Talk:Floyd–Warshall algorithm

 Question: Is the main function correct?

Where it appears:

   next[i][j] ← next[i][k]

Should it be?:

   next[i][j] ← next[k][j]

Clearly, it seems to be a problem for many people. In fact, as in the web reference that goes with the pseudocode, the modification of the array next is path[i][j] := path[k][j]. I did the modification yesterday without looking the talk page (my mistake) thinking it was a minor error, but my modification was reverted by @MfR: with this explanation :

Pseudocode in this page computes the second node of the path from i to j, not the penultimate (as in reference).

which I don't really understand... In Introduction to Algorithms, Cormen et al., MIT Press, Third Edition at page 697, again, we see [k][j]... Raphaelbwiki (talk) 13:49, 4 June 2019 (UTC)

Can confirm that this doesn't work if implemented as written in the article right now. I am busy doing something and won't be fixing the article; whoever's reverting it when other people fix it needs to stop. — Preceding unsigned comment added by 174.34.23.247 (talk) 23:11, 29 November 2019 (UTC)

To answer the question: Yes, the main function is correct. The web reference that goes with the pseudocode uses a path array, but this article uses a next array. This is not just a difference in naming; the variables are used for different things. Specifically, path[u][v] answers the question "on the shortest path from u to v, which vertex will be visited last (before arriving at v)?" Whereas next[u][v] answers the question "on the shortest path from u to v, which vertex will be visited first (after leaving u)?"

This is why path from the web reference is initialized as path[u][v] = u for all edges ("on a direct connection from u to v, we must have come from u"), but next in the article is initialized as next[u][v] = v ("on a direct connection from u to v, we first go to v").

This difference in data representation has two consequences when it comes to reconstructing the full path:

1. The main loop in the PrintPath procedure from the web reference keeps checking Path[source][destination]. The value of source is constant, but destination is updated continuously. This is because Path essentially encodes the path information backwards: At each step we have to ask, "on the shortest path from source to destination, what was the last vertex we visited (before reaching destination)? And before that? And before that? And before that?" ... until we have retraced our steps all the way back to source, at which point the loop stops.
   On the other hand, the main loop in the Path procedure from the article keeps checking next[u][v]. Here the value of v (the destination) is constant, but u (the source) is updated continuously. This is because next encodes the path information forwards: At each step we ask, "on the shortest path from u to v, what is the first vertex we have to visit (after leaving u)? And after that? And after that?" ... until we reach our destination v, at which point the loop stops.
2. Since PrintPath from the web reference reconstructs the path backwards, it has to use a stack to reverse the order of visited vertices (LIFO). That's why there is a second loop in that code. But the Path procedure in the article works forwards, so it can just append each segment to the path variable as it goes.

84.149.142.109 (talk) 11:30, 28 December 2022 (UTC)

Pseudocode contains end-ifs but no end-fors

The pseudocode contains end-ifs but no end-fors:

I think it makes sense to have it be consistent: either no end-fors and no end-ifs or every for-loop terminated with an end-for and every if-statement terminated with end-if

1 let dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)
2 for each edge (u,v)
3    dist[u][v] ← w(u,v)  // the weight of the edge (u,v)
4 for each vertex v
5    dist[v][v] ← 0
6 for k from 1 to |V|
7    for i from 1 to |V|
8       for j from 1 to |V|
9          if dist[i][j] > dist[i][k] + dist[k][j] 
10             dist[i][j] ← dist[i][k] + dist[k][j]
11         end if  — Preceding unsigned comment added by 2601:282:8001:2E4A:845:D12F:594F:7A77 (talk) 22:43, 4 October 2018 (UTC) 

Could Use an Example Graph Not Containing a Negative Cycle

The Floyd-Warshall algorithm only finds shortest paths if there are no negative cycles. It is illustrative to see what happens if there is a negative cycle, so the existing example provided in the article is useful. However, it would also be nice to have a graph not containing any negative-cycles.

Below are some notes on a step-through of the algorithm for the given example:

upper-bound on cost from node 2 to node 1 is +4.
upper-bound on cost from node 1 to node 3 is -2.
cost on the path from 2 to 3 (going through 1 has) new upper bound of -2.
old upper-bound on cost from node 2 to 3 was +3

Next path: (4 --> 2 --> 3)
OBSERVE cost(4, 2) <= -1
OBSERVE cost(2, 3) <= -2
UPDATE cost(4, 3) <= -3
old cost(4, 3) was +inf

Next path: (1 --> 3 --> 4)
OBSERVE cost(1, 3) <= -2
OBSERVE cost(3, 4) <= +2
UPDATE cost(1, 4) <= 0

Next path: (2 --> 3 --> 4)
OBSERVE cost(2, 3) <= -2
OBSERVE cost(3, 4) <= +2
UPDATE cost(2, 4) <= 0

Next path: (4 --> 3 --> 4)
OBSERVE cost(4, 3) <= -3
OBSERVE cost(3, 4) <= +2
UPDATE cost(4, 4) <= -1

Negative cycle found. can leave 4 and return to 4 with net negative cost

This comment needs a signature and date. 128.226.2.54 (talk) 19:58, 23 January 2024 (UTC)

Attribution

The names are justified by the assertion that "it is essentially the same as". No justification is given for this assertion. I suspect it conceals some nontriviality.

Also, the citation to MathWorld is unwise. MathWorld is notably unreliable. A citation to a real publication is needed. 128.226.2.54 (talk) 20:01, 23 January 2024 (UTC)

The transitive closure and shortest path algorithms are all using a dynamic program to compute aggregate information about the same subsets of paths of a graph, in the same order. They differ only in what aggregate information they compute: whether it is the existence of a path or whether it is the minimum weight of the path. That is, where one has an OR, the other has a MIN. That is the only difference. The regular expression conversion algorithm has the same structure but replaces the OR of Boolean values or the MIN of numbers with the OR of regular expressions. The fact that these algorithms are really doing the same thing with different operations was formalized in the late 1960s and early 1970s using the theory of semirings; the semiring article has two relevant footnotes and I think this can also be sourced to the textbook of Aho, Hopcroft, and Ullman. —David Eppstein (talk) 21:02, 23 January 2024 (UTC)