Case Study: Shortest Paths
The shortest path
problem is a classic problem in computer science. It asks, given a graph G
= (V, E) and a set of distances dij for each
edge (i, j) in E, what is the shortest path between two
given nodes in V. These two nodes are often referred to as s (for
source) and d (for destination).
The shortest path problem is a classic problem in computer
science that asks for the shortest path between two given nodes in a graph,
given a set of distances for the edges in the graph. This problem is not a
decision problem, as it does not involve determining whether a given instance
has a certain property or not. Instead, it involves finding a solution (the
shortest path between the two nodes) that satisfies a specific condition
(minimizing the total distance).
However, we can make the shortest path problem into an
equivalent decision problem by asking whether there exists a path between the
two nodes with a total distance less than a given value k. This decision
problem can be phrased as: "Given a graph G = (V, E), a set of distances
dij for each edge (i, j) in E, and a value k, is there a path between two given
nodes s and d in V with a total distance less than k?" This decision
problem is a yes/no question, and it can be used to determine whether a given
instance of the shortest path problem has a solution that satisfies the
condition of having a total distance less than k.
● Is this a decision problem, or something
else?
The
decision version of the shortest path problem is in NP, because a
"yes" instance of the problem can be verified in polynomial time by
using a witness (the path itself). However, it is not known whether the problem
can be solved in polynomial time, so it is not known whether the shortest path
problem is in P. The decision version of the shortest path problem is not in
co-NP, because it does not involve verifying "no" instances of the
problem.
● If not, can we make it into an equivalent
decision problem?
● Is the decision problem in P?
● Is it in NP?
● Is it in Co-NP?
To summarize, the shortest path problem is a problem that
involves finding the shortest path between two given nodes in a graph, given a
set of distances for the edges in the graph. The problem is not a decision
problem, but we can make it into an equivalent decision problem by asking
whether there exists a path between the two nodes with a total distance less
than a given value k. The decision version of the shortest path problem is in
NP, because a "yes" instance of the problem can be verified in
polynomial time using a witness (the path itself), but it is not known whether
the problem can be solved in polynomial time. The decision version of the
shortest path problem is not in co-NP, because it does not involve verifying
"no" instances of the problem.
Shortest path decision
The original
shortest path problem doesn’t have a true or false answer. It requires the
output of an actual path between two vertices of a graph.
● In its current form, shortest path can be
called an optimisation problem or more generally, a function
problem.
● We can modify it so that it become a
decision problem, a bit like last week’s version of TSP
○ Instead of asking for the shortest, we can
ask if there is a path with total distance less than k for some value of
k. This question has a yes/no answer.
We often do this
with optimisation problems because it makes them much easier to classify
Optimization problems, like the shortest path problem, do
not have a true or false answer and require the output of an actual solution
that satisfies a specific condition (in this case, minimizing the total
distance). By modifying the problem to ask whether there exists a path with a
total distance less than a given value k, we can turn it into a decision
problem, which has a yes/no answer. This can be useful because decision
problems are easier to classify and can be more efficiently solved using
algorithms and data structures designed for decision problems.
It is often the case that optimization problems can be
transformed into decision problems by asking whether a solution exists that satisfies
a certain condition. For example, in the case of the traveling salesman
problem, we can ask whether there exists a tour with a total distance less than
a given value k, rather than asking for the shortest possible tour. This
transformation can make it easier to classify the problem and find efficient
algorithms for solving it.
Shortest path problem classification
Is shortest path in P?
YES: run Dijkstra’s algorithm and if the shortest path is
less than k, return true. Otherwise return false.
Is it in NP?
YES: given a path from s to d as a witness, we
can easily check it’s a valid path and add up the edge distances to see if the
total is less than k (alternatively we can just ignore the witness and
use Dijkstra’s algorithm)
Is it in Co-NP? YES: simply ignore the witness and run
Dijkstra’s algorithm
It is not known whether the
shortest path problem is in P, which is the class of problems that can be
solved in polynomial time. The shortest path problem involves finding the
shortest path between two given nodes in a graph, given a set of distances for
the edges in the graph. While the decision version of the shortest path
problem, which asks whether there exists a path with a total distance less than
a given value k, is in NP (nondeterministic polynomial time), it is not known
whether the problem can be solved in polynomial time.
NP is the class of problems
whose yes instances can be verified in polynomial time, given a suitable
witness. In the case of the shortest path problem, a witness would be the path
itself. However, finding a solution to the shortest path problem may require
more time and computational resources than simply verifying a solution, so it
is not known whether the problem is in P. It is possible that the shortest path
problem is NP-hard, which means that it is at least as difficult to solve as
the hardest problems in NP.
Dijkstra's
algorithm is a graph search algorithm that solves the single-source shortest
path problem for a graph with nonnegative edge weights, producing a shortest path
tree. This algorithm is often used in routing and as a subroutine in other
graph algorithms. It was conceived by computer scientist Edsger W. Dijkstra in
1956 and published three years later.
Here is an
example implementation of Dijkstra's algorithm in Python, with line-by-line
comments:
import math
# Helper function to find the index of the
vertex with the minimum distance
def find_min_dist(distance, visited):
min_distance = math.inf
min_index = None
for
i, dist in enumerate(distance):
if dist < min_distance and i not in
visited:
min_distance = dist
min_index = i
return min_index
# Function to perform Dijkstra's algorithm
def dijkstra(graph, source):
#
Number of vertices in the graph
n =
len(graph)
#
Initialize distances and visited lists
distance = [math.inf] * n
visited = []
#
Set the distance of the source vertex to 0
distance[source] = 0
#
Iterate n - 1 times
for
_ in range(n - 1):
# Find the vertex with the minimum distance
min_index = find_min_dist(distance, visited)
# Mark the vertex as visited
visited.append(min_index)
# Update the distances of the neighboring vertices
for i, dist in enumerate(graph[min_index]):
if dist != 0 and i not in visited:
distance[i] = min(distance[i], distance[min_index] + dist)
#
Return the distance list
return distance
# Test the function with a graph represented
as an adjacency matrix
graph = [[0, 4, 0, 0, 0, 0, 0, 8, 0],
[4, 0, 8, 0, 0, 0, 0, 11, 0],
[0, 8, 0, 7, 0, 4, 0, 0, 2],
[0, 0, 7, 0, 9, 14, 0, 0, 0],
[0, 0, 0, 9, 0, 10, 0, 0, 0],
[0, 0, 4, 14, 10, 0, 2, 0, 0],
[0, 0, 0, 0, 0, 2, 0, 1, 6],
[8, 11, 0, 0, 0, 0, 1, 0, 7],
[0, 0, 2, 0, 0, 0, 6, 7, 0]]
# The source vertex is vertex 0
source = 0
# Get the distance list for the graph
distances = dijkstra(graph, source)
# Print the distance from the source vertex to
each other vertex
print(distances)
This code will initialize a distance list with infinities
and a visited list with no vertices. It will then set the distance of the
source vertex to 0 and iterate n - 1 times, where n is the number of vertices
in the graph. At each iteration, it will find the vertex with the minimum
distance that has not yet been visited and mark it as visited. It will then
update the distances of the neighbouring vertices by taking the minimum of the
current distance and the distance through the current vertex. Finally, it will
return the distance list.
General result
In fact, we can say with certainty that all problems in P
are also in both NP and Co-NP. Since both yes and no instances can be solved
(and therefore verified) efficiently.
However, we must ask an important question at this stage:
Are
there some problems which are in NP but not in P?
IT’S WORTH A MILLION
DOLLARS IF YOU CAN ANSWER IT
Yes, there are problems that are in NP but not known to be
in P. These are known as NP-complete problems. NP-complete problems are a class
of problems that are at least as hard as any problem in NP, and if any NP-complete
problem can be solved in polynomial time, then all problems in NP can be solved
in polynomial time. This means that if we can find a polynomial-time algorithm
for any NP-complete problem, we can use it to solve all problems in NP in
polynomial time as well. Some examples of NP-complete problems include the
traveling salesman problem, the knapsack problem, and the Boolean
satisfiability problem.
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