Lecture 30: ADJACENCY-Matrix based Graph

CSC 213 – Large Scale Programming Lecture 30:ADJACENCY-Matrixbased Graph

Today’s Goals • Review first two implementation for Graph ADT • What fields & data used in edge-list based approach • Operationsadjacency-list improves & how it does this • Consider when Graph used in real-life problems • For these cases, what operations are important? • How can we speed them up to make work go faster? • Could new implementation use arrays O(1) time? • Consider changes needed to enable using matrices

Edge-List Implementation • Base for all Graphimplementations • Sequences of vertices & edges • Each instance ofEdge refers to end vertices v b a u u w w vertices u v w a b edges

Adjacency-List Implementation • Vertex maintains Sequence of Edges • Only change needed • Methods which use incident edges faster • Costs some space • Improves few methods to O(1) v b a u u w w vertices w u v a b edges

Graph ADT • Accessor methods • vertices(): iterable for vertices • edges(): iterable for edges • endVertices(e): array with endpoints of edge e • opposite(v,e): e’sendpoint that is not v • areAdjacent(v,w): check if v and w are adjacent • replace(v,x): make x new element at vertex v • replace(e,x): make x new element at edge e • Update methods • insertVertex(x): create vertex storing elementx • insertEdge(v,w,x): add edge (v,w) with elementx • removeVertex(v): remove v(& incident edges) • removeEdge(e):remove e • Retrieval methods • incidentEdges(v): get edges incident tov

Graph ADT • Accessor methods • vertices(): iterable for vertices • edges(): iterable for edges • endVertices(e): array with endpoints of edgee • opposite(v,e):e’sendpoint that is not v • areAdjacent(v,w): check ifvand w are adjacent • replace(v,x): makexnew element at vertexv • replace(e,x): makexnew element at edgee • Update methods • insertVertex(x): create vertex storing elementx • insertEdge(v,w,x): add edge(v,w)with elementx • removeVertex(v): remove v(& incident edges) • removeEdge(e): remove e • Retrieval methods • incidentEdges(v): get edges incident tov

Can This Be Made Faster? • Testing for adjacency is very common • Often check how or if vertices are connected • Checking in O(1) time speeds up Graph algorithms

Can This Be Made Faster? • Testing for adjacency is very common • Often check how or if vertices are connected • Checking in O(1) time speeds up Graph algorithms • Can trade off lots of space to make faster • Unique integer ID assigned to each Vertex • Matrix is created as doubly-subscripted array of Edge • matrix[sourceID][targetID] refers to Edge or null

Adjacency Matrix Structure v b a • Edge-List structurestill used as base u w vertices 0 1 2 w v u a b edges

Adjacency Matrix Structure v b a • Edge-List structurestill used as base • Vertex stores int • Index found in matrix u w vertices 0 1 2 w v u a b edges

Adjacency Matrix Structure v b a • Edge-List structurestill used as base • Vertex stores int • Index found in matrix • Adjacency matrix in Graph class u w vertices 0 1 2 w v u a b edges

Adjacency Matrix Structure v b a • Edge-List structurestill used as base • Vertex stores int • Index found in matrix • Adjacency matrix in Graph class • nullif not adjacent u w vertices 0 1 2 w v u a b edges

Adjacency Matrix Structure v b a • Edge-List structurestill used as base • Vertex stores int • Index found in matrix • Adjacency matrix in Graph class • nullif not adjacent -or- • Edgeincidentto both vertices u w vertices 0 1 2 w v u a b edges

Adjacency Matrix Structure v b a • Undirected edgesstored in both array locations u w vertices 0 1 2 w v u a b edges

Adjacency Matrix Structure v b a • Undirected edgesstored in both array locations • Directed edgesonly in array from source to target u w vertices 0 1 2 w v u a b edges

Vertex in the Matrix • Another Verteximplementation • Only change is a field for this Vertex • Make subclass of existing Vertex class • Have 2 classes, which should we use? Does it matter? class AMVertex<V>extends Vertex<V>{ -or- class AMVertex<V,E>extends ALVertex<V,E> {private intrank;// Also need to define getRank, but not setRank }

Inserting/Removing Vertex • Reallocates & copy adjacency matrix • Insertion grows array creating locations for vertex • But we have choices when Vertex removed • Resize adjacency-matrix to prevent “bubbles” • Only good when vertices are constant

Inserting/Removing Vertex • Reallocates & copy adjacency matrix • Insertion grows array creating locations for vertex • But we have choices when Vertex removed • Resize adjacency-matrix to prevent “bubbles” • Only good when vertices are constant • What else could we do & when is it useful?

Asymptotic Performance

For Next Lecture • Finish up your coding of program #2; due today • Can use virtual extension, if you still have it • Midterm #2 in class week from today • Test will include all material through today • Lab on graphs & implementations, so get chance to use

Lecture 30: ADJACENCY-Matrix based Graph