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/
bron_kerbosch.go
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/
bron_kerbosch.go
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// Copyright ©2015 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package topo
import (
"github.com/ArkaGPL/gonum/graph"
"github.com/ArkaGPL/gonum/graph/internal/ordered"
"github.com/ArkaGPL/gonum/graph/internal/set"
)
// DegeneracyOrdering returns the degeneracy ordering and the k-cores of
// the undirected graph g.
func DegeneracyOrdering(g graph.Undirected) (order []graph.Node, cores [][]graph.Node) {
order, offsets := degeneracyOrdering(g)
ordered.Reverse(order)
cores = make([][]graph.Node, len(offsets))
offset := len(order)
for i, n := range offsets {
cores[i] = order[offset-n : offset]
offset -= n
}
return order, cores
}
// KCore returns the k-core of the undirected graph g with nodes in an
// optimal ordering for the coloring number.
func KCore(k int, g graph.Undirected) []graph.Node {
order, offsets := degeneracyOrdering(g)
var offset int
for _, n := range offsets[:k] {
offset += n
}
core := make([]graph.Node, len(order)-offset)
copy(core, order[offset:])
return core
}
// degeneracyOrdering is the common code for DegeneracyOrdering and KCore. It
// returns l, the nodes of g in optimal ordering for coloring number and
// s, a set of relative offsets into l for each k-core, where k is an index
// into s.
func degeneracyOrdering(g graph.Undirected) (l []graph.Node, s []int) {
nodes := graph.NodesOf(g.Nodes())
// The algorithm used here is essentially as described at
// http://en.wikipedia.org/w/index.php?title=Degeneracy_%28graph_theory%29&oldid=640308710
// Initialize an output list L in return parameters.
// Compute a number d_v for each vertex v in G,
// the number of neighbors of v that are not already in L.
// Initially, these numbers are just the degrees of the vertices.
dv := make(map[int64]int, len(nodes))
var (
maxDegree int
neighbours = make(map[int64][]graph.Node)
)
for _, n := range nodes {
id := n.ID()
adj := graph.NodesOf(g.From(id))
neighbours[id] = adj
dv[id] = len(adj)
if len(adj) > maxDegree {
maxDegree = len(adj)
}
}
// Initialize an array D such that D[i] contains a list of the
// vertices v that are not already in L for which d_v = i.
d := make([][]graph.Node, maxDegree+1)
for _, n := range nodes {
deg := dv[n.ID()]
d[deg] = append(d[deg], n)
}
// Initialize k to 0.
k := 0
// Repeat n times:
s = []int{0}
for range nodes {
// Scan the array cells D[0], D[1], ... until
// finding an i for which D[i] is nonempty.
var (
i int
di []graph.Node
)
for i, di = range d {
if len(di) != 0 {
break
}
}
// Set k to max(k,i).
if i > k {
k = i
s = append(s, make([]int, k-len(s)+1)...)
}
// Select a vertex v from D[i]. Add v to the
// beginning of L and remove it from D[i].
var v graph.Node
v, d[i] = di[len(di)-1], di[:len(di)-1]
l = append(l, v)
s[k]++
delete(dv, v.ID())
// For each neighbor w of v not already in L,
// subtract one from d_w and move w to the
// cell of D corresponding to the new value of d_w.
for _, w := range neighbours[v.ID()] {
dw, ok := dv[w.ID()]
if !ok {
continue
}
for i, n := range d[dw] {
if n.ID() == w.ID() {
d[dw][i], d[dw] = d[dw][len(d[dw])-1], d[dw][:len(d[dw])-1]
dw--
d[dw] = append(d[dw], w)
break
}
}
dv[w.ID()] = dw
}
}
return l, s
}
// BronKerbosch returns the set of maximal cliques of the undirected graph g.
func BronKerbosch(g graph.Undirected) [][]graph.Node {
nodes := graph.NodesOf(g.Nodes())
// The algorithm used here is essentially BronKerbosch3 as described at
// http://en.wikipedia.org/w/index.php?title=Bron%E2%80%93Kerbosch_algorithm&oldid=656805858
p := set.NewNodesSize(len(nodes))
for _, n := range nodes {
p.Add(n)
}
x := set.NewNodes()
var bk bronKerbosch
order, _ := degeneracyOrdering(g)
ordered.Reverse(order)
for _, v := range order {
neighbours := graph.NodesOf(g.From(v.ID()))
nv := set.NewNodesSize(len(neighbours))
for _, n := range neighbours {
nv.Add(n)
}
bk.maximalCliquePivot(g, []graph.Node{v}, set.IntersectionOfNodes(p, nv), set.IntersectionOfNodes(x, nv))
p.Remove(v)
x.Add(v)
}
return bk
}
type bronKerbosch [][]graph.Node
func (bk *bronKerbosch) maximalCliquePivot(g graph.Undirected, r []graph.Node, p, x set.Nodes) {
if len(p) == 0 && len(x) == 0 {
*bk = append(*bk, r)
return
}
neighbours := bk.choosePivotFrom(g, p, x)
nu := set.NewNodesSize(len(neighbours))
for _, n := range neighbours {
nu.Add(n)
}
for _, v := range p {
if nu.Has(v) {
continue
}
vid := v.ID()
neighbours := graph.NodesOf(g.From(vid))
nv := set.NewNodesSize(len(neighbours))
for _, n := range neighbours {
nv.Add(n)
}
var found bool
for _, n := range r {
if n.ID() == vid {
found = true
break
}
}
var sr []graph.Node
if !found {
sr = append(r[:len(r):len(r)], v)
}
bk.maximalCliquePivot(g, sr, set.IntersectionOfNodes(p, nv), set.IntersectionOfNodes(x, nv))
p.Remove(v)
x.Add(v)
}
}
func (*bronKerbosch) choosePivotFrom(g graph.Undirected, p, x set.Nodes) (neighbors []graph.Node) {
// TODO(kortschak): Investigate the impact of pivot choice that maximises
// |p ⋂ neighbours(u)| as a function of input size. Until then, leave as
// compile time option.
if !tomitaTanakaTakahashi {
for _, n := range p {
return graph.NodesOf(g.From(n.ID()))
}
for _, n := range x {
return graph.NodesOf(g.From(n.ID()))
}
panic("bronKerbosch: empty set")
}
var (
max = -1
pivot graph.Node
)
maxNeighbors := func(s set.Nodes) {
outer:
for _, u := range s {
nb := graph.NodesOf(g.From(u.ID()))
c := len(nb)
if c <= max {
continue
}
for n := range nb {
if _, ok := p[int64(n)]; ok {
continue
}
c--
if c <= max {
continue outer
}
}
max = c
pivot = u
neighbors = nb
}
}
maxNeighbors(p)
maxNeighbors(x)
if pivot == nil {
panic("bronKerbosch: empty set")
}
return neighbors
}