HW 6: due Tues, December 6th
Suppose in our city we have ndestinations connected by roads whose dis-
tances are represented by weighted edges where weights satisfy the triangle
inequality (i.e. d(u, v )≤d(u, w) + d(w, v), for all possible destinations u,v,
and w. (You can assume the city is connected).
Define a destination’s “local neighborhood” to be the √nnodes that are
closest to the destination, including the destination itself and breaking ties
arbitrarily.
1. A landmark is just a chosen destination. Show that you can always
choose a set of O(√nlog n)landmarks so that every destination has at
least one landmark in its local neighborhood.
2. Consider the following route from destination uto destination v. If
uis in v’s local neighborhood, we route directly along the shortest
path. Otherwise, instead of routing from uto v, we route from uto v’s
closest landmark, and then from v’s closest landmark to v. Show that
the length of this route is never more than 3 times the distance of the
optimal route.
3. A k-spanner of a network is a subset of the edges of the original network
such that shortest distances in the spanner are at most ktimes shortest
distances in the original network. Show that any (weighted) graph has
a 3-spanner with at most O(n3/2log n) edges.