Implementation of Floyd-Warshall for all pair shortest distance #27518

Hrishabh-yadav · 2019-03-20T19:28:37Z

Currently sage has implemented Floyd-Warshall algorithm only for graphs with constant edge weight of 1 in cython. Implementation of Floyd warshall algorithm in cython for finding all pair shortest distances(ASAD) can be done. This algorithm can be further improved to find all pair shortest paths(ASAP)(Later).

We can also improve all pair shortest distance algorithm using the technique described in the following paper:-
https://pdfs.semanticscholar.org/568f/58737e752bf831a3e48036cf5205facce769.pdf
Usually complexity for finding ASAD with matrix multiplication comes with large constants that results in slow running time, But the implementation that is described in the paper can faster the running time for algorithm.

CC: @dcoudert

Component: graph theory

Author: Georgios Giapitzakis Tzintanos

Branch/Commit: dc082bb

Reviewer: David Coudert

Issue created by migration from https://trac.sagemath.org/ticket/27518

The text was updated successfully, but these errors were encountered:

giorgosgiapis · 2019-03-21T12:25:41Z

comment:1

I am working on this. I think we should take care of the case when edge_label is not a number (Probably set weight to 1).

giorgosgiapis · 2019-03-21T17:29:16Z

Branch: u/gh-giorgosgiapis/weighted_floyd_warshall

giorgosgiapis · 2019-03-21T17:33:23Z

Commit: ffcea9e

giorgosgiapis · 2019-03-21T17:33:23Z

New commits:

`ffcea9e`	`Modified Floyd-Warshall to work with weighted graphs with non-negative integer edge weights.`

dcoudert · 2019-03-21T17:40:40Z

comment:5

The usage of edge weights is not fully unified in the graph module. However, many methods have as input parameter wfunction, a function that inputs an edge and outputs a number. You should consider adding this parameter as input of the method.
There is an implementation of Floyd-Warshall algorithm in boost https://www.boost.org/doc/libs/1_69_0/libs/graph/doc/floyd_warshall_shortest.html

and we have an interface with boost src/sage/graphs/base/boost_*.

So you should first expose the boost implementation in Sagemath.
Moreover, Johnson algorithm is considered faster for sparse graphs (and most of the graphs are sparse).
- we can use the implementation of boost: 'Johnson_Boost'
- we can also use the implementation of igraph (an optional package), although not directly exposed. In fact method shortest_paths of igraph call Johnson algorithm and not Floyd-Warshall for the reason explained here: https://lists.nongnu.org/archive/html/igraph-help/2009-06/msg00029.html
  
  when igraph is installed, we can do H = G.igraph_graph() and then H.shortest_paths() with appropriate parameters.

Hrishabh-yadav · 2019-03-21T21:28:08Z

comment:6

Does the algorithm only work if given graph has all possible edges defined, Like if I input a Graph with 100 edges and 11 vertices :

 sage: 
....: from sage.graphs.distances_all_pairs import floyd_warshall
....: G= Graph()
....: G.allow_multiple_edges(True)
....: G.allow_loops(True)
....: G.weighted(True)
....: for i in range(100):
....:     u=randint(0,10)
....:     v=randint(0,10)
....:     l=randint(0,1000)
....:     G.add_edge(u,v,l)
....:     G.add_edge(v,u,l)
....:     
....: print(floyd_warshall(G,paths=False,distances=True))
....: 
....:     
---------------------------------------------------------------------------
LookupError                               Traceback (most recent call last)
<ipython-input-13-c620edb96b9c> in <module>()
     12     G.add_edge(v,u,l)
     13 
---> 14 print(floyd_warshall(G,paths=False,distances=True))
     15 
     16 

/home/hrishabh/sage/local/lib/python2.7/site-packages/sage/graphs/distances_all_pairs.pyx in sage.graphs.distances_all_pairs.floyd_warshall (build/cythonized/sage/graphs/distances_all_pairs.c:15993)()
   1623         for u_int in g.out_neighbors(v_int):
   1624             if gg.weighted() == True:
-> 1625                 label = str(gg.edge_label(v_int, u_int))
   1626                 try:
   1627                     weight = int(label)

/home/hrishabh/sage/local/lib/python2.7/site-packages/sage/graphs/generic_graph.pyc in edge_label(self, u, v)
  11310             True
  11311         """
> 11312         return self._backend.get_edge_label(u,v)
  11313 
  11314     def edge_labels(self):

/home/hrishabh/sage/local/lib/python2.7/site-packages/sage/graphs/base/sparse_graph.pyx in sage.graphs.base.sparse_graph.SparseGraphBackend.get_edge_label (build/cythonized/sage/graphs/base/sparse_graph.c:19894)()
   1643         cdef int v_int = self.get_vertex(v)
   1644         if not (<SparseGraph>self._cg).has_arc_unsafe(u_int, v_int):
-> 1645             raise LookupError("({0}, {1}) is not an edge of the graph.".format(repr(u),repr(v)))
   1646         if self.multiple_edges(None):
   1647             return [self.edge_labels[l_int] if l_int != 0 else None

LookupError: (1, 2) is not an edge of the graph.

But the same algorithm works for 500 edges and 11 vertices:-

....: from sage.graphs.distances_all_pairs import floyd_warshall
....: G= Graph()
....: G.allow_multiple_edges(True)
....: G.allow_loops(True)
....: G.weighted(True)
....: for i in range(500):
....:     u=randint(0,10)
....:     v=randint(0,10)
....:     l=randint(0,1000)
....:     G.add_edge(u,v,l)
....:     G.add_edge(v,u,l)
....:     
....: print(floyd_warshall(G,paths=False,distances=True))
....: 
....:     
{0: {0: 0, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1}, 1: {0: 1, 1: 0, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1}, 2: {0: 1, 1: 1, 2: 0, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1}, 3: {0: 1, 1: 1, 2: 1, 3: 0, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1}, 4: {0: 1, 1: 1, 2: 1, 3: 1, 4: 0, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1}, 5: {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 0, 6: 1, 7: 1, 8: 1, 9: 1, 10: 1}, 6: {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 0, 7: 1, 8: 1, 9: 1, 10: 1}, 7: {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 0, 8: 1, 9: 1, 10: 1}, 8: {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 0, 9: 1, 10: 1}, 9: {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 0, 10: 1}, 10: {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 1, 9: 1, 10: 0}}

Any particular reason?

giorgosgiapis · 2019-03-21T23:24:03Z

comment:7

Well, I think I figured it out. If we work on the original graph (gg) and not in the graph created in line

cdef CGraph g = <CGraph> gg._backend.c_graph()[0]

the problem seems to get fixed. I don't really know what causes the issue in the first place but in general, the above line seems to add to the graph g some edges which are not present in gg.

dcoudert · 2019-03-22T14:58:32Z

comment:8

cdef CGraph g = <CGraph> gg._backend.c_graph()[0]
the problem seems to get fixed. I don't really know what causes the issue in the first place but in general, the above line seems to add to the graph g some edges which are not present in gg.

g is in underlying backend graph format. This is a different format than Graph in which nodes and edges have different labels than in the gg. Although you can make much more efficient code using the backend directly, don't start playing with this if you are not an advanced user !

giorgosgiapis · 2019-03-26T17:54:49Z

comment:9

The boost implementation of Floyd-Warshall and Johnson algorithm returns only the distances and not the paths. Would it be a good idea to use it and retrieve the paths from the distance values? Personally, I can't think of a way to do this faster than O(V³).

dcoudert · 2019-03-27T10:41:14Z

comment:10

What is your objective: get distances or get paths ? For distances, you can use boost. For paths, this is another story, and you will have to code a specific algorithm. However, I'm not convinced that using Floyd-Warshall is a good idea if you want paths.

giorgosgiapis · 2019-03-27T10:47:19Z

comment:11

The current implementation has the option to also return the shortest paths between every two vertices. My objective is to preserve this feature. Would it be a good idea to reimplement Johnson's algorithm in Cython to be able to do this efficiently?

dcoudert · 2019-03-27T16:34:42Z

comment:12

Unless I'm mistaken, the methods don't return paths but the matrix of predecessors.
Given a matrix (or dict of dict) of distances, you can find the predecessors. For a source vertex s, for a vertex v, you find the neighbor u of v that is on a shortest s-v path by checking which vertex u is such that d(s, u) + w(u, v) = d(s, v). So it should be easy to add the functionality to the method implemented with boost.

giorgosgiapis · 2019-03-27T19:19:17Z

comment:13

Replying to @dcoudert:

Unless I'm mistaken, the methods don't return paths but the matrix of predecessors.
Given a matrix (or dict of dict) of distances, you can find the predecessors. For a source vertex s, for a vertex v, you find the neighbor u of v that is on a shortest s-v path by checking which vertex u is such that d(s, u) + w(u, v) = d(s, v). So it should be easy to add the functionality to the method implemented with boost.

But isn't this going to be O(V³)?

dcoudert · 2019-03-27T22:40:51Z

comment:14

It will be O(N.M) with a small constants...

giorgosgiapis · 2019-03-27T23:41:49Z

comment:15

Ok. Here is what I am planning to do in the next couple of days:

Expose the boost implementation of Floyd-Warshall in Sagemath
Modify johnson_shortest_paths() to return the predecessors matrix
Create a new method for all pairs shortest paths where the user will be able to select the algorithm used (like in shortest_paths method in src/sage/graphs/base/boost_graph.pyx).

Please let me know if there are any problems or pitfalls with what I proposed above.

dcoudert · 2019-03-28T07:35:36Z

comment:16

Expose the boost implementation of Floyd-Warshall in Sagemath

Modify johnson_shortest_paths() to return the predecessors matrix

OK

Create a new method for all pairs shortest paths where the user will be able to select the algorithm used (like in shortest_paths method in src/sage/graphs/base/boost_graph.pyx).

Have you checked the documentation of method shortest_path_all_pairs ? it already returns distances and predecessors matrix for different algorithms.

sagetrac-git · 2019-03-30T17:47:10Z

Branch pushed to git repo; I updated commit sha1. New commits:

`5aceae0`	`Added predecessors support in boost johnson_shortest_paths method`
`e87b06b`	`Merge branch 'u/gh-giorgosgiapis/weighted_floyd_warshall' of git://trac.sagemath.org/sage into johnson_pred`
`ef91486`	`Added predecessors support in boost johnson_shortest_paths method`

sagetrac-git · 2019-03-30T17:47:10Z

Changed commit from ffcea9e to ef91486

sagetrac-git · 2019-03-30T23:34:25Z

Branch pushed to git repo; I updated commit sha1. New commits:

`41358f9`	`Fixed minor issues`

sagetrac-git · 2019-03-30T23:34:25Z

Changed commit from ef91486 to 41358f9

dcoudert · 2019-03-31T16:59:04Z

comment:19

please ensure that comments and documentation are in 80 column modes. See the developer manual
for building the distance dict, why not using the previous code ?

     dist = {int_to_v[v]: {int_to_v[w]: correct_type(result[v][w])
                     for w in range(N) if result[v][w] != sys.float_info.max}
             for v in range(N)}

for building the predecessor dict, it's a pitty not using the boost graph that has already been build and that contains the correct edge weights. I agree it's much more complicated and requires to extend boost_interface.cpp to expose other methods in order to iterate over weighted edges.

Also, the pseudo code I have in mind is more like

for i, j, w in g.edges():
    for k in range(N):
        if result[k][j] == result[k][i] + w:
            pred[k][j] = i
        if not directed and result[k][i] == result[k][j] + w:
            pred[k][i] = j

It's only the pseudo-code and it must be improved, but it should give a simpler code.

So please check what can be done with boost graphs. It might not be so difficult to use, and should lead to much faster code.

sagetrac-git · 2019-03-31T17:51:33Z

Branch pushed to git repo; I updated commit sha1. New commits:

`8e8799d`	`Fixed comments, documentation and code for predecessors`

sagetrac-git · 2019-03-31T17:51:33Z

Changed commit from 41358f9 to 8e8799d

giorgosgiapis · 2019-03-31T17:57:22Z

comment:21

Replying to @dcoudert:

please ensure that comments and documentation are in 80 column modes. See the developer manual

Fixed all errors

for building the distance dict, why not using the previous code ?

     dist = {int_to_v[v]: {int_to_v[w]: correct_type(result[v][w])
                     for w in range(N) if result[v][w] != sys.float_info.max}
             for v in range(N)}

Changed back to the previous code

Also, the pseudo code I have in mind is more like
for i, j, w in g.edges():
    for k in range(N):
        if result[k][j] == result[k][i] + w:
            pred[k][j] = i
        if not directed and result[k][i] == result[k][j] + w:
            pred[k][i] = j
It's only the pseudo-code and it must be improved, but it should give a simpler code.

I adapted the pseudocode to my implementation. I will do some research on boost graphs and try to come up with a faster implementation using boost interface.

sagetrac-git · 2019-05-04T13:25:13Z

Branch pushed to git repo; I updated commit sha1. New commits:

`ef38d91`	`Fixed typo`

sagetrac-git · 2019-05-04T13:25:13Z

Changed commit from ee720d4 to ef38d91

dcoudert · 2019-05-06T16:55:50Z

Changed commit from ef38d91 to 803af4e

dcoudert · 2019-05-06T16:55:50Z

Changed branch from u/gh-giorgosgiapis/weighted_floyd_warshall to public/graphs/27518_floyd_warshall_boost

dcoudert · 2019-05-06T16:55:50Z

comment:57

I have rebased the branch on the last beta version and fixed the merge conflicts.
I pushed a new branch. It is in public/, so you can access and modify it.

Please check that it's working well.

New commits:

`803af4e`	`trac #27518: fix merge conflict with 8.8.beta4`

sagetrac-git · 2019-05-08T15:48:55Z

Branch pushed to git repo; I updated commit sha1. New commits:

`2a26a65`	`trac #27518: alignement`

sagetrac-git · 2019-05-08T15:48:55Z

Changed commit from 803af4e to 2a26a65

dcoudert · 2019-05-08T15:50:36Z

comment:59

For me this patch is good to go. I'm waiting confirmation from your side.

giorgosgiapis · 2019-05-08T15:55:58Z

comment:60

Replying to @dcoudert:

For me this patch is good to go. I'm waiting confirmation from your side.

After some testing everything seems to work as expected so it is indeed good to go.

dcoudert · 2019-05-09T07:05:04Z

Reviewer: David Coudert

dcoudert · 2019-05-09T07:05:04Z

comment:61

Please add your real name in Authors field.

LGTM.

giorgosgiapis · 2019-05-09T07:10:44Z

Author: Georgios Giapitzakis Tzintanos

vbraun · 2019-05-12T21:45:19Z

comment:63

Docbuild fails, see patchbot

kiwifb · 2019-05-13T00:35:34Z

comment:64

Just to make things clear, that is the failure I see

Error building the documentation.
multiprocessing.pool.RemoteTraceback:
"""
Traceback (most recent call last):
  File "/usr/lib/python3.7/multiprocessing/pool.py", line 121, in worker
    result = (True, func(*args, **kwds))
  File "/usr/lib/python3.7/multiprocessing/pool.py", line 44, in mapstar
    return list(map(*args))
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 79, in build_ref_doc
    getattr(ReferenceSubBuilder(doc, lang), format)(*args, **kwds)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 761, in _wrapper
    getattr(DocBuilder, build_type)(self, *args, **kwds)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 133, in f
    runsphinx()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/sphinxbuild.py", line 317, in runsphinx
    sys.stderr.raise_errors()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/sphinxbuild.py", line 252, in raise_errors
    raise OSError(self._error)
OSError: docstring of sage.graphs.base.boost_graph.johnson_shortest_paths:18: WARNING: Definition list ends without a blank line; unexpected unindent.
"""

The above exception was the direct cause of the following exception:

Traceback (most recent call last):
  File "sage_setup/docbuild/__main__.py", line 2, in <module>
    main()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 1730, in main
    builder()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 351, in _wrapper
    getattr(get_builder(document), 'inventory')(*args, **kwds)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 547, in _wrapper
    build_many(build_ref_doc, L)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 288, in build_many
    ret = x.get(99999)
  File "/usr/lib/python3.7/multiprocessing/pool.py", line 657, in get
    raise self._value
OSError: docstring of sage.graphs.base.boost_graph.johnson_shortest_paths:18: WARNING: Definition list ends without a blank line; unexpected unindent.

it looks like a stray space at the start of line 1085 of sage/graphs/base/boost_graph.pyx https://github.com/sagemath/sagetrac-mirror/blob/develop/src/sage/graphs/base/boost_graph.pyx?id=136cae2e3fb5f6dec8173f427da3321d49b0a835#n1085 is the cause of the problem. Removing it makes the documentation build here.

sagetrac-git · 2019-05-13T08:11:41Z

Changed commit from 2a26a65 to dc082bb

sagetrac-git · 2019-05-13T08:11:41Z

Branch pushed to git repo; I updated commit sha1. New commits:

`dc082bb`	`Fixed typo`

giorgosgiapis · 2019-05-13T08:16:38Z

comment:66

Replying to @kiwifb:

Just to make things clear, that is the failure I see

Error building the documentation.
multiprocessing.pool.RemoteTraceback:
"""
Traceback (most recent call last):
  File "/usr/lib/python3.7/multiprocessing/pool.py", line 121, in worker
    result = (True, func(*args, **kwds))
  File "/usr/lib/python3.7/multiprocessing/pool.py", line 44, in mapstar
    return list(map(*args))
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 79, in build_ref_doc
    getattr(ReferenceSubBuilder(doc, lang), format)(*args, **kwds)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 761, in _wrapper
    getattr(DocBuilder, build_type)(self, *args, **kwds)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 133, in f
    runsphinx()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/sphinxbuild.py", line 317, in runsphinx
    sys.stderr.raise_errors()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/sphinxbuild.py", line 252, in raise_errors
    raise OSError(self._error)
OSError: docstring of sage.graphs.base.boost_graph.johnson_shortest_paths:18: WARNING: Definition list ends without a blank line; unexpected unindent.
"""

The above exception was the direct cause of the following exception:

Traceback (most recent call last):
  File "sage_setup/docbuild/__main__.py", line 2, in <module>
    main()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 1730, in main
    builder()
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 351, in _wrapper
    getattr(get_builder(document), 'inventory')(*args, **kwds)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 547, in _wrapper
    build_many(build_ref_doc, L)
  File "/dev/shm/portage/sci-mathematics/sage-9999/work/sage-9999/src-python3_7/sage_setup/docbuild/__init__.py", line 288, in build_many
    ret = x.get(99999)
  File "/usr/lib/python3.7/multiprocessing/pool.py", line 657, in get
    raise self._value
OSError: docstring of sage.graphs.base.boost_graph.johnson_shortest_paths:18: WARNING: Definition list ends without a blank line; unexpected unindent.

it looks like a stray space at the start of line 1085 of sage/graphs/base/boost_graph.pyx https://github.com/sagemath/sagetrac-mirror/blob/develop/src/sage/graphs/base/boost_graph.pyx?id=136cae2e3fb5f6dec8173f427da3321d49b0a835#n1085 is the cause of the problem. Removing it makes the documentation build here.

Thank you for the info. Fixed it!

dcoudert · 2019-05-13T16:08:15Z

comment:67

@kiwifb: can you check if it's ok ? I have an issue building the documentation that is not related to this ticket. Thank you.

giorgosgiapis · 2019-05-14T12:39:21Z

comment:68

All tests seem to pass now!

dcoudert · 2019-05-16T13:24:11Z

comment:69

LGTM.

vbraun · 2019-05-21T12:15:53Z

Changed branch from public/graphs/27518_floyd_warshall_boost to dc082bb

Hrishabh-yadav mannequin added this to the sage-8.8 milestone Mar 20, 2019

Hrishabh-yadav mannequin added c: graph theory labels Mar 20, 2019

giorgosgiapis added the s: needs review label Mar 21, 2019

giorgosgiapis self-assigned this Mar 21, 2019

dcoudert added s: positive review and removed s: needs review labels May 9, 2019

vbraun added s: needs work and removed s: positive review labels May 12, 2019

dcoudert added s: positive review and removed s: needs work labels May 16, 2019

vbraun removed the s: positive review label May 21, 2019

vbraun closed this as completed in f8b9a2a May 21, 2019

dcoudert mentioned this issue May 25, 2019

py3: fix doctest in boost_graph.pyx #27862

Closed

Implementation of Floyd-Warshall for all pair shortest distance #27518

Implementation of Floyd-Warshall for all pair shortest distance #27518

Comments

Hrishabh-yadav mannequin commented Mar 20, 2019

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