Numbering Paths

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 solu- tions. 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. 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 modeled by specifying two one-way streets: j k and k j. Consider a city of four intersections connected by the following one-way streets: 01 02 12 23 There is one route from intersection 0 to 1, two routes from 0 to 2 (the routes are 0 → 1 → 2 and 0 → 2), one route 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. 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. 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).

2/2 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 701020424233143 5 02 01152521 9 010203 041421 20 30 31 Sample Output matrix for city 0 04132 00000 02021 01000 01010 matrix for city 1 021003 000001 010002 000000 000000 000000 matrix for city 2 -1 -1 -1 -1 -1 00001 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 00000