Algorithm for finding the longest path of a DAG (#209)

* Algorithm for finding the longest path of a DAG

* Automated test issue

* Including longest path test

* Test fix

* Simplify

* Fix JET warning

* Fixing doctest

* Adjustments

* Simplify

---------

Co-authored-by: Matheus Diógenes Andrade <matheusdiogenesandrade@pop-os.localdomain>
Co-authored-by: Guillaume Dalle <22795598+gdalle@users.noreply.github.com>

Author: 3 people

docs/src/algorithms/shortestpaths.md

@@ -1,6 +1,6 @@
 # Shortest paths
 
-*Graphs.jl* includes standard algorithms for [shortest paths](https://en.wikipedia.org/wiki/Shortest_path_problem).
+*Graphs.jl* includes standard algorithms for [shortest paths](https://en.wikipedia.org/wiki/Shortest_path_problem) and longest paths.
 
 ## Index
 
@@ -19,6 +19,7 @@ Pages = [
     "shortestpaths/dijkstra.jl",
     "shortestpaths/floyd-warshall.jl",
     "shortestpaths/johnson.jl",
+    "shortestpaths/longest_path.jl",
     "shortestpaths/spfa.jl",
     "shortestpaths/yen.jl",
 ]

src/Graphs.jl

@@ -426,7 +426,10 @@ export
     independent_set,
 
     # vertexcover
-    vertex_cover
+    vertex_cover,
+
+    # longestpaths
+    dag_longest_path
 
 """
     Graphs
@@ -496,6 +499,7 @@ include("traversals/eulerian.jl")
 include("connectivity.jl")
 include("distance.jl")
 include("editdist.jl")
+include("shortestpaths/utils.jl")
 include("shortestpaths/astar.jl")
 include("shortestpaths/bellman-ford.jl")
 include("shortestpaths/dijkstra.jl")
@@ -504,6 +508,7 @@ include("shortestpaths/desopo-pape.jl")
 include("shortestpaths/floyd-warshall.jl")
 include("shortestpaths/yen.jl")
 include("shortestpaths/spfa.jl")
+include("shortestpaths/longest_path.jl")
 include("linalg/LinAlg.jl")
 include("operators.jl")
 include("persistence/common.jl")

src/shortestpaths/longest_path.jl

@@ -0,0 +1,35 @@
+"""
+    dag_longest_path(g, distmx=weights(g); topological_order=topological_sort_by_dfs(g))
+
+Return a longest path within the directed acyclic graph `g`, with distance matrix `distmx` and using `topological_order` to iterate on vertices.
+"""
+function dag_longest_path end
+
+@traitfn function dag_longest_path(
+    g::::IsDirected,
+    distmx::AbstractMatrix=weights(g);
+    topological_order=topological_sort_by_dfs(g),
+)
+    U = eltype(g)
+    T = eltype(distmx)
+
+    dists = zeros(T, nv(g))
+    parents = zeros(U, nv(g))
+
+    for v in topological_order
+        for u in inneighbors(g, v)
+            newdist = dists[u] + distmx[u, v]
+            if newdist > dists[v]
+                dists[v] = newdist
+                parents[v] = u
+            end
+        end
+    end
+
+    if isempty(dists)
+        return U[]
+    else
+        v = argmax(dists)
+        return path_from_parents(v, parents)
+    end
+end

src/shortestpaths/utils.jl

@@ -0,0 +1,9 @@
+function path_from_parents(target::Integer, parents::AbstractVector)
+    v = target
+    path = [v]
+    while parents[v] != v && parents[v] != zero(v)
+        v = parents[v]
+        pushfirst!(path, v)
+    end
+    return path
+end

test/runtests.jl

@@ -91,6 +91,7 @@ tests = [
     "edit_distance",
     "connectivity",
     "persistence/persistence",
+    "shortestpaths/utils",
     "shortestpaths/astar",
     "shortestpaths/bellman-ford",
     "shortestpaths/desopo-pape",
@@ -99,6 +100,7 @@ tests = [
     "shortestpaths/floyd-warshall",
     "shortestpaths/yen",
     "shortestpaths/spfa",
+    "shortestpaths/longest_path",
     "traversals/bfs",
     "traversals/bipartition",
     "traversals/greedy_color",

test/shortestpaths/longest_path.jl

@@ -0,0 +1,20 @@
+@testset "Longest path" begin
+    # empty DAG
+    g = DiGraph()
+    @test dag_longest_path(g) == Int[]
+
+    # unweighted DAG
+    g = SimpleDiGraphFromIterator(Edge.([(1, 2), (2, 3), (2, 4), (3, 5), (5, 6), (3, 7)]))
+    @test dag_longest_path(g) == [1, 2, 3, 5, 6]
+
+    # weighted DAG
+    n = 6
+    g = DiGraph(n)
+    A = [(1, 2, -5), (2, 3, 1), (3, 4, 1), (4, 5, 0), (3, 5, 4), (1, 6, 2)]
+    distmx = fill(NaN, n, n)
+    for (i, j, dist) in A
+        add_edge!(g, (i, j))
+        distmx[i, j] = dist
+    end
+    @test dag_longest_path(g, distmx) == [2, 3, 5]
+end

test/shortestpaths/utils.jl

@@ -0,0 +1,9 @@
+@testset "Path from parents" begin
+    using Graphs: path_from_parents
+    parents = [3, 0, 2, 5, 5]
+    @test path_from_parents(1, parents) == [2, 3, 1]
+    @test path_from_parents(2, parents) == [2]
+    @test path_from_parents(3, parents) == [2, 3]
+    @test path_from_parents(4, parents) == [5, 4]
+    @test path_from_parents(5, parents) == [5]
+end