Problem 1 (Pizza Delivery, 30 points)
Alice, who lives at a vertex s of a directed, weighted graph G=(V,E) (with non-negative
weights), is going to her friend's house at vertex h for dinner. Naturally, Alice wants to get
from s to h as soon as possible, but it appears that her friend is terrible at cooking, so along
the way she wants to get a pizza from a pizza store just in case her friend cooks something
inedible. Let's say the pizza stores form a subset of the vertices BsubV. Thus, starting at
s, Alice must go to some vertex binB of her choice, and then head from b to h using the
shortest overall route possible (assuming she wastes no time at the pizza store). We can help
Alice reach h as soon as possible, by solving the following sub-problems.
Compute the shortest distance from s to all pizza stores binB.
Compute the shortest distance from every pizza store binB to h. Note that this is the
dual of the single-source shortest path where we are now asking for the shortest path
from every node to a particular destination.
Combine part 1 and 2 to solve the full problem.
A straightforward solution is to run Dijkstra's algorithm twice (once in part 1 and once in
part 2). In this problem, you will improve this solution by running Dijkstra's algorithm only
once. Specifically, you should define a new graph G^(') on 2|V| vertices and at most 2|E|+|V|
edges (and appropriate weights for these edges), so that the original problem can be solved
using a single Dijkstra call on G^('). Briefly argue correctness of your proposed solution. (Hint:
This is another problem on reductions. You might want to refresh your memories on Problem
1 in Homework 8, which is also a reduction problem.)
Problem 1(Pizza Delivery30 points)
Alice, who lives at a vertex s of a directed,weighted graph G=(VE)(with non-negative weights), is going to her friend's house at vertex h for dinner. Naturally, Alice wants to get from s to h as soon as possible, but it appears that her friend is terrible at cooking, so along the way she wants to get a pizza from a pizza store just in case her friend cooks something inedible. Let's say the pizza stores form a subset of the vertices B C V. Thus, starting at s, Alice must go to some vertex b e B of her choice, and then head from b to h using the
shortest overall route possible (assuming she wastes no time at the pizza store). We can help Alice reach h as soon as possible, by solving the following sub-problems
1. Compute the shortest distance from s to all pizza stores b e B
2. Compute the shortest distance from every pizza store b e B to h. Note that this is the dual of the single-source shortest path where we are now asking for the shortest path
from every node to a particular destination.
3. Combine part 1 and 2 to solve the full problem
A straightforward solution is to run Dijkstra's algorithm twice (once in part 1 and once in part 2). In this problem, you will improve this solution by running Dijkstra's algorithm only once.Specifically,you should define a new graph Gon 2Vvertices and at most 2E+V edges (and appropriate weights for these edges), so that the original problem can be solved
using a single Dijkstra call on G. Briefly argue correctness of your proposed solution.Hint. This is another problem on reductions. You might want to refresh your memories on Problem 1 in Homework 8, which is also a reduction problem.)
