Write me a program in C, with this info, "Problems that process input and generate a simple ``yes'' or ``no'' answer are called
decision problems. One class of decision problems, the NP-complete problems, are not
amenable to general efficient solutions. Other problems may be simple as decision
problems, but enumerating all possible ``yes'' answers may be very difficult (or at least
time-consuming).
This problem involves determining the number of routes available to an emergency
vehicle operating in a city of one-way streets.
The Problem
Given the intersections connected by one-way streets in a city, you are to write a program
that determines the number of different routes between each intersection. A route is a
sequence of one-way streets connecting two intersections.
Intersections are identified by non-negative integers. A one-way street is specified by a
pair of intersections. For example, j k indicates that there is a one-way street from
intersection j to intersection k. Note that two-way streets can be modelled by specifying
two one-way streets: j k and k j.
Consider a city of four intersections connected by the following one-way streets:
0 1
0 2
1 2
2 3
There is one route from intersection 0 to 1, two routes from 0 to 2 (the routes are 0 > 1 > 2
and 0 > 2), two routes from 0 to 3, one route from 1 to 2, one route from 1
to 3, one route from 2 to 3, and no other routes.
It is possible for an infinite number of different routes to exist. For example if the
intersections above are augmented by the street 3 2, there is still only one route from 0 to
1, but there are infinitely many different routes from 0 to 2. This is because the street
from 2 to 3 and back to 2 can be repeated yielding a different sequence of streets and
hence a different route. Thus the route 0 > 2 > 3 > 2 > 3 > 2 is a different route
than 0 > 2 > 3 > 2.
The Input
The input is a sequence of city specifications. Each specification begins with the number
of one-way streets in the city followed by that many one-way streets given as pairs of
intersections. Each pair j k represents a one-way street from intersection j to intersection
k. In all cities, intersections are numbered sequentially from 0 to the ``largest''
intersection. All integers in the input are separated by whitespace. The input is terminated
by end-of-file.
There will never be a one-way street from an intersection to itself. No city will have more
than 30 intersections.
The Output
For each city specification, a square matrix of the number of different routes from
intersection j to intersection k is printed. If the matrix is denoted M, then M[j][k] is the
number of different routes from intersection j to intersection k. The matrix M should be
printed in row-major order, one row per line. Each matrix should be preceded by the
string ``matrix for city k'' (with k appropriately instantiated, beginning with 0).
If there are an infinite number of different paths between two intersections a -1 should be
printed. DO NOT worry about justifying and aligning the output of each matrix. All
entries in a row should be separated by whitespace.
Sample Input
7 0 1 0 2 0 4 2 4 2 3 3 1 4 3
5
0 2
0 1 1 5 2 5 2 1
9
0 1 0 2 0 3
0 4 1 4 2 1
2 0
3 0
3 1
Sample Output
matrix for city 0
0 4 1 3 2
0 0 0 0 0
0 2 0 2 1
0 1 0 0 0
0 1 0 1 0
matrix for city 1
0 2 1 0 0 3
0 0 0 0 0 1
0 1 0 0 0 2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
matrix for city 2
-1 -1 -1 -1 -1
0 0 0 0 1
-1 -1 -1 -1 -1
-1 -1 -1 -1 -1
0 0 0 0 0"