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# Static Sparse Graphs

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 out-neighbours of a vertex (which is precisely what BFS, DFS, distance computations and the flow-related stuff waste their life on).

The code contained in this module is written C-style. While Sage needs a class for static graphs (not available today, i.e. 2012-01-13) it is not what we try to address here. The purpose is efficiency and simplicity.

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 graph.
> - `m` (`int`) – the number of edges in the graph.
> - `edges` (`uint32_t *`) – array whose length is the number of edges of the graph.
> - `neighbors` (`uint32_t **`) – this array has size \(n+1\), and describes how the data of `edges` should be read : the neighbors of vertex \(i\) are the elements of `edges` addressed by `neighbors[i]...neighbors[i+1]-1`. The element `neighbors[n]`, which corresponds to no vertex (they are numbered from \(0\) to \(n-1\)) is present so that it remains easy to enumerate the neighbors of vertex \(n-1\) : the last of them is the element addressed by `neighbors[n]-1`.
> - `edge_labels` – this cython list associates a label to each edge of the graph. If a given edge is represented by `edges[i]`, this its associated label can be found at `edge_labels[i]`. This object is usually NULL, unless the call to `init_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 out-neighbors, but not 4, which is an out-neighbour of vertex 1. Vertex \(n-1\) has 2, 5, 8 and 9 as out-neighbors. \(\text{neighbors[n]}\) points toward the cell immediately after the end of \(\text{edges}\), hence outside of the allocated memory. It is used to indicate the end of the outneighbors of vertex \(n-1\)

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",str(i),"to",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 out-neighbors in sparse graphs ! :-)

## Technical details

> - When creating a `fast_digraph` from a `Graph` or `DiGraph` named `G`, the \(i^{\text{th}}\) vertex corresponds to `G.vertices()[i]`
> - Some methods return `bitset_t` objets when lists could be expected. There is a very useful `bitset_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 the `g.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)` | Initializes `short_digraph g` from a Sage (Di)Graph.
`int n_edges(short_digraph g)` | Returns the number of edges in `g`
`int out_degree(short_digraph g, int i)` | Returns the out-degree of vertex \(i\) in `g`
`has_edge(short_digraph g, int u, int v)` | Tests the existence of an edge.
`edge_label(short_digraph g, int * edge)` | Returns the label associated with a given edge
`init_empty_copy(short_digraph dst, short_digraph src)` | Allocates `dst` so that it can contain as many vertices and edges as `src`.
`init_reverse(short_digraph dst, short_digraph src)` | Initializes `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)`

> Assuming `bitset_t reached` has size at least `g.n`, this method updates `reached` so that it represents the set of vertices that can be reached from `src` in `g`.

`strongly_connected_component_containing_vertex(short_digraph g, short_digraph g_reversed, int v, bitset_t scc)`

> Assuming `bitset_t reached` has size at least `g.n`, this method updates `scc` so that it represents the vertices of the strongly connected component containing `v` in `g`. The variable `g_reversed` is assumed to represent the reverse of `g`.

## What is this module used for ?

At the moment, it is only used in the `sage.graphs.distances_all_pairs` module.

## 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.``strongly_connected_components`(G)

  Returns the strongly connected components of the given DiGraph.

  INPUT:

  - `G` – a DiGraph.

  Note

  This method has been written as an attempt to solve the slowness reported in trac ticket #12235. It is not the one used by `sage.graphs.digraph.DiGraph.strongly_connected_components()` as saving some time on the computation of the strongly connected components is not worth copying the whole graph, but it is a nice way to test this module’s functions. It is also tested in the doctest or `sage.graphs.digraph.DiGraph.strongly_connected_components()`.

  EXAMPLE:

      sage: from sage.graphs.base.static_sparse_graph import strongly_connected_components
      sage: g = digraphs.ButterflyGraph(2)
      sage: strongly_connected_components(g)
      [[('00', 0)], [('00', 1)], [('00', 2)], [('01', 0)], [('01', 1)], [('01', 2)],
      [('10', 0)], [('10', 1)], [('10', 2)], [('11', 0)], [('11', 1)], [('11', 2)]]

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