# First construct the vertex set v for each vertex v v

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Unformatted text preview: valid path from a city s to a city t, or report that no such path exists. Solution: We will solve this problem by transforming G = (V, E, w), the graph of cities, roads, and toll/gas costs, into a new graph G = (V , E , w ). We will then use Dijkstra’s algorithm to ﬁnd the desired path. First construct the vertex set V . For each vertex v ∈ V , create three vertices in V : vf , vh , and ve . These stand for a full tank, a half-full tank, and an empty tank, respectively. This operation takes O(V ) time, since we iterate |V | times and create three new vertices each time. Next we will discuss the construction of E and w . For each edge (u, v ) = e ∈ E , create edges in E as follows. Create edges (uf , vh ) and (uh , ve ) to represent the emptying of half the tank during the move from u to v . The weights on both of these edges will be the original weight w(u, v ). This operation takes O(E ) time, since we iterate |E | times and insert two new edges into E each time. Finally, we will represent the cost of fueling up by creating additional edges as follows. For every vertex v ∈ V , we ch...
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## This document was uploaded on 03/17/2014 for the course ELECTRICAL 6.006 at MIT.

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