Static Sparse Graphs¶
What is the point ?¶
This class implements a Cython (di)graph structure made for efficiency. The graphs are static, i.e. no add/remove vertex/edges methods are available, nor can they easily or efficiently be implemented within this data structure.
The data structure, however, is made to save the maximum amount of computations for graph algorithms whose main operation is to list the outneighbours of a vertex (which is precisely what BFS, DFS, distance computations and the flowrelated stuff waste their life on).
The code contained in this module is written Cstyle. The purpose is efficiency and simplicity.
For an overview of graph data structures in sage, see
overview
.
Author:
 Nathann Cohen (2011)
Data structure¶
The data structure is actually pretty simple and compact. short_digraph
has
five fields
n
(int
); the number of vertices in the graphm
(int
); the number of edges in the graphedges
(uint32_t *
); array whose length is the number of edges of the graphneighbors
(uint32_t **
); this array has size \(n+1\), and describes how the data ofedges
should be read : the neighbors of vertex \(i\) are the elements ofedges
addressed byneighbors[i]...neighbors[i+1]1
. The elementneighbors[n]
, which corresponds to no vertex (they are numbered from \(0\) to \(n1\)) is present so that it remains easy to enumerate the neighbors of vertex \(n1\) : the last of them is the element addressed byneighbors[n]1
.edge_labels
(list
); this cython list associates a label to each edge of the graph. If a given edge is represented byedges[i]
, this its associated label can be found atedge_labels[i]
. This object is usually NULL, unless the call toinit_short_digraph
explicitly requires the labels to be stored in the data structure.
In the example given above, vertex 0 has 2,3,5,7,8 and 9 as outneighbors, but
not 4, which is an outneighbour of vertex 1. Vertex \(n1\) has 2, 5, 8 and 9 as
outneighbors. neighbors[n]
points toward the cell immediately after the
end of edges
, hence outside of the allocated memory. It is used to
indicate the end of the outneighbors of vertex \(n1\)
Iterating over the edges
This is the one thing to have in mind when working with this data structure:
cdef list_edges(short_digraph g):
cdef int i, j
for i in range(g.n):
for j in range(g.neighbors[i+1]g.neighbors[i]):
print("There is an edge from {} to {}".format(i, g.neighbors[i][j]))
Advantages
Two great points :
 The neighbors of a vertex are C types, and are contiguous in memory.
 Storing such graphs is incredibly cheaper than storing Python structures.
Well, I think it would be hard to have anything more efficient than that to enumerate outneighbors in sparse graphs ! :)
Technical details¶
 When creating a
short_digraph
from aGraph
orDiGraph
namedG
, the \(i^{\text{th}}\) vertex corresponds by default toG.vertices()[i]
. Using optional parametervertex_list
, you can specify the order of the vertices. Then \(i^{\text{th}}\) vertex will corresponds tovertex_list[i]
.  Some methods return
bitset_t
objets when lists could be expected. There is a very usefulbitset_list
function for this kind of problems :)  When the edges are labelled, most of the space taken by this graph is taken by edge labels. If no edge is labelled then this space is not allocated, but if any edge has a label then a (possibly empty) label is stored for each edge, which can double the memory needs.
 The data structure stores the number of edges, even though it appears that
this number can be reconstructed with
g.neighbors[n]g.neighbors[0]
. The trick is that not all elements of theg.edges
array are necessarily used : when an undirected graph contains loops, only one entry of the array of size \(2m\) is used to store it, instead of the expected two. Storing the number of edges is the only way to avoid an uselessly costly computation to obtain the number of edges of an undirected, looped, AND labelled graph (think of several loops on the same vertex with different labels).  The codes of this module are well documented, and many answers can be found directly in the code.
Cython functions¶
init_short_digraph(short_digraph g, G) 
Initialize short_digraph g from a Sage (Di)Graph. 
int n_edges(short_digraph g) 
Return the number of edges in g 
int out_degree(short_digraph g, int i) 
Return the outdegree of vertex \(i\) in g 
has_edge(short_digraph g, int u, int v) 
Test the existence of an edge. 
edge_label(short_digraph g, int * edge) 
Return the label associated with a given edge 
init_empty_copy(short_digraph dst, short_digraph src) 
Allocate dst so that it can contain as many vertices and edges as src . 
init_reverse(short_digraph dst, short_digraph src) 
Initialize dst to a copy of src with all edges in the opposite direction. 
free_short_digraph(short_digraph g) 
Free the ressources used by g 
Connectivity
can_be_reached_from(short_digraph g, int src, bitset_t reached)
Assumingbitset_t reached
has size at leastg.n
, this method updatesreached
so that it represents the set of vertices that can be reached fromsrc
ing
.
strongly_connected_component_containing_vertex(short_digraph g, short_digraph g_reversed, int v, bitset_t scc)
Assumingbitset_t reached
has size at leastg.n
, this method updatesscc
so that it represents the vertices of the strongly connected component containingv
ing
. The variableg_reversed
is assumed to represent the reverse ofg
.
tarjan_strongly_connected_components_C(short_digraph g, int *scc)
Assumingscc
is already allocated and has size at leastg.n
, this method computes the strongly connected components ofg
, and outputs inscc[v]
the number of the strongly connected component containingv
. It returns the number of strongly connected components.
strongly_connected_components_digraph_C(short_digraph g, int nscc, int *scc, short_digraph output):
Assumingnscc
andscc
are the outputs oftarjan_strongly_connected_components_C
ong
, this routine setsoutput
to the strongly connected component digraph ofg
, that is, the vertices ofoutput
are the strongly connected components ofg
(numbers are provided byscc
), andoutput
contains an arc(C1,C2)
ifg
has an arc from a vertex inC1
to a vertex inC2
.
What is this module used for ?¶
It is for instance used in the sage.graphs.distances_all_pairs
module,
and in the strongly_connected_components()
method.
Python functions¶
These functions are available so that Python modules from Sage can call the Cython routines this module implements (as they can not directly call methods with C arguments).

sage.graphs.base.static_sparse_graph.
spectral_radius
(G, prec=1e10)¶ Return an interval of floating point number that encloses the spectral radius of this graph
The input graph
G
must be strongly connected.INPUT:
prec
– (default1e10
) an upper bound for the relative precision of the interval
The algorithm is iterative and uses an inequality valid for nonnegative matrices. Namely, if \(A\) is a nonnegative square matrix with PerronFrobenius eigenvalue \(\lambda\) then the following inequality is valid for any vector \(x\)
\[\min_i \frac{(Ax)_i}{x_i} \leq \lambda \leq \max_i \frac{(Ax)_i}{x_i}\]Note
The speed of convergence of the algorithm is governed by the spectral gap (the distance to the second largest modulus of other eigenvalues). If this gap is small, then this function might not be appropriate.
The algorithm is not smart and not parallel! It uses basic interval arithmetic and native floating point arithmetic.
EXAMPLES:
sage: from sage.graphs.base.static_sparse_graph import spectral_radius sage: G = DiGraph([(0,0),(0,1),(1,0)], loops=True) sage: phi = (RR(1) + RR(5).sqrt() ) / 2 sage: phi # abs tol 1e14 1.618033988749895 sage: e_min, e_max = spectral_radius(G, 1e14) sage: e_min, e_max # abs tol 1e14 (1.618033988749894, 1.618033988749896) sage: (e_max  e_min) # abs tol 1e14 1e14 sage: e_min < phi < e_max True
This function also works for graphs:
sage: G = Graph([(0,1),(0,2),(1,2),(1,3),(2,4),(3,4)]) sage: e_min, e_max = spectral_radius(G, 1e14) sage: e = max(G.adjacency_matrix().charpoly().roots(AA, multiplicities=False)) sage: e_min < e < e_max True sage: G.spectral_radius() # abs tol 1e9 (2.48119430408, 2.4811943041)
A larger example:
sage: G = DiGraph() sage: G.add_edges((i,i+1) for i in range(200)) sage: G.add_edge(200,0) sage: G.add_edge(1,0) sage: e_min, e_max = spectral_radius(G, 0.00001) sage: p = G.adjacency_matrix(sparse=True).charpoly() sage: p x^201  x^199  1 sage: r = p.roots(AA, multiplicities=False)[0] sage: e_min < r < e_max True
A much larger example:
sage: G = DiGraph(100000) sage: r = list(range(100000)) sage: while not G.is_strongly_connected(): ....: shuffle(r) ....: G.add_edges(enumerate(r), loops=False) sage: spectral_radius(G, 1e10) # random (1.9997956006500042, 1.9998043797692782)
The algorithm takes care of multiple edges:
sage: G = DiGraph(2,loops=True,multiedges=True) sage: G.add_edges([(0,0),(0,0),(0,1),(1,0)]) sage: spectral_radius(G, 1e14) # abs tol 1e14 (2.414213562373094, 2.414213562373095) sage: max(G.adjacency_matrix().eigenvalues(AA)) 2.414213562373095?
Some bipartite graphs:
sage: G = Graph([(0,1),(0,3),(2,3)]) sage: G.spectral_radius() # abs tol 1e10 (1.6180339887253428, 1.6180339887592732) sage: G = DiGraph([(0,1),(0,3),(2,3),(3,0),(1,0),(1,2)]) sage: G.spectral_radius() # abs tol 1e10 (1.5537739740270458, 1.553773974033029) sage: G = graphs.CompleteBipartiteGraph(1,3) sage: G.spectral_radius() # abs tol 1e10 (1.7320508075688772, 1.7320508075688774)

sage.graphs.base.static_sparse_graph.
strongly_connected_components_digraph
(G)¶ Return the digraph of the strongly connected components (SCCs).
This routine is used to test
strongly_connected_components_digraph_C
, but it is not used by the Sage digraph. It outputs a pair[g_scc,scc]
, whereg_scc
is the SCC digraph of g,scc
is a dictionary associating to each vertexv
the number of the SCC ofv
, as it appears ing_scc
.EXAMPLES:
sage: from sage.graphs.base.static_sparse_graph import strongly_connected_components_digraph sage: strongly_connected_components_digraph(digraphs.Path(3)) (Digraph on 3 vertices, {0: 2, 1: 1, 2: 0}) sage: strongly_connected_components_digraph(DiGraph(4)) (Digraph on 4 vertices, {0: 0, 1: 1, 2: 2, 3: 3})

sage.graphs.base.static_sparse_graph.
tarjan_strongly_connected_components
(G)¶ Return the lists of vertices in each strongly connected components (SCCs).
This method implements the Tarjan algorithm to compute the strongly connected components of the digraph. It returns a list of lists of vertices, each list of vertices representing a strongly connected component.
The basic idea of the algorithm is this: a depthfirst search (DFS) begins from an arbitrary start node (and subsequent DFSes are conducted on any nodes that have not yet been found). As usual with DFSes, the search visits every node of the graph exactly once, declining to revisit any node that has already been explored. Thus, the collection of search trees is a spanning forest of the graph. The strongly connected components correspond to the subtrees of this spanning forest that have no edge directed outside the subtree.
To recover these components, during the DFS, we keep the index of a node, that is, the position in the DFS tree, and the lowlink: as soon as the subtree rooted at \(v\) has been fully explored, the lowlink of \(v\) is the smallest index reachable from \(v\) passing from descendants of \(v\). If the subtree rooted at \(v\) has been fully explored, and the index of \(v\) equals the lowlink of \(v\), that whole subtree is a new SCC.
For more information, see the Wikipedia article Tarjan’s_strongly_connected_components_algorithm.
EXAMPLES:
sage: from sage.graphs.base.static_sparse_graph import tarjan_strongly_connected_components sage: tarjan_strongly_connected_components(digraphs.Path(3)) [[2], [1], [0]] sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: D.connected_components() [[0, 1, 2, 3], [4, 5, 6]] sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: D.strongly_connected_components() [[3], [2], [1], [0], [6], [5], [4]] sage: D.add_edge([2,0]) sage: D.strongly_connected_components() [[3], [0, 1, 2], [6], [5], [4]] sage: D = DiGraph([('a','b'), ('b','c'), ('c', 'd'), ('d', 'b'), ('c', 'e')]) sage: [sorted(scc) for scc in D.strongly_connected_components()] [['e'], ['b', 'c', 'd'], ['a']]

sage.graphs.base.static_sparse_graph.
triangles_count
(G)¶ Return the number of triangles containing \(v\), for every \(v\).
INPUT:
 \(G\)– a graph
EXAMPLES:
sage: from sage.graphs.base.static_sparse_graph import triangles_count sage: triangles_count(graphs.PetersenGraph()) {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0} sage: sum(triangles_count(graphs.CompleteGraph(15)).values()) == 3*binomial(15,3) True