All Pairs Shortest Path

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Points: 20 (partial)

Time limit: 0.4s

Java 2.0s

Memory limit: 512M

Author:

wleung_bvg

Problem types

You are given a graph with $N$ ~N~ vertices and $M$ ~M~ edges. Each edge is a directed edge from vertex $u_i$ ~u_i~ to vertex $v_i$ ~v_i~ with weight $w_i$ ~w_i~. You are asked to find the shortest path between all pairs of vertices. The graph may contain multiple edges between any pair of vertices, as well as self-loops. There may also be negative edge weights. It is not guaranteed that the graph is connected.

Constraints

For all subtasks:

$0 \le M \le 4\,000$ ~0 \le M \le 4\,000~

$1 \le u_i, v_i \le N$ ~1 \le u_i, v_i \le N~

Subtask 1 [10%]

$1 \le N \le 300$ ~1 \le N \le 300~

$-10^9 \le w_i \le 10^9$ ~-10^9 \le w_i \le 10^9~

Subtask 2 [10%]

$1 \le N \le 1\,000$ ~1 \le N \le 1\,000~

$1 \le w_i \le 10^9$ ~1 \le w_i \le 10^9~

Subtask 3 [80%]

$1 \le N \le 1\,000$ ~1 \le N \le 1\,000~

$-10^9 \le w_i \le 10^9$ ~-10^9 \le w_i \le 10^9~

Input Specification

The first line contains 2 integers $N$ ~N~ and $M$ ~M~, subject to the constraints above.

The next $M$ ~M~ lines describe the edges of the graph. Each line contains 3 integers, $u_i$ ~u_i~, $v_i$ ~v_i~, $w_i$ ~w_i~, indicating a directed edge from vertex $u_i$ ~u_i~ to vertex $v_i$ ~v_i~ with weight $w_i$ ~w_i~, subject to the constraints above.

Output Specification

This problem is graded with an identical checker. This includes whitespace characters. Ensure that every line of output is terminated with a \n character.

The output consists of $N$ ~N~ lines, each with $N$ ~N~ space separated integers. Each line should end with a new line character. Integer $j$ ~j~ on line $i$ ~i~ contains the distance of the shortest path from vertex $i$ ~i~ to vertex $j$ ~j~.

If there is no path from $i$ ~i~ to $j$ ~j~, INF should be printed instead of an integer.

If there is no lower bound on the length of the shortest path from $i$ ~i~ to $j$ ~j~ (or equivalently, there is a path from $i$ ~i~ to $j$ ~j~ that contains a negative edge cycle), -INF should be printed instead of an integer.

A negative edge cycle is a path that starts and ends on the same vertex, and the sum of the weights of those edges on that path is less than $0$ ~0~.

Sample Input 1

Sample Output 1

0 9 INF 8 INF
INF 0 INF 2 INF
INF INF 0 INF 4
INF INF INF 0 INF
INF INF INF INF 0

Sample Input 2

Sample Output 2

0 2 -8 INF -INF -INF
INF 0 -10 INF -INF -INF
INF INF 0 INF -INF -INF
INF INF INF 0 INF INF
INF INF INF INF -INF -INF
INF INF INF INF -INF -INF

Comments

4

Dingledooper commented on March 10, 2020, 4:59 a.m.

Floyd-Warshall passes, which is probably not intended.

5

wleung_bvg commented on March 10, 2020, 8:52 a.m.

Time limits have been adjusted.