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Unformatted text preview: using that road. Your goal is to ﬁnd the
leastexpensive path from your start to your destination.
You represent the city network using a directed graph G = (V, E, w) with weights w deﬁned on
both edges and vertices. The vertices V represent the cities and the edges E represent the roads.
The weight w(e) of an edge e represents the toll amount on that road. The weight w(v ) of a vertex
v is the price of ﬁlling your gas tank in that city (which is a ﬁxed price independent of how much
gas you have left, or ∞ if there is no gas available to purchase). You are allowed (but not obligated)
to end your journey with an empty tank, and you may assume that you always start your journey
with a full tank.
Below is an example graph that we will use to answer part (a). One seemingly cheap path from s
to t is (s, u1 , u2 , t) at a cost of $8. Unfortunately, this path is not valid, because our leaky gas tank
won’t permit moving across three edges without reﬁlling our gas tank.
One valid path is (s, u3 , u2 , t) at a cost of $22. (This is a valid path: we begin with a full tank,
trav...
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 Fall '11
 ErikDemaine

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