<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, n: int, edges: List[List[int]], source: int, destination: int, target: int) -> List[List[int]]:
        def modifyShortestPath(n,edges,source,destination,modify=True):
            adj = [[] for _ in range(n)]
            m = len(edges)
            for i in range(m):
                a,b,w = edges[i]
                adj[a].append((b,w,i))
                adj[b].append((a,w,i))

            h = []
            heapq.heappush(h,(0,source,[]))
            visited={}
            visited[source]=0
            while h:
                (dist, v, edgesList) = heapq.heappop(h)
                
                
                if v==destination : 
                    return dist,edgesList
                else:
                    for u,w,i in adj[v]:
                        if w==-1 and modify==True:                            
                            w=1
                        if w > 0:
                            if u not in visited:
                                heapq.heappush(h,(dist+w,u,edgesList+[i]))
                                visited[u]=dist+w
                            else: # u in visited
                                if dist+w < visited[u]:
                                    heapq.heappush(h,(dist+w,u,edgesList+[i]))
                                    visited[u]=dist+w
            return -1,[]
            
        def shortestPath(n,edges,source,destination):
            adj = [[] for _ in range(n)]
            m = len(edges)            
            for i in range(m):
                a,b,w = edges[i]
                adj[a].append((b,w))
                adj[b].append((a,w))
                        
            h = []
            heapq.heappush(h,(0,source))
            visited={}
            visited[source]=0
            while h:
                (dist, v) = heapq.heappop(h)

                if v==destination :                
                    return dist
                else:
                    for u,w in adj[v]:
                        if w > 0:
                            if u not in visited:
                                heapq.heappush(h,(dist+w,u))
                                visited[u]=dist+w
                            else: # u in visited
                                if dist+w < visited[u]:
                                    heapq.heappush(h,(dist+w,u))
                                    visited[u]=dist+w                            
            return -1

        m = len(edges)
        neg = {}
        for i in range(m):
            if edges[i][2]==-1:
                neg[i] = True

        dist,edgesList = modifyShortestPath(n,edges,source,destination,modify=False) 
        # modify==Fasle represents ignoring the negtive edges when searching the shortest distance.
        if dist >= 0 :
            if dist < target: # the shorst distance without considering the negtive edges is less than target
                return []
            elif dist == target:
                for i in neg: # change all negtive edges to a large number
                    edges[i][2]=1000000005                
                return edges


        dist,edgesList = modifyShortestPath(n,edges,source,destination,modify=True)
        # EdgesList represents the set of edges in the shortest path, and dist is the shorest distance when the weights of the negtive edges are changed to 1.
        # We adjust the weights as follows: the first negative edge in edgesList is set to target-dist+1, the other negative edges in edgesList are set to 1, and the weights of edges not in edgesList are set to a large number, such as 1,000,000,005.
        if dist <= target:
            modify = False
            for i in range(m):
                if edges[i][2] == -1:                            
                    if i in edgesList:
                        if not modify:
                            edges[i][2] = target-dist+1
                            modify = True
                        else:
                            edges[i][2] = 1
                    else:
                        edges[i][2] = 1000000005
        else: # dist > target
            return []

        # We recheck whether the shortest distance, based on the updated edge weights, equals the target value.
        while True:
            dist2,edgesList2 = modifyShortestPath(n,edges,source,destination)
            if target == dist2:
                return edges
            elif dist2 < target: # change the weight of the first negitive edge again
                if len(neg) > 0:
                    for i in edgesList2:
                        if i in neg:
                            edges[i][2]=target-dist2+1
                            del neg[i]
                            break                        
                else:
                    break

        return []


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