King Ordering
Reduces the bandwidth of a graph by reordering the indices assigned to each vertex using local minimization of i-th bandwidths.
Complexity: O(m2 log(m) |E|) where m = max { degree(v) | v in V }
Defined in: <boost/graph/king_ordering.hpp>
Example
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/king_ordering.hpp>
#include <iostream>
#include <vector>
using namespace boost;
using Graph = adjacency_list<vecS, vecS, undirectedS>;
using Vertex = graph_traits<Graph>::vertex_descriptor;
int main() {
Graph g(6);
add_edge(0, 3, g); add_edge(0, 5, g);
add_edge(1, 2, g); add_edge(1, 4, g);
add_edge(2, 5, g); add_edge(3, 4, g);
std::vector<Vertex> order;
king_ordering(g, std::back_inserter(order));
std::cout << "King ordering:";
for (auto v : order)
std::cout << " " << v;
std::cout << "\n";
}King ordering: 1 2 4 5 3 0(1) Single starting vertex
template <class IncidenceGraph, class OutputIterator,
class ColorMap, class DegreeMap, class VertexIndexMap>
OutputIterator king_ordering(
const IncidenceGraph& g,
typename graph_traits<Graph>::vertex_descriptor s,
OutputIterator inverse_permutation,
ColorMap color, DegreeMap degree, VertexIndexMap index_map);| Direction | Parameter | Description |
|---|---|---|
IN |
|
An undirected graph. The graph’s type must be a model of IncidenceGraph. |
IN |
|
The starting vertex. |
OUT |
|
The new vertex ordering. The vertices are written to the output iterator in their new order. |
WORK |
|
Used internally to keep track of the progress of the algorithm (to avoid visiting the same vertex twice). |
IN |
|
This must map vertices to their degree. |
IN |
|
This maps each vertex to an integer in the range |
(2) Automatic starting vertex selection
template <class IncidenceGraph, class OutputIterator>
OutputIterator king_ordering(
const IncidenceGraph& g,
OutputIterator inverse_permutation);template <class IncidenceGraph, class OutputIterator, class VertexIndexMap>
OutputIterator king_ordering(
const IncidenceGraph& g,
OutputIterator inverse_permutation,
VertexIndexMap index_map);template <class VertexListGraph, class OutputIterator,
class ColorMap, class DegreeMap, class VertexIndexMap>
OutputIterator king_ordering(
const VertexListGraph& G,
OutputIterator inverse_permutation,
ColorMap color, DegreeMap degree, VertexIndexMap index_map);| Direction | Parameter | Description |
|---|---|---|
IN |
|
An undirected graph. The graph’s type must be a model of VertexListGraph. |
OUT |
|
The new vertex ordering. The vertices are written to the output iterator in their new order. |
WORK |
|
Used internally to keep track of the progress of the algorithm (to avoid visiting the same vertex twice). |
IN |
|
This must map vertices to their degree. |
IN |
|
This maps each vertex to an integer in the range |
(3) Explicit starting vertex queue
template <class IncidenceGraph, class OutputIterator,
class ColorMap, class DegreeMap, class VertexIndexMap>
OutputIterator king_ordering(
const IncidenceGraph& g,
std::deque<typename graph_traits<Graph>::vertex_descriptor> vertex_queue,
OutputIterator permutation,
ColorMap color, DegreeMap degree, VertexIndexMap index_map);| Direction | Parameter | Description |
|---|---|---|
IN |
|
An undirected graph. The graph’s type must be a model of IncidenceGraph. |
IN |
|
The deque containing the starting vertices for each component. |
OUT |
|
The new vertex ordering. The vertices are written to the output iterator in their new order. |
WORK |
|
Used internally to keep track of the progress of the algorithm (to avoid visiting the same vertex twice). |
IN |
|
This must map vertices to their degree. |
IN |
|
This maps each vertex to an integer in the range |
Description
The goal of the King ordering algorithm [56] is to reduce the bandwidth of a graph by reordering the indices assigned to each vertex. The King ordering algorithm works by a local minimization of the i-th bandwidths. The vertices are basically assigned a breadth-first search order, except that at each step, the adjacent vertices are placed in the queue in order of increasing pseudo-degree, where pseudo-degree is defined as the number of outgoing edges with white endpoints (vertices yet to be examined).
Overload (1) lets the user choose the "starting vertex". Overload (2) finds a good starting vertex using the pseudo-peripheral pair heuristic (among each component). Overload (3) contains the starting nodes for each vertex in the deque. The choice of the "starting vertex" can have a significant effect on the quality of the ordering.
The output of the algorithm are the vertices in the new ordering. Storing the output into a vector gives you the permutation from the new ordering to the old ordering.
inv_perm[new_index[u]] == u
Often times, it is the opposite permutation that you want, the permutation from the old index to the new index. This can easily be computed in the following way.
for (size_type i = 0; i != inv_perm.size(); ++i)
perm[old_index[inv_perm[i]]] = i;