<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 dijkstra(self, G, s, t, n):
        dist = [float("inf")] * n
        dist[s] = 0
        pq = [(0, s)]

        while pq:
            dist_u, u = heapq.heappop(pq)
            # if dist_u > dist[u]:
            #     continue
            for v, w_uv in G[u]:
                if dist_u + w_uv < dist[v]:
                    dist[v] = dist_u + w_uv
                    heapq.heappush(pq, (dist[v], v))

        return dist[t]

    def not_possible(self, n, edges, s, t, target):
        G = collections.defaultdict(list)
        for u, v, w in edges:
            if w == -1:
                continue
            G[u].append((v, w))
            G[v].append((u, w))

        sp = self.dijkstra(G, s, t, n)
        if sp < target:
            return True

        for u, v, w in edges:
            if w != -1:
                continue
            G[u].append((v, 1))
            G[v].append((u, 1))

        sp = self.dijkstra(G, s, t, n)
        if sp > target:
            return True

        return False

    def modifiedGraphEdges(
        self, n: int, edges: List[List[int]], s: int, t: int, target: int
    ) -> List[List[int]]:
        # two cases where the modification is NOT possible
        # SP is already < target (check by deleting the negative edges)
        # SP > target after setting negative edge weights to 1

        # ok we know the modification is possible, now what?
        # initial idea
        # set the negative edge weights to 1
        # find SP
        # set weight of any initially negative edge to target - SP.weight
        # return
        # i think this should work
        # jk nope
        # consider this graph
        # A --(-1)-- B --(-1)-- C --(-1)-- D
        #  \_______5____________/
        # initially all the become 1, SP = 3
        # set AB to 5, ABCD has weight 7
        # but ACD has weight 6 oops

        # idea 2
        # instead of setting ALL modifiable edges to -1 and going from there
        # do it one edge at a time
        # as soon as we find a SP < target, modify edges
        if self.not_possible(n, edges, s, t, target):
            return []

        G = collections.defaultdict(list)
        for u, v, w in edges:
            if w == -1:
                continue
            G[u].append((v, w))
            G[v].append((u, w))
        
        
        if self.dijkstra(G, s, t, n) == target:
            for e in edges:
                if e[2] == -1:
                    e[2] = target + 1
            return edges

        for i, (u, v, w) in enumerate(edges):
            if w != -1:
                continue
            # modify the -1 edges
            edges[i][2] = 1
            G[u].append((v, 1))
            G[v].append((u, 1))
     
            # see if a SP exists, if it doesn't exist, sp = infinity
            sp = self.dijkstra(G, s, t, n)
      
            if sp <= target:
                edges[i][2] += target - sp
                # modify all other -1 edges and return
                for j in range(len(edges)):
                    if i == j:
                        continue
                    if edges[j][2] == -1:
                        edges[j][2] = target + 1
                return edges

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