<p>You are given an <strong>undirected weighted</strong> <strong>connected</strong> graph containing <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code>, and an integer array <code>edges</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>, w<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code> with weight <code>w<sub>i</sub></code>.</p>

<p>Some edges have a weight of <code>-1</code> (<code>w<sub>i</sub> = -1</code>), while others have a <strong>positive</strong> weight (<code>w<sub>i</sub> &gt; 0</code>).</p>

<p>Your task is to modify <strong>all edges</strong> with a weight of <code>-1</code> by assigning them <strong>positive integer values </strong>in the range <code>[1, 2 * 10<sup>9</sup>]</code> so that the <strong>shortest distance</strong> between the nodes <code>source</code> and <code>destination</code> becomes equal to an integer <code>target</code>. If there are <strong>multiple</strong> <strong>modifications</strong> that make the shortest distance between <code>source</code> and <code>destination</code> equal to <code>target</code>, any of them will be considered correct.</p>

<p>Return <em>an array containing all edges (even unmodified ones) in any order if it is possible to make the shortest distance from </em><code>source</code><em> to </em><code>destination</code><em> equal to </em><code>target</code><em>, or an <strong>empty array</strong> if it&#39;s impossible.</em></p>

<p><strong>Note:</strong> You are not allowed to modify the weights of edges with initial positive weights.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/18/graph.png" style="width: 300px; height: 300px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 5, edges = [[4,1,-1],[2,0,-1],[0,3,-1],[4,3,-1]], source = 0, destination = 1, target = 5
<strong>Output:</strong> [[4,1,1],[2,0,1],[0,3,3],[4,3,1]]
<strong>Explanation:</strong> The graph above shows a possible modification to the edges, making the distance from 0 to 1 equal to 5.
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/18/graph-2.png" style="width: 300px; height: 300px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 3, edges = [[0,1,-1],[0,2,5]], source = 0, destination = 2, target = 6
<strong>Output:</strong> []
<strong>Explanation:</strong> 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.
</pre>

<p><strong class="example">Example 3:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/19/graph-3.png" style="width: 300px; height: 300px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 4, edges = [[1,0,4],[1,2,3],[2,3,5],[0,3,-1]], source = 0, destination = 2, target = 6
<strong>Output:</strong> [[1,0,4],[1,2,3],[2,3,5],[0,3,1]]
<strong>Explanation:</strong> The graph above shows a modified graph having the shortest distance from 0 to 2 as 6.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 100</code></li>
	<li><code><font face="monospace">1 &lt;= edges.length &lt;= n * (n - 1) / 2</font></code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i&nbsp;</sub>&lt;&nbsp;n</code></li>
	<li><code><font face="monospace">w<sub>i</sub>&nbsp;= -1&nbsp;</font></code>or <code><font face="monospace">1 &lt;= w<sub>i&nbsp;</sub>&lt;= 10<sup><span style="font-size: 10.8333px;">7</span></sup></font></code></li>
	<li><code>a<sub>i&nbsp;</sub>!=&nbsp;b<sub>i</sub></code></li>
	<li><code>0 &lt;= source, destination &lt; n</code></li>
	<li><code>source != destination</code></li>
	<li><code><font face="monospace">1 &lt;= target &lt;= 10<sup>9</sup></font></code></li>
	<li>The graph is connected, and there are no self-loops or repeated edges</li>
</ul>

class Solution:
    def modifiedGraphEdges(self,num_nodes: int,edges: List[List[int]],start: int,end: int,target_distance: int) -> List[List[int]]:
        graph = [[] for _ in range(num_nodes)]

        # Build the graph with known weights
        for node1, node2, weight in edges:
            if weight != -1:
                graph[node1].append((node2, weight))
                graph[node2].append((node1, weight))

        # Helper function to perform Dijkstra's algorithm without considering -1 weights
        def dijkstra(source, destination, graph):
            pq = [[0, source]]
            distances = [float('inf')] * num_nodes
            distances[source] = 0
            
            while pq:
                current_dist, current_node = heappop(pq)
                if current_dist > distances[current_node]:
                    continue
                for neighbor, weight in graph[current_node]:
                    if weight == -1:
                        continue
                    new_dist = current_dist + weight
                    if new_dist < distances[neighbor]:
                        distances[neighbor] = new_dist
                        heappush(pq, (new_dist, neighbor))
            return distances[destination]

        # Step 2: Compute the shortest path distance with existing weights
        initial_distance = dijkstra(start, end, graph)
        
        # Step 3: Check if the target distance is achievable
        if initial_distance < target_distance:
            return []
        
        # Step 4: Check if the shortest path distance is exactly equal to the target
        is_target_reachable = initial_distance == target_distance

        for i, (node1, node2, weight) in enumerate(edges):
            if weight != -1:
                continue

            if is_target_reachable:
                edges[i][2] = 2 * ( 10 ** 9)
                continue
            
            # Try setting the edge weight to 1 initially
            edges[i][2] = 1
            graph[node1].append((node2, 1))
            graph[node2].append((node1, 1))

            new_distance = dijkstra(start, end, graph)
            if new_distance <= target_distance:
                is_target_reachable = True

                # Increase the edge weight to exactly hit the target distance
                edges[i][2] += target_distance - new_distance

                # Remove the temporary edge with weight 1
                graph[node1].remove((node2, 1))
                graph[node2].remove((node1, 1))

                # Add the edge with the adjusted weight
                graph[node1].append((node2, edges[i][2]))
                graph[node2].append((node1, edges[i][2]))

        if is_target_reachable:
            return edges
        return []
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