<p>You are given an array <code>start</code> where <code>start = [startX, startY]</code> represents your initial position <code>(startX, startY)</code> in a 2D space. You are also given the array <code>target</code> where <code>target = [targetX, targetY]</code> represents your target position <code>(targetX, targetY)</code>.</p>

<p>The <strong>cost</strong> of going from a position <code>(x1, y1)</code> to any other position in the space <code>(x2, y2)</code> is <code>|x2 - x1| + |y2 - y1|</code>.</p>

<p>There are also some <strong>special roads</strong>. You are given a 2D array <code>specialRoads</code> where <code>specialRoads[i] = [x1<sub>i</sub>, y1<sub>i</sub>, x2<sub>i</sub>, y2<sub>i</sub>, cost<sub>i</sub>]</code> indicates that the <code>i<sup>th</sup></code> special road goes in <strong>one direction</strong> from <code>(x1<sub>i</sub>, y1<sub>i</sub>)</code> to <code>(x2<sub>i</sub>, y2<sub>i</sub>)</code> with a cost equal to <code>cost<sub>i</sub></code>. You can use each special road any number of times.</p>

<p>Return the <strong>minimum</strong> cost required to go from <code>(startX, startY)</code> to <code>(targetX, targetY)</code>.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">start = [1,1], target = [4,5], specialRoads = [[1,2,3,3,2],[3,4,4,5,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

<p><strong>Explanation:</strong></p>

<ol>
	<li>(1,1) to (1,2) with a cost of |1 - 1| + |2 - 1| = 1.</li>
	<li>(1,2) to (3,3). Use <code><span class="example-io">specialRoads[0]</span></code><span class="example-io"> with</span><span class="example-io"> the cost 2.</span></li>
	<li><span class="example-io">(3,3) to (3,4) with a cost of |3 - 3| + |4 - 3| = 1.</span></li>
	<li><span class="example-io">(3,4) to (4,5). Use </span><code><span class="example-io">specialRoads[1]</span></code><span class="example-io"> with the cost</span><span class="example-io"> 1.</span></li>
</ol>

<p><span class="example-io">So the total cost is 1 + 2 + 1 + 1 = 5.</span></p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">start = [3,2], target = [5,7], specialRoads = [[5,7,3,2,1],[3,2,3,4,4],[3,3,5,5,5],[3,4,5,6,6]]</span></p>

<p><strong>Output:</strong> <span class="example-io">7</span></p>

<p><strong>Explanation:</strong></p>

<p>It is optimal not to use any special edges and go directly from the starting to the ending position with a cost |5 - 3| + |7 - 2| = 7.</p>

<p>Note that the <span class="example-io"><code>specialRoads[0]</code> is directed from (5,7) to (3,2).</span></p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">start = [1,1], target = [10,4], specialRoads = [[4,2,1,1,3],[1,2,7,4,4],[10,3,6,1,2],[6,1,1,2,3]]</span></p>

<p><strong>Output:</strong> <span class="example-io">8</span></p>

<p><strong>Explanation:</strong></p>

<ol>
	<li>(1,1) to (1,2) with a cost of |1 - 1| + |2 - 1| = 1.</li>
	<li>(1,2) to (7,4). Use <code><span class="example-io">specialRoads[1]</span></code><span class="example-io"> with the cost</span><span class="example-io"> 4.</span></li>
	<li>(7,4) to (10,4) with a cost of |10 - 7| + |4 - 4| = 3.</li>
</ol>
</div>

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

<ul>
	<li><code>start.length == target.length == 2</code></li>
	<li><code>1 &lt;= startX &lt;= targetX &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= startY &lt;= targetY &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= specialRoads.length &lt;= 200</code></li>
	<li><code>specialRoads[i].length == 5</code></li>
	<li><code>startX &lt;= x1<sub>i</sub>, x2<sub>i</sub> &lt;= targetX</code></li>
	<li><code>startY &lt;= y1<sub>i</sub>, y2<sub>i</sub> &lt;= targetY</code></li>
	<li><code>1 &lt;= cost<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def minimumCost(self, start: List[int], target: List[int], specialRoads: List[List[int]]) -> int:
        cs=int(1e5)+1
        def key(x,y):
            return x*cs+y
        def value(z):
            return (z//cs,z%cs)
        def dist(x1,y1):
            (a1,b1),(a2,b2)=value(x1),value(y1)
            return abs(a1-a2)+abs(b1-b2)

        set_nodes=set()
        set_nodes.add(key(*start))
        set_nodes.add(key(*target))
        for x1,y1,x2,y2,z in specialRoads:
            set_nodes.add(key(x1,y1))
            set_nodes.add(key(x2,y2))

        mat=defaultdict(dict)
        for i in set_nodes:
            for j in set_nodes:
                if i==j: continue
                mat[i][j]=dist(i,j)

        # print(mat)
        for x1,y1,x2,y2,z in specialRoads:
            key1=key(x1,y1)
            key2=key(x2,y2)
            # print(x1,y1,x2,y2)
            if key1==key2: continue
            mat[key1][key2]=min(z,mat[key1][key2])
        
        # for ke,ro in mat.items():
        #     print(value(ke),ke,ro)

        hp=[(0,key(*start))]
        target=key(*target)
        vis=set()
        min_time=defaultdict(lambda:float("inf"))
        while hp:
            while hp[0][1] in vis: heappop(hp)
            dis,cur=heappop(hp)
            vis.add(cur)
            min_time[cur]=dis
            if cur==target: break
            for neigh,wt in mat[cur].items():
                if dis+wt<min_time[neigh]: 
                    heappush(hp,(dis+wt,neigh))
                    min_time[neigh]=dis+wt
        return min_time[target]
540