Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Find
Find the longest path between two vertices in a directed acyclic graph
Contributed by:
Charles Pooh
ResourceFunction["FindLongestPath"][g,s,t] finds the longest path from source vertex s to target vertex t in the directed acyclic graph g. | |
ResourceFunction["FindLongestPath"][{v→w,…},…] uses rules v→w to specify the graph g. |
Details and Options
ResourceFunction["FindLongestPath"][g,s,t] gives a path from s to t.
ResourceFunction["FindLongestPath"] can only be used for directed acyclic graphs (DAGs).
ResourceFunction["FindLongestPath"] uses the Bellman–Ford algorithm on a DAG with negative weights.
Examples
Basic Examples (2)
Find the longest path from the root of a directed acyclic graph (DAG) to a leaf:
| In[1]:= |
![g = \!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {4, 7}, {8, 6}, {8, 7}, {8, 3}, {8, 2}, {4, 8}, {8, 9}, {4, 9}, {5, 10}}, Null}, {
GraphLayout -> {"Dimension" -> 2}, VertexLabels -> {Automatic}}]]},
TagBox[GraphicsGroupBox[{
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{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[
0.7]}], {DiskBox[{0., 3.}, 0.04847715736040609], InsetBox["1", Offset[{2, 2}, {0.04847715736040609, 3.048477157360406}],
ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{-1., 0.}, 0.04847715736040609], InsetBox["2", Offset[{2, 2}, \
{-0.9515228426395939, 0.04847715736040609}], ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{0., 0.}, 0.04847715736040609], InsetBox["3", Offset[{2, 2}, \
{0.04847715736040609, 0.04847715736040609}], ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{3., 2.}, 0.04847715736040609], InsetBox["4", Offset[{2, 2}, {3.048477157360406, 2.048477157360406}], ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{-2., 2.}, 0.04847715736040609], InsetBox["5", Offset[{2, 2}, {-1.9515228426395939, 2.048477157360406}],
ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{2., 0.}, 0.04847715736040609], InsetBox["6", Offset[{2, 2}, {2.048477157360406, 0.04847715736040609}],
ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{3., 0.}, 0.04847715736040609], InsetBox["7", Offset[{2, 2}, {3.048477157360406, 0.04847715736040609}],
ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{2., 1.}, 0.04847715736040609], InsetBox["8", Offset[{2, 2}, {2.048477157360406, 1.0484771573604061}], ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{4., 0.}, 0.04847715736040609], InsetBox["9", Offset[{2, 2}, {4.048477157360406, 0.04847715736040609}],
ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}, {DiskBox[{-2., 1.}, 0.04847715736040609], InsetBox["10", Offset[{2, 2}, \
{-1.9515228426395939, 1.0484771573604061}], ImageScaled[{0, 0}],
BaseStyle->"Graphics"]}}}],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->{
"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]},
FormatType->TraditionalForm,
FrameTicks->None,
ImageSize->{217.95703125, Automatic}]\);](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/211dec4e0c024214.png)
|
| In[2]:= |
![ResourceFunction["FindLongestPath"][g, 1, 3]](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/01ff2a52ec1696c9.png){kind=small}
|
| Out[2]= |
|
Highlight the longest path:
| In[3]:= |
![HighlightGraph[g, PathGraph[%, DirectedEdges -> True]]](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/7db7e38a35a0a3e7.png){kind=small}
|
| Out[3]= |
|
Scope (2)
FindLongestPath works with directed acyclic graphs:
| In[4]:= |
![g = \!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{"A", "B", "C", "D", "E", "F", "G", "H"}, {{{1, 2}, {1, 3}, {2, 3}, {2, 4}, {3, 4}, {2, 5}, {4, 5}, {4, 6}, {4, 7}, {3, 7}, {5, 8}, {6, 8}, {7, 8}}, Null}, {
GraphLayout -> {
"LayeredDigraphEmbedding", "Orientation" -> Left}, VertexLabels -> {
Placed["Name", Center]}, VertexSize -> {Medium}}]]},
TagBox[GraphicsGroupBox[{
{Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{-4.999999999999998, 1.339744851455892*^-7}, {-3.9999999999999987`, 1.0717958811647137`*^-7}}, 0.09999999999999999], ArrowBox[BezierCurveBox[CompressedData["
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"]], 0.09999999999999999], ArrowBox[{{-3.9999999999999987`, 1.0717958811647137`*^-7}, {-2.999999973205102, 1.0000000803846907`}}, 0.09999999999999999], ArrowBox[BezierCurveBox[CompressedData["
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"]], 0.09999999999999999], ArrowBox[BezierCurveBox[CompressedData["
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"]], 0.09999999999999999], ArrowBox[{{-2.999999973205102, 1.0000000803846907`}, {-1.9999999999999993`, 5.3589794058235685`*^-8}}, 0.09999999999999999], ArrowBox[BezierCurveBox[CompressedData["
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"]], 0.09999999999999999], ArrowBox[{{-1.9999999999999993`, 5.3589794058235685`*^-8}, {-1.0000000267948967`, \
-0.9999999732051026}}, 0.09999999999999999], ArrowBox[{{-1.9999999999999993`, 5.3589794058235685`*^-8}, {-0.9999999999999997, 2.6794897029117842`*^-8}}, 0.09999999999999999], ArrowBox[{{-1.9999999999999993`, 5.3589794058235685`*^-8}, {-0.9999999732051026, 1.0000000267948967`}}, 0.09999999999999999], ArrowBox[{{-1.0000000267948967`, -0.9999999732051026}, {0.,
0.}}, 0.09999999999999999], ArrowBox[{{-0.9999999999999997, 2.6794897029117842`*^-8}, {
0., 0.}}, 0.09999999999999999], ArrowBox[{{-0.9999999732051026, 1.0000000267948967`}, {0., 0.}}, 0.09999999999999999]},
{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[
0.7]}], {
DiskBox[{-4.999999999999998, 1.339744851455892*^-7}, 0.09999999999999999], InsetBox["\<\"A\"\>", {-4.999999999999998, 1.339744851455892*^-7},
BaseStyle->"Graphics"]}, {
DiskBox[{-3.9999999999999987, 1.0717958811647137*^-7}, 0.09999999999999999], InsetBox["\<\"B\"\>", {-3.9999999999999987, 1.0717958811647137*^-7},
BaseStyle->"Graphics"]}, {
DiskBox[{-2.999999973205102, 1.0000000803846907}, 0.09999999999999999], InsetBox["\<\"C\"\>", {-2.999999973205102, 1.0000000803846907},
BaseStyle->"Graphics"]}, {
DiskBox[{-1.9999999999999993, 5.3589794058235685*^-8}, 0.09999999999999999], InsetBox["\<\"D\"\>", {-1.9999999999999993, 5.3589794058235685*^-8},
BaseStyle->"Graphics"]}, {
DiskBox[{-1.0000000267948967, -0.9999999732051026}, 0.09999999999999999], InsetBox["\<\"E\"\>", {-1.0000000267948967, -0.9999999732051026},
BaseStyle->"Graphics"]}, {
DiskBox[{-0.9999999999999997, 2.6794897029117842*^-8}, 0.09999999999999999], InsetBox["\<\"F\"\>", {-0.9999999999999997, 2.6794897029117842*^-8},
BaseStyle->"Graphics"]}, {
DiskBox[{-0.9999999732051026, 1.0000000267948967}, 0.09999999999999999], InsetBox["\<\"G\"\>", {-0.9999999732051026, 1.0000000267948967},
BaseStyle->"Graphics"]}, {DiskBox[{0., 0.}, 0.09999999999999999], InsetBox["\<\"H\"\>", {0., 0.},
BaseStyle->"Graphics"]}}}],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->{
"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]},
FormatType->TraditionalForm,
FrameTicks->None,
ImageSize->{288., Automatic}]\);](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/16a4e2699c79ee6a.png)
|
| In[5]:= |
![HighlightGraph[g, PathGraph[ResourceFunction["FindLongestPath"][g, "A", "H"], DirectedEdges -> True]]](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/33cda20eb4b17f15.png){kind=small}
|
| Out[5]= |
|
Use rules to specify the graph:
| In[6]:= |
|
| In[7]:= |
![path = ResourceFunction["FindLongestPath"][g, 0, 5]](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/75bb4ef79158d338.png){kind=small}
|
| Out[7]= |
|
| In[8]:= |
![HighlightGraph[g, PathGraph[path, DirectedEdges -> True]]](https://www.wolframcloud.com/obj/resourcesystem/images/f83/f83b39a2-ac8f-4fae-8c4f-638534efd2b2/106e343db5de2dfc.png){kind=small}
|
| Out[8]= |
|
Possible Issues (1)
FindLongestPath is left unevaluated if the input graph is not a DAG:
| In[9]:= |
![ResourceFunction["FindLongestPath"][\!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4, 5}, {{{1, 2}, {2, 3}, {3, 4}, {4, 5}, {3, 5}}, {{3, 1}}}, {
EdgeStyle -> {
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DefaultBaseStyle->{
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GridLinesStyle->Directive[
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Version History
- 1.0.0 – 18 December 2020
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This work is licensed under a Creative Commons Attribution 4.0 International License