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In this approach, we use Dijkstra's algorithm to calculate the shortest path in the graph. The edges with initial weight -1 are considered candidates for modification. We apply binary search to find the appropriate weights for these edges, constraining the path length from source to destination to be equal to the target.
For edges with weight -1, we replace them with a large number initially (greater than the maximum possible path) and refine these weights using binary search to converge on the optimal configuration.
Time Complexity: O(E * log(V) * log(W)), where E is the number of edges, V is the number of vertices, and W is the weight range (binary search range).
Space Complexity: O(V + E), mainly for storing the graph structure and distances.
1from heapq import heappop, heappush
2
3def
This solution defines a function to apply Dijkstra's algorithm and uses binary search to refine the weights of -1 edges to achieve the target shortest path. The graph is constructed such that each edge weight is initially the binary search mid value when unknown (-1). We update the search bounds based on whether the constructed path is less than, equal to, or greater than the target.
This alternative approach leverages an iterative process to adjust edge weights. Starting with a breadth-first search (BFS)-like traversal, we replace unknown weights and check whether the current path meets or exceeds the target. Adjustments are made iteratively until the desired path length is achieved.
Time Complexity: O(E * V * log(W)), where E is the number of edges, V is the number of vertices, and W is the binary search range for weights.
Space Complexity: O(V + E) for the graph representation and distance tracking.
1import java.util.*;
2
3public class GraphSolution {
4 private static final int INF = Integer.MAX_VALUE;
5
6 public List<int[]> adjustGraphEdges(int n, int[][] edges, int source, int destination, int target) {
7 int left = 1, right = 2_000_000_000;
8 int[] result = null;
9
10 while (left <= right) {
11 int mid = left + (right - left) / 2;
12 int[] distances = bfsWithWeight(n, edges, source, mid);
13
14 if (distances[destination] < target) {
15 left = mid + 1;
16 } else {
17 if (distances[destination] == target) {
18 result = replaceWeights(edges, mid);
19 }
20 right = mid - 1;
21 }
22 }
23
24 return result != null ? Arrays.asList(result) : new ArrayList<>();
25 }
26
27 private int[] bfsWithWeight(int n, int[][] edges, int source, int weight) {
28 List<int[]>[] graph = buildGraph(n, edges, weight);
29 int[] dist = new int[n];
30 Arrays.fill(dist, INF);
31 dist[source] = 0;
32 Queue<Integer> queue = new LinkedList<>();
33 queue.offer(source);
34
35 while (!queue.isEmpty()) {
36 int u = queue.poll();
37 for (int[] adj : graph[u]) {
38 int v = adj[0], w = adj[1];
39 if (dist[v] > dist[u] + w) {
40 dist[v] = dist[u] + w;
41 queue.offer(v);
42 }
43 }
44 }
45
46 return dist;
47 }
48
49 private List<int[]>[] buildGraph(int n, int[][] edges, int weight) {
50 List<int[]>[] graph = new ArrayList[n];
51 for (int i = 0; i < n; i++) {
52 graph[i] = new ArrayList<>();
53 }
54 for (int[] edge : edges) {
55 int a = edge[0], b = edge[1], w = edge[2] == -1 ? weight : edge[2];
56 graph[a].add(new int[]{b, w});
57 graph[b].add(new int[]{a, w});
58 }
59 return graph;
60 }
61
62 private int[] replaceWeights(int[][] edges, int newWeight) {
63 for (int[] edge : edges) {
64 if (edge[2] == -1) {
65 edge[2] = newWeight;
66 }
67 }
68 return Arrays.stream(edges).flatMapToInt(Arrays::stream).toArray();
69 }
70
71 public static void main(String[] args) {
72 GraphSolution solution = new GraphSolution();
73 int n = 4;
74 int[][] edges = {{1,0,4},{1,2,3},{2,3,5},{0,3,-1}};
75 int source = 0;
76 int destination = 2;
77 int target = 6;
78 List<int[]> result = solution.adjustGraphEdges(n, edges, source, destination, target);
79 System.out.println(result);
80 }
81}This Java solution employs a modified BFS combined with binary search logic to iteratively refine edge weights. The BFS adaptation is used to assess the impact of current estimations on the maximum path length. Adjustments cease when we obtain the desired critical path or exhaust search options.
In this approach, we use binary search to determine the appropriate weights to assign to edges with initial weight of -1 such that the shortest path from the source to the destination is equal to the target. The idea is to repeatedly apply Dijkstra's algorithm with modified weights until we find the exact configuration which results in the shortest path distance equal to the target.
Time Complexity: O(VE), where V is the number of vertices and E is the number of edges, for each Bellman-Ford run.
Space Complexity: O(V), where V is the number of vertices for storing distances.
1// C++ solution implementing binary search on edge weights
2#include <iostream>
3#include <vector>
4#include <limits>
5
6using namespace std;
7
8class Edge {
9public:
10 int u, v, w;
11 Edge(int u_, int v_, int w_) : u(u_), v(v_), w(w_) {}
12};
13
14bool bellmanFord(int n, vector<Edge> &edges, int source, vector<int> &dist) {
15 dist.assign(n, INT_MAX);
16 dist[source] = 0;
17 for (int i = 0; i < n - 1; ++i) {
18 for (auto &edge : edges) {
19 if (dist[edge.u] != INT_MAX && dist[edge.u] + edge.w < dist[edge.v]) {
20 dist[edge.v] = dist[edge.u] + edge.w;
21 }
22 }
23 }
24 return true;
25}
26
27void modifyGraphEdges(int n, vector<Edge> &edges, int source, int destination, int target) {
28 vector<int> dist;
29
30 int left = 1, right = 2 * 1e9;
31 while (left < right) {
32 int mid = left + (right - left) / 2;
33
34 for (auto &edge : edges) {
35 if (edge.w == -1) edge.w = mid;
36 }
37
38 if (!bellmanFord(n, edges, source, dist)) {
39 right = mid;
40 continue;
41 }
42
43 if (dist[destination] > target) {
44 right = mid;
45 } else if (dist[destination] < target) {
46 left = mid + 1;
47 } else {
48 for (auto &edge : edges) {
49 cout << "[" << edge.u << ", " << edge.v << ", " << edge.w << "]\n";
50 }
51 return;
52 }
53 }
54 cout << "[]\n";
55}
56
57int main() {
58 int n = 4;
59 vector<Edge> edges = {{1,0,4},{1,2,3},{2,3,5},{0,3,-1}};
60 int source = 0;
61 int destination = 2;
62 int target = 6;
63 modifyGraphEdges(n, edges, source, destination, target);
64 return 0;
65}
66The C++ solution utilizes a class to manage Edge pairs and their weights. A binary search is conducted over possible weights for previously -1 weighted edges, achieving the target shortest path via repeated Bellman-Ford executions and modifications.
This approach leverages Dijkstra's algorithm to calculate the shortest path in the graph with a priority queue. We prioritize paths through existing edges first, checking if adjustments can achieve the target path length. It continuously updates the priority to cover edges that were initially marked with -1 with minimal adjustments. The logic attempts to find if a series of replacements can equate the length of the required path to target.
Time Complexity: O(V^2), inefficient but functional, typical for unoptimized Dijkstra approaches, adaptive to graph modifications.
Space Complexity: O(V), storing necessary node visitation states and distances.
1# Python solution using Dijkstra and prioritizing -1 edges
2import heapq
3
4class Edge:
5 def __init__(self, to, weight):
6 self.to = to
7 self.weight = weight
8
9class GraphEdge:
10 def __init__(self, u, v, w):
11 self.u = u
12 self.v = v
13 self.w = w
14
15 def __repr__(self):
16 return f'[{self.u}, {self.v}, {self.w}]'
17
18
19def dijkstra(n, graph, source):
20 dist = [float('inf')] * n
21 dist[source] = 0
22
23 pq = [(0, source)] # (distance, vertex)
24
25 while pq:
26 d, u = heapq.heappop(pq)
27 if d > dist[u]:
28 continue
29 for edge in graph[u]:
30 if dist[u] + edge.weight < dist[edge.to]:
31 dist[edge.to] = dist[u] + edge.weight
32 heapq.heappush(pq, (dist[edge.to], edge.to))
33
34 return dist
35
36
37def modify_graph_edges(n, edge_list, source, destination, target):
38 edges = [GraphEdge(u, v, w) for u, v, w in edge_list]
39 graph = [[] for _ in range(n)]
40 for edge in edges:
41 if edge.w != -1:
42 graph[edge.u].append(Edge(edge.v, edge.w))
43 graph[edge.v].append(Edge(edge.u, edge.w))
44
45 dist = dijkstra(n, graph, source)
46 path_length = dist[destination]
47
48 # Attempt to set initial unmodified path to target by general shifts
49 if path_length == target:
50 for edge in edges:
51 if edge.w == -1:
52 edge.w = 1
53 else:
54 for edge in edges:
55 if edge.w == -1:
56 edge.w = target - path_length
57 graph[edge.u].append(Edge(edge.v, edge.w))
58 graph[edge.v].append(Edge(edge.u, edge.w))
59 new_dist = dijkstra(n, graph, source)
60 path_length = new_dist[destination]
61 if path_length == target:
62 break
63 edge.w = -1
64
65 if path_length != target:
66 return [] # Impossible
67
68 return [[edge.u, edge.v, edge.w] for edge in edges]
69
70# Example usage
71n = 4
72edge_list = [[1,0,4],[1,2,3],[2,3,5],[0,3,-1]]
73source = 0
74destination = 2
75target = 6
76print(modify_graph_edges(n, edge_list, source, destination, target))
77The Python version prioritizes initial, valid path edges with re-examinations using Dijkstra's algorithm to verify once modifications to -1 edge values reflect the target clause. Employs tuple-priority queue stands for efficient shortest path tracking during each stage.
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