# OpenGraph DrawingFramework

v. 2022.02 (Dogwood)

ogdf::DynamicBCTree Class Reference

Dynamic BC-trees. More...

#include <ogdf/decomposition/DynamicBCTree.h>

Inheritance diagram for ogdf::DynamicBCTree:

## Public Member Functions

DynamicBCTree (Graph &G, bool callInitConnected=false)

DynamicBCTree (Graph &G, node vG, bool callInitConnected=false)
A constructor. More...

node bcproper (node vG) const override

node bcproper (edge eG) const override
Returns the BC-tree-vertex representing the biconnected component which a given edge of the original graph is belonging to. More...

node repVertex (node uG, node vB) const override

node cutVertex (node uB, node vB) const override
Returns the copy of a cut-vertex in the biconnected components graph which belongs to a certain B-component and leads to another B-component. More...

virtual edge updateInsertedEdge (edge eG)

virtual node updateInsertedNode (edge eG, edge fG)
Update of the dynamic BC-tree after vertex insertion into the original graph. More...

edge insertEdge (node sG, node tG)
Inserts a new edge into the original graph and updates the BC-tree. More...

node insertNode (edge eG)
Inserts a new vertex into the original graph and updates the BC-tree. More...

node bComponent (node uG, node vG) const

Public Member Functions inherited from ogdf::BCTree
BCTree (Graph &G, bool callInitConnected=false)
A constructor. More...

BCTree (Graph &G, List< node > &vG)
Constructor for not connected graphs. More...

BCTree (Graph &G, node vG, bool callInitConnected=false)
A constructor. More...

virtual ~BCTree ()
Virtual destructor. More...

const GraphauxiliaryGraph () const
Returns the biconnected components graph. More...

const GraphbcTree () const
Returns the BC-tree graph. More...

int numberOfBComps () const
Returns the number of B-components. More...

int numberOfCComps () const
Returns the number of C-components. More...

const GraphoriginalGraph () const
Returns the original graph. More...

GNodeType typeOfGNode (node vG) const

node rep (node vG) const
Returns a vertex of the biconnected components graph corresponding to a given vertex of the original graph. More...

edge rep (edge eG) const
Returns the edge of the biconnected components graph corresponding to a given edge of the original graph. More...

node original (node vH)

edge original (edge eH) const
Returns the edge of the original graph which a given edge of the biconnected components graph is corresponding to. More...

BNodeType typeOfBNode (node vB) const

const SList< edge > & hEdges (node vB) const
Returns a linear list of the edges of the biconnected components graph belonging to the biconnected component represented by a given BC-tree-vertex. More...

int numberOfEdges (node vB) const
Returns the number of edges belonging to the biconnected component represented by a given BC-tree-vertex. More...

int numberOfNodes (node vB) const
Returns the number of vertices belonging to the biconnected component represented by a given BC-tree-vertex. More...

node bComponent (node uG, node vG) const

SList< node > & findPath (node sG, node tG) const
Calculates a path in the BC-tree. More...

SList< node > * findPathBCTree (node sB, node tB) const
Calculates a path in the BC-tree. More...

## Protected Member Functions

void init ()

node unite (node uB, node vB, node wB)

node find (node vB) const
The FIND function of the UNION/FIND structure. More...

node parent (node vB) const override

node condensePath (node sG, node tG)
Performs path condensation. More...

Protected Member Functions inherited from ogdf::BCTree
Generates the BC-tree and the biconnected components graph recursively. More...

void init (node vG)
Initialization. More...

void initNotConnected (List< node > &vG)
Initialization for not connected graphs. More...

void initNotConnected (node vG)
Initialization for not connected graphs. More...

node findNCA (node uB, node vB) const
Calculates the nearest common ancestor of two vertices of the BC-tree. More...

## Protected Attributes

NodeArray< int > m_bNode_degree
Array that contains for each proper BC-tree-vertex its degree. More...

NodeArray< nodem_bNode_owner
Array that contains for each BC-tree-vertex its parent in its UNION/FIND-tree structure. More...

Protected Attributes inherited from ogdf::BCTree
Graph m_B
The BC-tree. More...

Graphm_G
The original graph. More...

Graph m_H
The biconnected components graph. More...

int m_numB
The number of B-components. More...

int m_numC
The number of C-components. More...

NodeArray< bool > m_gNode_isMarked

NodeArray< nodem_gNode_hNode
An injective mapping vertices(G) -> vertices(H). More...

EdgeArray< edgem_gEdge_hEdge
A bijective mapping edges(G) -> edges(H). More...

NodeArray< BNodeTypem_bNode_type
Array that contains the type of each BC-tree-vertex. More...

NodeArray< bool > m_bNode_isMarked
Array of marks for the BC-tree-vertices. More...

NodeArray< nodem_bNode_hRefNode
Array that contains for each BC-tree-vertex the representantive of its parent within the subgraph in the biconnected components graph belonging to the biconnected component represented by the respective BC-tree-vertex. More...

NodeArray< nodem_bNode_hParNode
Array that contains for each BC-tree-vertex the representant of itself within the subgraph in the biconnected components graph belonging to the biconnected component represented by the parent of the respective BC-tree-vertex. More...

NodeArray< SList< edge > > m_bNode_hEdges
Array that contains for each BC-tree-vertex a linear list of the edges of the biconnected components graph belonging to the biconnected component represented by the respective BC-tree-vertex. More...

NodeArray< int > m_bNode_numNodes
Array that contains for each BC-tree-vertex the number of vertices belonging to the biconnected component represented by the respective BC-tree-vertex. More...

NodeArray< nodem_hNode_bNode

EdgeArray< nodem_hEdge_bNode
A surjective mapping edges(H) -> vertices(B). More...

NodeArray< nodem_hNode_gNode
A surjective mapping vertices(H) -> vertices(G). More...

EdgeArray< edgem_hEdge_gEdge
A bijective mapping edges(H) -> edges(G). More...

int m_count

NodeArray< int > m_number
Temporary array. More...

NodeArray< int > m_lowpt
Temporary array. More...

Temporary stack. More...

NodeArray< nodem_gtoh
Temporary array. More...

SList< nodem_nodes
Temporary list. More...

## Friends

class PlanarAugmentation

class PlanarAugmentationFix

Public Types inherited from ogdf::BCTree
enum  BNodeType { BNodeType::BComp, BNodeType::CComp }
Enumeration type for characterizing the BC-tree-vertices. More...

enum  GNodeType { GNodeType::Normal, GNodeType::CutVertex }
Enumeration type for characterizing the vertices of the original graph. More...

## Detailed Description

Dynamic BC-trees.

This class provides dynamic BC-trees.
The main difference of the dynamic BC-tree structure compared to the static one implemented by the class BCTree is, that B- and C-components are not any longer represented by single vertices of a BC-tree graph structure but by root vertices of UNION/FIND-trees. This allows path condensation within the BC-tree, when edges are inserted into the original graph. Path condensation is done by gathering BC-tree-vertices into a UNION/FIND-tree. However, the original vertices of the BC-tree remain in the m_B graph, but only those being the roots of their respective UNION/FIND-tree are proper representants of the biconnected components of the original graph.

Definition at line 54 of file DynamicBCTree.h.

## ◆ DynamicBCTree() [1/2]

 ogdf::DynamicBCTree::DynamicBCTree ( Graph & G, bool callInitConnected = false )
inlineexplicit

A constructor.

This constructor does only call BCTree::BCTree() and DynamicBCTree::init(). DynamicBCTree(G) is equivalent to DynamicBCTree(G, G.firstNode()).

Parameters
 G is the original graph. callInitConnected decides which init is called, default call is init().

Definition at line 146 of file DynamicBCTree.h.

## ◆ DynamicBCTree() [2/2]

 ogdf::DynamicBCTree::DynamicBCTree ( Graph & G, node vG, bool callInitConnected = false )
inline

A constructor.

This constructor does only call BCTree::BCTree() and DynamicBCTree::init().

Parameters
 G is the original graph. vG is the vertex of the original graph which the DFS algorithm starts with. callInitConnected decides which init is called, default call is init().

Definition at line 156 of file DynamicBCTree.h.

## ◆ bComponent()

 node ogdf::DynamicBCTree::bComponent ( node uG, node vG ) const

Returns the BC-tree-vertex representing the B-component which two given vertices of the original graph are belonging to.

Parameters
 uG is a vertex of the original graph. vG is a vertex of the original graph.
Returns
If uG and vG are belonging to the same B-component, the very vertex of the BC-tree representing this B-component is returned. Otherwise, nullptr is returned. This member function returns the representant of the correct B-component even if uG or vG or either are cut-vertices and are therefore belonging to C-components, too.

The difference between BCTree::bComponent() and DynamicBCTree::bComponent() is, that the latter one considers the UNION/FIND-tree structures.

## ◆ bcproper() [1/2]

 node ogdf::DynamicBCTree::bcproper ( edge eG ) const
overridevirtual

Returns the BC-tree-vertex representing the biconnected component which a given edge of the original graph is belonging to.

Parameters
 eG is an edge of the original graph.
Returns
the vertex of the BC-tree representing the B-component which eG is belonging to.

The difference between BCTree::bcproper() and DynamicBCTree::bcproper() is, that the latter one considers the UNION/FIND-tree structures.

Reimplemented from ogdf::BCTree.

## ◆ bcproper() [2/2]

 node ogdf::DynamicBCTree::bcproper ( node vG ) const
overridevirtual

Returns a BC-tree-vertex representing a biconnected component which a given vertex of the original graph is belonging to.

Parameters
 vG is a vertex of the original graph.
Returns
a vertex of the BC-tree:
• If vG is not a cut-vertex, then typeOfGNode(vG) returns the very vertex of the BC-tree representing the unambiguous B-component which vG is belonging to.
• If vG is a cut-vertex, then typeOfGNode(vG) returns the very vertex of the BC-tree representing the unambiguous C-component which vG is belonging to.

The difference between BCTree::bcproper() and DynamicBCTree::bcproper() is, that the latter one considers the UNION/FIND-tree structures.

Reimplemented from ogdf::BCTree.

## ◆ condensePath()

 node ogdf::DynamicBCTree::condensePath ( node sG, node tG )
protected

Performs path condensation.

This member function condenses the path from bcproper(sG) to bcproper(tG) in the BC-tree into one single B-component by calling findPath() and subsequently unite().

Parameters
 sG is a vertex of the original graph. tG is a vertex of the original graph.
Returns
the proper representant of the resulting B-component.

## ◆ cutVertex()

 node ogdf::DynamicBCTree::cutVertex ( node uB, node vB ) const
inlineoverridevirtual

Returns the copy of a cut-vertex in the biconnected components graph which belongs to a certain B-component and leads to another B-component.

If two BC-tree-vertices are neighbours, then the biconnected components represented by them have exactly one cut-vertex in common. But there are several copies of this cut-vertex in the biconnected components graph, namely one copy for each biconnected component which the cut-vertex is belonging to. The member function rep() had been designed for returning the very copy of the cut-vertex belonging to the copy of the unambiguous C-component which it is belonging to, whereas this member function is designed to return the very copy of the cut-vertex connecting two biconnected components which belongs to the copy of the second one.

Parameters
 uB is any vertex of m_B. vB is any vertex of m_B.
Returns
a vertex of the biconnected components graph:
• If uB == vB and they are representing a B-component, then cutVertex(uB,vB) returns nullptr.
• If uB == vB and they are representing a C-component, then cutVertex(uB,vB) returns the single isolated vertex in the biconnected components graph which is the copy of the C-component.
• If uB and vB are neighbours in the BC-tree, then there exists a cut-vertex leading from the biconnected component represented by vB to the biconnected component represented by uB. cutVertex(uB,vB) returns the very copy of this vertex within the biconnected components graph which belongs to the copy of the biconnected component represented by vB.
• Otherwise, cutVertex(uB,vB) returns nullptr.

The difference between BCTree::cutVertex() and DynamicBCTree::cutVertex() is, that the latter one considers the UNION/FIND-tree structures.

Reimplemented from ogdf::BCTree.

Definition at line 239 of file DynamicBCTree.h.

## ◆ find()

 node ogdf::DynamicBCTree::find ( node vB ) const
protected

The FIND function of the UNION/FIND structure.

Parameters
 vB is any vertex of m_B.
Returns
the owner of vB properly representing a biconnected component, i.e. the root of the UNION/FIND-tree of vB.

## ◆ init()

 void ogdf::DynamicBCTree::init ( )
protected

Initialization of m_bNode_owner and m_bNode_degree.

## ◆ insertEdge()

 edge ogdf::DynamicBCTree::insertEdge ( node sG, node tG )
inline

Inserts a new edge into the original graph and updates the BC-tree.

This member function inserts a new edge between sG and tG into the original graph and then calls updateInsertedEdge().

Parameters
 sG is a vertex of the original graph. tG is a vertex of the original graph.
Returns
the new edge of the original graph.

Definition at line 288 of file DynamicBCTree.h.

## ◆ insertNode()

 node ogdf::DynamicBCTree::insertNode ( edge eG )
inline

Inserts a new vertex into the original graph and updates the BC-tree.

This member function inserts a new vertex into the original graph by splitting the edge eG and then calls updateInsertedNode().

Parameters
 eG is an edge of the original graph.
Returns
the new vertex of the original graph.

Definition at line 298 of file DynamicBCTree.h.

## ◆ parent()

 node ogdf::DynamicBCTree::parent ( node vB ) const
overrideprotectedvirtual

Returns the parent of a given BC-tree-vertex.

Parameters
 vB is any vertex of m_B or nullptr.
Returns
the parent of vB in the BC-tree structure, if vB is not the root of the BC-tree, and nullptr, if vB is nullptr or the root of the BC-tree. The UNION/FIND-tree structures are considered.

Reimplemented from ogdf::BCTree.

## ◆ repVertex()

 node ogdf::DynamicBCTree::repVertex ( node uG, node vB ) const
inlineoverridevirtual

Returns a vertex of the biconnected components graph corresponding to a given vertex of the original graph and belonging to the representation of a certain biconnected component given by a vertex of the BC-tree.

Parameters
 uG is a vertex of the original graph. vB is any vertex of m_B.
Returns
a vertex of the biconnected components graph:
• If uG is belonging to the biconnected component represented by vB, then rep(uG,vB) returns the very vertex of the biconnected components graph corresponding to uG within the representation of vB.
• Otherwise, nullptr is returned.

The difference between BCTree::repVertex() and DynamicBCTree::repVertex() is, that the latter one considers the UNION/FIND-tree structures.

Reimplemented from ogdf::BCTree.

Definition at line 205 of file DynamicBCTree.h.

## ◆ unite()

 node ogdf::DynamicBCTree::unite ( node uB, node vB, node wB )
protected

The UNION function of the UNION/FIND structure.

Parameters
 uB is a vertex of the BC-tree representing a B-component. vB is a vertex of the BC-tree representing a C-component. wB is a vertex of the BC-tree representing a B-component.
Precondition
uB and vB and wB have to be proper representants of their B-components, i.e. they have to be the root vertices of their respective UNION/FIND-trees.
uB and wB have to be adjacent to vB.
Returns
the vertex properly representing the condensed B-component.

## ◆ updateInsertedEdge()

 virtual edge ogdf::DynamicBCTree::updateInsertedEdge ( edge eG )
virtual

Update of the dynamic BC-tree after edge insertion into the original graph.

This member function performs on-line maintenance of the dynamic BC-tree according to J. Westbrook and R. E. Tarjan, Maintaining Bridge-Connected and Biconnected Components On-Line, Algorithmica (1992) 7:433-464.

Parameters
 eG is a newly inserted edge of the original graph.

After a new edge has been inserted into the original graph by calling Graph::newEdge(), this member function updates the corresponding BC-tree in $$O(\alpha(k,n))$$ amortized time and the coponents graph in O(1 + n/k) amortized time per insertEdge() operation, where k is the number of such operations.

Returns
the new edge of the original graph.

Reimplemented in ogdf::DynamicSPQRForest, and ogdf::DynamicSPQRTree.

## ◆ updateInsertedNode()

 virtual node ogdf::DynamicBCTree::updateInsertedNode ( edge eG, edge fG )
virtual

Update of the dynamic BC-tree after vertex insertion into the original graph.

This member function performs on-line maintenance of the dynamic BC-tree according to J. Westbrook and R. E. Tarjan, Maintaining Bridge-Connected and Biconnected Components On-Line, Algorithmica (1992) 7:433-464.

Parameters
 eG is the incoming edge of the newly inserted vertex which has been generated by a Graph::split() operation. fG is the outgoing edge of the newly inserted vertex which has been generated by a Graph::split() operation.

After a new vertex has been inserted into an edge of the original graph by splitting the edge, all data structures of the DynamicBCTree class are updated by this member funtion. It takes O(1) time.

Returns
the new vertex of the original graph.

Reimplemented in ogdf::DynamicSPQRForest, and ogdf::DynamicSPQRTree.

## ◆ PlanarAugmentation

 friend class PlanarAugmentation
friend

Definition at line 55 of file DynamicBCTree.h.

## ◆ PlanarAugmentationFix

 friend class PlanarAugmentationFix
friend

Definition at line 56 of file DynamicBCTree.h.

## ◆ m_bNode_degree

 NodeArray ogdf::DynamicBCTree::m_bNode_degree
protected

Array that contains for each proper BC-tree-vertex its degree.

For each vertex vB of the BC-tree structure:

• If vB is representing a biconnected component, then m_bNode_degree[vB] is the degree of the vertex of the BC-tree.
• If vB is not any longer representing a biconnected component due to path condensation, then m_bNode_degree[vB] is undefined. This array is necessary, because the edges of the BC-tree are not updated during path condensation for efficiency reasons. Thus, vB->degree() != m_bNode_degree[vB]

Definition at line 84 of file DynamicBCTree.h.

## ◆ m_bNode_owner

 NodeArray ogdf::DynamicBCTree::m_bNode_owner
mutableprotected

Array that contains for each BC-tree-vertex its parent in its UNION/FIND-tree structure.

For each vertex vB of the BC-tree structure:

• If vB is representing a biconnected component, then m_bNode_owner[vB] points to the vertex vB itself.
• If vB is not any longer representing a biconnected component due to path condensation, then m_bNode_owner[vB] points to the parent of vB in its UNION/FIND-tree.

Definition at line 70 of file DynamicBCTree.h.

The documentation for this class was generated from the following file: