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3).. . Also at each step I disregard the roads that has been travelled already for that particular path. In order to do that, you must be running Dijkstra on a graph with exponentially more vertices than the input graph.


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The Grassfire algorithm only works on a particular kind of graph; Dijkstra's algorithm works on any graph.


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Your dynamic program is a step in the right direction. My idea would be to maintain two values for a node v: $L_v$: The longest path starting from v going downwards (these corresponds to your dynamic program and the first two cases) $U_v$: and the longest path in the subtree of v (i.e., case 3). You already gave the formula for $L_v$. For $U_v$ you take ...


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