Problem Statement
You are provided with an undirected weighted connected graph comprising n nodes, labeled from 0 to n - 1. The graph is represented using an integer array edges, where each element edges[i] = [ai, bi, wi] represents an edge linking node ai to node bi with a weight wi. Some edges have a specified weight, wi > 0, while others are unspecified with wi = -1.
Your objective is to assign new positive integer weights to all the unspecified edges (wi = -1). The weights should be assigned such that the shortest path from a given source node to a destination node precisely matches a specified integer target. If multiple assignments of weights to the unspecified edges result in the shortest path matching the target, all such configurations are valid. Your function must return an array of the edges, which includes the reassigned weights if a solution exists, or returns an empty array when it is infeasible to match the target distance with any assignment.
This task must be executed without altering the weights of the already specified positive weight edges and ensuring the solution adheres to prescribed constraints.
Examples
Example 1
Input:
n = 5, edges = [[4,1,-1],[2,0,-1],[0,3,-1],[4,3,-1]], source = 0, destination = 1, target = 5
Output:
[[4,1,1],[2,0,1],[0,3,3],[4,3,1]]
Explanation:
The graph above shows a possible modification to the edges, making the distance from 0 to 1 equal to 5.
Example 2
Input:
n = 3, edges = [[0,1,-1],[0,2,5]], source = 0, destination = 2, target = 6
Output:
[]
Explanation:
The graph above contains the initial edges. It is not possible to make the distance from 0 to 2 equal to 6 by modifying the edge with weight -1. So, an empty array is returned.
Example 3
Input:
n = 4, edges = [[1,0,4],[1,2,3],[2,3,5],[0,3,-1]], source = 0, destination = 2, target = 6
Output:
[[1,0,4],[1,2,3],[2,3,5],[0,3,1]]
Explanation:
The graph above shows a modified graph having the shortest distance from 0 to 2 as 6.
Constraints
1 <= n <= 1001 <= edges.length <= n * (n - 1) / 2edges[i].length == 30 <= ai, bi < nwi = -1or1 <= wi <= 107ai != bi0 <= source, destination < nsource != destination1 <= target <= 109- The graph is connected, and there are no self-loops or repeated edges
Approach and Intuition
When approaching this problem, the key is leveraging graph algorithms tailored toward finding and manipulating shortest paths. Given the mixed nature of specified and unspecified weights, the challenges primarily involve decision-making on how to determine the values for unspecified weights such that the shortest path conditions are met:
Understand the Basic Graph Structure: Recognize that the graph is undirected and weights can be both unspecified (
-1) or positive. Nodes and edges are indexed starting from 0, according to the given constraints.Identify the Goal: The main objective is to adjust the weight of edges with
-1so that the shortest distance betweensourceanddestinationis exactlytarget.Use of Graph Algorithms: Since the problem involves finding shortest paths, algorithms like Dijkstra’s or Bellman-Ford could be considered. Dijkstra’s algorithm is efficient but only works with non-negative weights. In initial scenarios, where weights are
-1and need to be determined, a strategy involving iterative adjustments and validations using these algorithms can be useful.Iterative Adjustment and Validation:
- Start by assigning a tentative positive weight (like 1 or the minimum required to achieve connectivity) to all
-1edges. - Calculate the shortest path.
- Adjust the weights iteratively while checking if the shortest path length matches the
target. Consider large enough weight adjustments to overcome local minimum traps.
- Start by assigning a tentative positive weight (like 1 or the minimum required to achieve connectivity) to all
Check for Multiple Solutions: If a valid path configuration is found, consider checking for alternative weight configurations that also satisfy the condition, as any valid solution is accepted.
Edge Cases and Constraints Handling: Ensure that the solution respects all given constraints like edge values' bounds and connectivity conditions. Remember, if no configuration can be found, return an empty array.
This problem beautifully combines elements of graph theory with an optimization puzzle, demanding both strategic application of algorithms and creative problem-solving for edge weight assignments.
Solutions
- C++
- Java
- Python
class Solution {
public:
const int MAX_VALUE = 2000000000;
vector<vector<int>> updateGraph(int vertexCount, vector<vector<int>>& edgeList,
int src, int dest, int targetDistance) {
vector<vector<pair<int, int>>> adjacencyList(vertexCount);
// Building graph excluding negative weight edges
for (const auto& edge : edgeList) {
if (edge[2] != -1) {
adjacencyList[edge[0]].emplace_back(edge[1], edge[2]);
adjacencyList[edge[1]].emplace_back(edge[0], edge[2]);
}
}
// Calculating shortest distance from src to dest
int initialDistance = shortestPath(vertexCount, src, dest, adjacencyList);
if (initialDistance < targetDistance) {
return vector<vector<int>>();
}
bool achievedTargetDistance = (initialDistance == targetDistance);
// Adjusting weights or adding new weighted edges
for (auto& edge : edgeList) {
if (edge[2] != -1) continue; // Skip if weight already set
edge[2] = achievedTargetDistance ? MAX_VALUE : 1;
adjacencyList[edge[0]].emplace_back(edge[1], edge[2]);
adjacencyList[edge[1]].emplace_back(edge[0], edge[2]);
if (!achievedTargetDistance) {
int updatedDistance = shortestPath(vertexCount, src, dest, adjacencyList);
if (updatedDistance <= targetDistance) {
achievedTargetDistance = true;
edge[2] += targetDistance - updatedDistance;
}
}
}
return achievedTargetDistance ? edgeList : vector<vector<int>>();
}
private:
int shortestPath(int numVertices, int start, int end,
const vector<vector<pair<int, int>>>& adjList) {
vector<int> distances(numVertices, MAX_VALUE);
priority_queue<pair<int, int>, vector<pair<int, int>>, greater<>> priorityQueue;
distances[start] = 0;
priorityQueue.emplace(0, start);
while (!priorityQueue.empty()) {
auto [currDistance, node] = priorityQueue.top();
priorityQueue.pop();
if (currDistance > distances[node]) continue;
for (const auto& [nextNode, weight] : adjList[node]) {
if (currDistance + weight < distances[nextNode]) {
distances[nextNode] = currDistance + weight;
priorityQueue.emplace(distances[nextNode], nextNode);
}
}
}
return distances[end];
}
};
Modify a given graph's edge weights to achieve a specific shortest path distance between two specified vertices using a C++ program. The provided solution class employs the following approach for this modification:
Initialize an adjacency list to represent the graph. Populate this adjacency list while excluding edges with negative weights.
Calculate the shortest path distance from the source vertex (
src) to the destination vertex (dest) using Dijkstra's algorithm.If the initial distance to the destination is less than the desired
targetDistance, return an empty list indicating that no adjustment can make the path reach the target distance.Handle cases where the current graph yields the exact target distance by marking as achieved and setting high weights for further added edges to prevent distance reduction.
For edges with undefined (-1) or adjustable weights, iteratively add or adjust weights. If adding a weight of 1 does not achieve the target distance, modify the weight further until either the target distance is achieved or exceeded.
Adjust weights by considering if the current weight configuration already satisfies the target path distance or if further adjustments are required without exceeding the minimum required path.
If after all possible adjustments, the target distance is achievable, return the adjusted edge list. Otherwise, return an empty list indicating that the target distance is not achievable with given modifications.
Key implementation details to note:
- Use a pair representation for the adjacency list to store connected nodes and respective weights.
- Implement Dijkstra’s algorithm for determining the shortest path using a priority queue to efficiently fetch the next node with the shortest cumulative distance.
- Manage edge weight adjustments carefully to achieve the exact target distance when possible or exceed it when necessary, ensuring correctness through iterative validation.
This method ensures that the graph modifications are minimal and only carried out when they contribute directly toward achieving the specified target shortest path.
class Solution {
List<int[]>[] adjacencyList;
private static final int LARGE_INT = (int) 2e9;
public int[][] processGraphEdges(
int nodeCount,
int[][] inputEdges,
int startPoint,
int endPoint,
int requiredDistance
) {
// Build the graph, omit edges marked with -1
adjacencyList = new ArrayList[nodeCount];
for (int i = 0; i < nodeCount; i++) {
adjacencyList[i] = new ArrayList<>();
}
for (int[] edge : inputEdges) {
if (edge[2] != -1) {
adjacencyList[edge[0]].add(new int[] { edge[1], edge[2] });
adjacencyList[edge[1]].add(new int[] { edge[0], edge[2] });
}
}
// Initial shortest path calculation
int initialShortestPath = runDijkstra(nodeCount, startPoint, endPoint);
if (initialShortestPath < requiredDistance) {
return new int[0][0];
}
boolean reachedTarget = (initialShortestPath == requiredDistance);
// Adjust edge weights accordingly
for (int[] edge : inputEdges) {
if (edge[2] != -1) continue; // Skip already processed edges
edge[2] = reachedTarget ? LARGE_INT : 1;
adjacencyList[edge[0]].add(new int[] { edge[1], edge[2] });
adjacencyList[edge[1]].add(new int[] { edge[0], edge[2] });
if (!reachedTarget) {
// Recalculate shortest paths after updating edge weights
int newShortestPath = runDijkstra(nodeCount, startPoint, endPoint);
if (newShortestPath <= requiredDistance) {
reachedTarget = true;
edge[2] += requiredDistance - newShortestPath;
}
}
}
// Return adjusted edges if we've matched the target distance
return reachedTarget ? inputEdges : new int[0][0];
}
// Implementation of Dijkstra's algorithm for shortest path finding
private int runDijkstra(int nodeCount, int sourceNode, int destinationNode) {
int[] distances = new int[nodeCount];
PriorityQueue<int[]> priorityQueue = new PriorityQueue<>(
(a, b) -> a[1] - b[1]
);
Arrays.fill(distances, LARGE_INT);
distances[sourceNode] = 0;
priorityQueue.add(new int[] { sourceNode, 0 });
while (!priorityQueue.isEmpty()) {
int[] current = priorityQueue.poll();
int currentNode = current[0];
int currentDist = current[1];
if (currentDist > distances[currentNode]) continue;
for (int[] neighbor : adjacencyList[currentNode]) {
int nextNode = neighbor[0];
int weight = neighbor[1];
if (currentDist + weight < distances[nextNode]) {
distances[nextNode] = currentDist + weight;
priorityQueue.add(new int[] { nextNode, distances[nextNode] });
}
}
}
return distances[destinationNode];
}
}
Modify and manage edge weights in a graph efficiently with Java. Use this solution when you need to adjust edge weights in a graph to match a specific required distance between two nodes. This method is particularly useful for scenarios where the graph needs dynamic updates after the initial construction.
Start by initiating an adjacency list to represent the graph, ensuring you use the node count to predefine list sizes. Consider ignoring any edges marked with a placeholder (e.g., -1), as these represent edges not initially considered.
Utilize Dijkstra's algorithm for the initial shortest path calculation from the start to the end node. If this initial path is less than the required distance, return an empty array to indicate no modifications are needed.
In cases where initial checks don't achieve the required distance, proceed to systematically adjust edge weights. For edges initially ignored (marked with -1), set weights to either a large integer if the current path meets the required distance or 1 if it doesn't, which helps in minimizing initial weights and exploring potential valid paths efficiently.
Continuously apply Dijkstra's algorithm after each adjustment to ensure new configurations meet or exceed the specified distance. Persist adjustments only if they help achieve or maintain the required distance. This iterative approach aids in fine-tuning the graph to meet specific criteria without excessive recomputation.
Return the array of input edges with their final weights for verification or further processing. This Java solution ensures a methodical and minimal adjustment to graph edge weights, enabling dynamic graph weight management based on specific path criteria.
class Solution:
def modifyGraphEdges(
self,
num_nodes: int,
mod_edges: List[List[int]],
start_point: int,
end_point: int,
desired_distance: int,
) -> List[List[int]]:
UNREACHABLE = int(2e9)
adjacency_list = [[] for _ in range(num_nodes)]
for node1, node2, weight in mod_edges:
if weight != -1:
adjacency_list[node1].append((node2, weight))
adjacency_list[node2].append((node1, weight))
initial_distance = self._calculateDijkstra(adjacency_list, start_point, end_point)
if initial_distance < desired_distance:
return []
if initial_distance == desired_distance:
for edge in mod_edges:
if edge[2] == -1:
edge[2] = UNREACHABLE
return mod_edges
for index, (node1, node2, weight) in enumerate(mod_edges):
if weight != -1:
continue
mod_edges[index][2] = 1
adjacency_list[node1].append((node2, 1))
adjacency_list[node2].append((node1, 1))
updated_distance = self._calculateDijkstra(adjacency_list, start_point, end_point)
if updated_distance <= desired_distance:
mod_edges[index][2] += desired_distance - updated_distance
for j in range(index + 1, len(mod_edges)):
if mod_edges[j][2] == -1:
mod_edges[j][2] = UNREACHABLE
return mod_edges
return []
def _calculateDijkstra(self, adjacency_list: List[List[Tuple[int, int]]], start_node: int, target_node: int) -> int:
distances = [float('inf')] * len(adjacency_list)
distances[start_node] = 0
priority_queue = [(0, start_node)]
while priority_queue:
current_dist, current_node = heapq.heappop(priority_queue)
if current_dist > distances[current_node]:
continue
for neighbor, weight in adjacency_list[current_node]:
if current_dist + weight < distances[neighbor]:
distances[neighbor] = current_dist + weight
heapq.heappush(priority_queue, (distances[neighbor], neighbor))
return distances[target_node]
The solution provided involves a graph modification strategy using Dijkstra's Algorithm to ensure a desired distance between a start and endpoint is met by potentially adjusting edge weights. Below are key concepts and summary of the approach utilized in the Python code:
- The function
modifyGraphEdgesaccepts the total number of nodes, a list of modified edges where each edge contains two nodes and a weight, start and end points, and the desired distance between the two points. - An
adjacency_listis created to represent the graph. - For edges with a defined weight (non-negative), they are added to the adjacency list.
- An initial maximum distance using the current graph structure is computed with
self._calculateDijkstra. - Based on the comparison between the initial distance and the desired distance, the following logic applies:
- If the initial distance is less than the desired distance, modifications are not feasible so an empty list is returned.
- If the initial distance equals the desired distance, any undefined weights in edges are assigned a very large number (UNREACHABLE) to signify no change is needed in those edges.
- If the initial distance is greater than the desired distance, attempt to adjust the weights of edges with
-1(undefined weight) to decrease the overall path distance.
- In this adjustment loop, each undefined weight is replaced first with
1to recalculate the shortest path. If the updated distance is less than or equal to the desired distance, the edge’s weight is further adjusted to match exactly the desired distance. - If no suitable adjustments can be achieved for exiting the loop, an empty list is returned signaling the target condition cannot be met.
The code applies Dijkstra’s algorithm to frequently recalculate the shortest paths which is computationally intensive but necessary for ensuring accuracy with each slight adjustment to edge weights. The solution design thoroughly explores the possibility of meeting the desired distance condition with strategic minimal adjustments to the graph.
No comments yet.