinDistance(List<List<Integer>> graph, int nodes) {
        int[] distances = new int[nodes];
        distances[nodes - 1] = 0;
        for (int i = nodes - 2; i >= 0; i--) {
            int shortest = nodes;
            for (int adj : graph.get(i)) {
                shortest = Math.min(shortest, distances[adj] + 1);
            }
            distances[i] = shortest;
        }
        return distances[0];
    }
    
    public int[] shortestPathQueries(int nodeCount, int[][] edges) {
        List<Integer> results = new ArrayList<>();
        List<List<Integer>> adjacencyList = new ArrayList<>();
    
        for (int i = 0; i < nodeCount; i++) {
            adjacencyList.add(new ArrayList<>());
        }
    
        for (int i = 0; i < nodeCount - 1; i++) {
            adjacencyList.get(i).add(i + 1);
        }
    
        for (int[] edge : edges) {
            int start = edge[0];
            int end = edge[1];
            adjacencyList.get(start).add(end);
            results.add(calculateMinDistance(adjacencyList, nodeCount));
        }
    
        int[] solution = new int[results.size()];
        for (int i = 0; i < results.size(); i++) {
            solution[i] = results.get(i);
        }
        return solution;
    }
}

The solution provided in the Java code defines two methods aimed at determining the shortest distance from the last index (node) to the starting index after each road (edge) addition in a directed graph. This approach involves dynamically updating the shortest paths and handling multiple queries which involve adding roads one at a time.

The structure of the solution is as follows:

• calculateMinDistance Method:

  • This method computes the shortest distances for all nodes to the starting node. The graph's adjacency list and the total number of nodes are parameters.
  • A shortest path logic using backward iteration from the last node to the first to determine the minimum distance is crucial here.

• shortestPathQueries Method:

  • Initiates by setting up a basic linear graph where each node points to its immediate next node.
  • Processes each road addition query by:
    • Updating the adjacency list with the new edge.
    • Calling calculateMinDistance to assess the shortest distance from the start after each addition.
    • Records the result after each query.

• Edge Structure and Graph Representation:

  • An adjacency list is used for representing the graph where each node's neighbors can be dynamically updated.
  • Road additions directly reflect changes in this adjacency list.

• Output:

  • The results of all queries are collected and returned in an array. Each entry corresponds to the shortest path from the start node to end node as the graph evolves with each edge addition.

This code supports efficient handling of graph updates through direct additions to the adjacency list and recalculating shortest paths using a straightforward approach. This makes it suitable for scenarios where graph dynamics are dependent on incremental updates.

class Solution:
    def compute_minimum_distance(self, graph, node_count):
        distances = [0] * node_count
        distances[node_count - 1] = 0  # Distance to itself is 0
    
        for node in range(node_count - 2, -1, -1):
            shortest = node_count
            for adjacent in graph[node]:
                shortest = min(shortest, distances[adjacent] + 1)
            distances[node] = shortest
    
        return distances[0]
    
    def calculateDistancesAfterAdjustments(self, node_total, adjustments):
        results = []
        graph_representation = [[] for _ in range(node_total)]
    
        for idx in range(node_total - 1):
            graph_representation[idx].append(idx + 1)
    
        for modification in adjustments:
            origin, destination = modification
            graph_representation[origin].append(destination)
            results.append(self.compute_minimum_distance(graph_representation, node_total))
    
        return results

The solution described involves calculating the shortest distance from the start node to the last node in a graph after making several adjustments as specified by queries. The approach uses the Python language and consists of two primary functions in a Solution class:

1. compute_minimum_distance(graph, node_count): This function calculates the shortest distance from within the graph. It initializes a list distances to store the distance from each node to the last node, setting the distance to itself as zero. The function then iterates backwards through the nodes, calculating the shortest distance for each node based on its adjacency list using a nested loop.

2. calculateDistancesAfterAdjustments(node_total, adjustments): This function manages the main implementation flow. It initializes the graph with a direct path from each node to the next. The function updates this graph based on the given adjustments, which define additional routes between nodes, then recalculates distances after each adjustment by invoking compute_minimum_distance.

The method employed handles graph adjustments dynamically, recalculating minimum distances as changes are applied, making it well-suited to scenarios where adjustments to the pathing in a graph need to be evaluated on the fly. Results from each query are stored and returned as a list, offering a straightforward output of minimum distances for verification or further processing.