Fast sparse graphs#
For an overview of graph data structures in sage, see
overview
.
Usage Introduction#
sage: from sage.graphs.base.sparse_graph import SparseGraph
Sparse graphs are initialized as follows:
sage: S = SparseGraph(nverts = 10, expected_degree = 3, extra_vertices = 10)
This example initializes a sparse graph with room for twenty vertices, the first
ten of which are in the graph. In general, the first nverts
are “active.”
For example, see that 9 is already in the graph:
sage: S.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: S.add_vertex(9)
9
sage: S.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
But 10 is not, until we add it:
sage: S.add_vertex(10)
10
sage: S.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
You can begin working with unlabeled arcs right away as follows:
sage: S.add_arc(0,1)
sage: S.add_arc(1,2)
sage: S.add_arc(1,0)
sage: S.has_arc(7,3)
False
sage: S.has_arc(0,1)
True
sage: S.in_neighbors(1)
[0]
sage: S.out_neighbors(1)
[0, 2]
sage: S.del_all_arcs(0,1)
sage: S.all_arcs(0,1)
[]
sage: S.all_arcs(1,2)
[0]
sage: S.del_vertex(7)
sage: S.all_arcs(7,3)
Traceback (most recent call last):
...
LookupError: vertex (7) is not a vertex of the graph
Sparse graphs support multiple edges and labeled edges, but requires that the labels be positive integers (the case label = 0 is treated as no label).
sage: S.add_arc_label(0,1,-1)
Traceback (most recent call last):
...
ValueError: Label (-1) must be a nonnegative integer.
sage: S.add_arc(0,1)
sage: S.arc_label(0,1)
0
Note that arc_label
only returns the first edge label found in the specified
place, and this can be in any order (if you want all arc labels, use
all_arcs
):
sage: S.add_arc_label(0,1,1)
sage: S.arc_label(0,1)
1
sage: S.all_arcs(0,1)
[0, 1]
Zero specifies only that there is no labeled arc:
sage: S.arc_label(1,2)
0
So do not be fooled:
sage: S.all_arcs(1,2)
[0]
sage: S.add_arc(1,2)
sage: S.arc_label(1,2)
0
Instead, if you work with unlabeled edges, be sure to use the right functions:
sage: T = SparseGraph(nverts = 3, expected_degree = 2)
sage: T.add_arc(0,1)
sage: T.add_arc(1,2)
sage: T.add_arc(2,0)
sage: T.has_arc(0,1)
True
Sparse graphs are by their nature directed. As of this writing, you need to do operations in pairs to treat the undirected case (or use a backend or a Sage graph):
sage: T.has_arc(1,0)
False
Multiple unlabeled edges are also possible:
sage: for _ in range(10): S.add_arc(5,4)
sage: S.all_arcs(5,4)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
The curious developer is encouraged to check out the unsafe
functions,
which do not check input but which run in pure C.
Underlying Data Structure#
The class SparseGraph
contains the following variables which are inherited
from CGraph
(for explanation, refer to the documentation there):
cdef int num_verts
cdef int num_arcs
cdef int *in_degrees
cdef int *out_degrees
cdef bitset_t active_vertices
It also contains the following variables:
cdef int hash_length
cdef int hash_mask
cdef SparseGraphBTNode **vertices
For each vertex u
, a hash table of length hash_length
is instantiated.
An arc (u, v)
is stored at u * hash_length + hash(v)
of the array
vertices
, where hash
should be thought of as an arbitrary but fixed hash
function which takes values in 0 <= hash < hash_length
. Each address may
represent different arcs, say (u, v1)
and (u, v2)
where
hash(v1) == hash(v2)
. Thus, a binary tree structure is used at this step to
speed access to individual arcs, whose nodes (each of which represents a pair
(u,v)
) are instances of the following type:
cdef struct SparseGraphBTNode:
int vertex
int number
SparseGraphLLNode *labels
SparseGraphBTNode *left
SparseGraphBTNode *right
Which range of the vertices
array the root of the tree is in determines
u
, and vertex
stores v
. The integer number
stores only the
number of unlabeled arcs from u
to v
.
Currently, labels are stored in a simple linked list, whose nodes are instances of the following type:
cdef struct SparseGraphLLNode:
int label
int number
SparseGraphLLNode *next
The int label
must be a positive integer, since 0 indicates no label, and
negative numbers indicate errors. The int number
is the number of arcs with
the given label.
TODO: Optimally, edge labels would also be represented by a binary tree, which
would help performance in graphs with many overlapping edges. Also, a more
efficient binary tree structure could be used, although in practice the trees
involved will usually have very small order, unless the degree of vertices
becomes significantly larger than the expected_degree
given, because this is
the size of each hash table. Indeed, the expected size of the binary trees is
\(\frac{\text{actual degree}}{\text{expected degree}}\). Ryan Dingman, e.g., is
working on a general-purpose Cython-based red black tree, which would be optimal
for both of these uses.
- class sage.graphs.base.sparse_graph.SparseGraph#
Bases:
CGraph
Compiled sparse graphs.
sage: from sage.graphs.base.sparse_graph import SparseGraph
Sparse graphs are initialized as follows:
sage: S = SparseGraph(nverts = 10, expected_degree = 3, extra_vertices = 10)
INPUT:
nverts
– non-negative integer, the number of vertices.expected_degree
– non-negative integer (default: 16), expected upperbound on degree of vertices.
extra_vertices
– non-negative integer (default: 0), how many extravertices to allocate.
verts
– optional list of vertices to addarcs
– optional list of arcs to add
The first
nverts
are created as vertices of the graph, and the nextextra_vertices
can be freely added without reallocation. See top level documentation for more details. The inputverts
andarcs
are mainly for use in pickling.- add_arc_label(u, v, l=0)#
Add arc
(u, v)
to the graph with labell
.INPUT:
u, v
– non-negative integers, must be in selfl
– a positive integer label, or zero for no label
EXAMPLES:
sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc_label(0,1) sage: G.add_arc_label(4,7) Traceback (most recent call last): ... LookupError: vertex (7) is not a vertex of the graph sage: G.has_arc(1,0) False sage: G.has_arc(0,1) True sage: G.add_arc_label(1,2,2) sage: G.arc_label(1,2) 2
- in_degree(v)#
Returns the in-degree of
v
INPUT:
v
– integer
EXAMPLES:
sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc(0,1) sage: G.add_arc(1,2) sage: G.add_arc(1,3) sage: G.in_degree(0) 0 sage: G.in_degree(1) 1
- is_directed()#
Return whether the graph is directed.
EXAMPLES:
sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.is_directed() True sage: G = SparseGraph(5, directed=False) sage: G.is_directed() False
- out_degree(u)#
Returns the out-degree of
v
INPUT:
u
– integer
EXAMPLES:
sage: from sage.graphs.base.sparse_graph import SparseGraph sage: G = SparseGraph(5) sage: G.add_arc(0,1) sage: G.add_arc(1,2) sage: G.add_arc(1,3) sage: G.out_degree(0) 1 sage: G.out_degree(1) 2
- realloc(total)#
Reallocate the number of vertices to use, without actually adding any.
INPUT:
total
– integer, the total size to make the array
Returns -1 and fails if reallocation would destroy any active vertices.
EXAMPLES:
sage: from sage.graphs.base.sparse_graph import SparseGraph sage: S = SparseGraph(nverts=4, extra_vertices=4) sage: S.current_allocation() 8 sage: S.add_vertex(6) 6 sage: S.current_allocation() 8 sage: S.add_vertex(10) 10 sage: S.current_allocation() 16 sage: S.add_vertex(40) Traceback (most recent call last): ... RuntimeError: requested vertex is past twice the allocated range: use realloc sage: S.realloc(50) sage: S.add_vertex(40) 40 sage: S.current_allocation() 50 sage: S.realloc(30) -1 sage: S.current_allocation() 50 sage: S.del_vertex(40) sage: S.realloc(30) sage: S.current_allocation() 30
- class sage.graphs.base.sparse_graph.SparseGraphBackend#
Bases:
CGraphBackend
Backend for Sage graphs using SparseGraphs.
sage: from sage.graphs.base.sparse_graph import SparseGraphBackend
This class is only intended for use by the Sage Graph and DiGraph class. If you are interested in using a SparseGraph, you probably want to do something like the following example, which creates a Sage Graph instance which wraps a SparseGraph object:
sage: G = Graph(30, sparse=True) sage: G.add_edges([(0,1), (0,3), (4,5), (9, 23)]) sage: G.edges(sort=True, labels=False) [(0, 1), (0, 3), (4, 5), (9, 23)]
Note that Sage graphs using the backend are more flexible than SparseGraphs themselves. This is because SparseGraphs (by design) do not deal with Python objects:
sage: G.add_vertex((0,1,2)) sage: sorted(list(G), ....: key=lambda x: (isinstance(x, tuple), x)) [0, ... 29, (0, 1, 2)] sage: from sage.graphs.base.sparse_graph import SparseGraph sage: SG = SparseGraph(30) sage: SG.add_vertex((0,1,2)) Traceback (most recent call last): ... TypeError: an integer is required
- get_edge_label(u, v)#
Return the edge label for
(u, v)
.INPUT:
u,v
– the vertices of the edge
EXAMPLES:
sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.add_edges([(0,1,1), (2,3,2), (4,5,3), (5,6,2)], False) sage: list(D.iterator_edges(range(9), True)) [(0, 1, 1), (2, 3, 2), (4, 5, 3), (5, 6, 2)] sage: D.get_edge_label(3,2) 2
- has_edge(u, v, l)#
Returns whether this graph has edge
(u,v)
with labell
. Ifl
isNone
, return whether this graph has an edge(u,v)
with any label.INPUT:
u, v
– the vertices of the edgel
– the edge label, orNone
EXAMPLES:
sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: D.add_edges([(0,1), (2,3), (4,5), (5,6)], False) sage: D.has_edge(0,1,None) True
- multiple_edges(new)#
Get/set whether or not
self
allows multiple edges.INPUT:
new
– boolean (to set) orNone
(to get)
EXAMPLES:
sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: G.multiple_edges(True) sage: G.multiple_edges(None) True sage: G.multiple_edges(False) sage: G.multiple_edges(None) False sage: G.add_edge(0,1,0,True) sage: G.add_edge(0,1,0,True) sage: list(G.iterator_out_edges(range(9), True)) [(0, 1, 0)]
- set_edge_label(u, v, l, directed)#
Label the edge
(u,v)
byl
.INPUT:
u,v
– the vertices of the edgel
– the edge labeldirected
– ifFalse
, also set(v,u)
with labell
EXAMPLES:
sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(9) sage: G.add_edge(1,2,None,True) sage: G.set_edge_label(1,2,'a',True) sage: list(G.iterator_out_edges(range(9), True)) [(1, 2, 'a')]
Note that it fails silently if there is no edge there:
sage: G.set_edge_label(2,1,'b',True) sage: list(G.iterator_out_edges(range(9), True)) [(1, 2, 'a')]