<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]]:
        import heapq

        tbl_neighbors = [[] for _ in range(n)]
        max_hop_bound = 1
        for u, v, wt in edges:
            if wt == -1:
                max_hop_bound += 1
            tbl_neighbors[u].append((v, wt))
            tbl_neighbors[v].append((u, wt))

        tbl_node_dist = [[None for _ in range(n)] for _ in range(max_hop_bound)]
        tbl_node_parent = [[None for _ in range(n)] for _ in range(max_hop_bound)]
        nodes_curr = [(0, (0, source), (0, None))]
        backtrace_init_hop = 0
        backtrace_ready = False
        for max_hop in range(max_hop_bound):
            nodes_next = []
            while nodes_curr:
                src_dist, (src_hop, src), parent = heapq.heappop(nodes_curr)
                if src_hop > max_hop:
                    continue
                if tbl_node_dist[src_hop][src] is not None:
                    continue
                tbl_node_dist[src_hop][src] = src_dist
                tbl_node_parent[src_hop][src] = parent
                if src == destination and src_dist <= target:
                    backtrace_init_hop = src_hop
                    backtrace_ready = True
                    break
                for node, node_dist in tbl_neighbors[src]:
                    node_hop = src_hop + (1 if node_dist == -1 else 0)
                    dist = src_dist + (1 if node_dist == -1 else node_dist)
                    if node_hop >= max_hop_bound or tbl_node_dist[node_hop][node] is not None:
                        continue
                    record = (dist, (node_hop, node), (src_hop, src))                        
                    if node_hop <= max_hop:
                        heapq.heappush(nodes_curr, record)
                    else:
                        heapq.heappush(nodes_next, record)
            if max_hop == 0:
                maybe_dist = tbl_node_dist[0][destination]
                if maybe_dist is not None and maybe_dist < target:
                    return []
            nodes_curr = nodes_next
            if backtrace_ready:
                break
        else:
            return []

        var_edge_set = set()
        diff_val = 0
        diff_val = target - tbl_node_dist[backtrace_init_hop][destination]
        assert diff_val >= 0
        curr_hop, curr_node = backtrace_init_hop, destination
        while curr_node is not None:
            prev_hop, prev_node = tbl_node_parent[curr_hop][curr_node]
            if curr_hop > prev_hop:
                var_edge_set.add((curr_node, prev_node))
                var_edge_set.add((prev_node, curr_node))
            curr_hop, curr_node = prev_hop, prev_node

        INFTY = int(2E9)
        result = []
        diff_assigned = False
        for u, v, wt in edges:
            if wt != -1:
                result.append((u, v, wt))
                continue
            if (u, v) not in var_edge_set:
                result.append((u, v, INFTY))
                continue
            if diff_assigned:
                result.append((u, v, 1))
                continue
            result.append((u, v, diff_val + 1))
            diff_assigned = True
        return result
        



            


        
        

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