Graphs ORD SFO LAX DFW 1843 802 1743 337 1233 Graphs 11/16/2018 2:34 AM Graphs 1843 ORD SFO 802 1743 337 LAX 1233 DFW

Biconnected Components Graphs 11/16/2018 2:34 AM Biconnected Components Since the link components (equivalence classes of edges with respect to the link relation) of G correspond to the biconnected components, to construct the biconnected componnets of G we need only compute the link components.

Biconnected Components Graphs 11/16/2018 2:34 AM Biconnected Components Since the link components (equivalence classes of edges with respect to the link relation) of G correspond to the biconnected components, to construct the biconnected componnets of G we need only compute the link components.

Biconnected Components Graphs 11/16/2018 2:34 AM Biconnected Components LinkComponents (G) Input: A connected graph G Output: The link components of G 1) Perform a DFS traversal T on G 2) Compute the auxiliary graph B 3) compute the connected components of B by performing a DFS traversal on B 4) For each connected component of B, output the vertices of B as a link component of G Biconnected components, separation edges, and separation vertexes can be identified after we know the link components.

Outline and Reading Graphs (§6.1) Definitions Applications Terminology 11/16/2018 2:34 AM Outline and Reading Graphs (§6.1) Definitions Applications Terminology Properties ADT Data structures for graphs (§6.2) Edge list structure Adjacency list structure Adjacency matrix structure

Position ADT- Recall from Chpt 2 Graphs 11/16/2018 2:34 AM Position ADT- Recall from Chpt 2 The Position ADT models the notion of the place within a data structure where a single object is stored It gives a unified view of diverse ways of storing data, such as a cell of an array a node of a linked list Just one method: object element(): returns the element stored at the position

Graph A graph is a pair (V, E), where Graphs 11/16/2018 2:34 AM Graph A graph is a pair (V, E), where V is a set of nodes, called vertices E is a collection of pairs of vertices, called edges Vertices and edges are positions and store elements Edges can be directed or undirected. If all the edges are directed, the graph is called a directed graph or digraph. 849 PVD 1843 ORD 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 LAX 1233 DFW 1120 MIA

Examples of Graphs Airline route map Flowcharts Binary relations Vertices: an airport identified by a three-letter airport code Edges: a flight route between two airports and specified by the mileage of the route Flowcharts These represent the flow of control in a procedure. Vertices: symbolic flowchart boxes and diamonds Edges: connecting lines Binary relations We say xRy if (x,y) ε SxS, where S is a set. Vertices: elements in S Edges: connect x and y iff xRy.

Examples of Graphs Computer networks Vertices: computers Edges: communication links Electric circuits Vertices: diodes, transistors, capacitors, switches, ... Edges: Connecting wires These examples suggest the type of questions that can be asked when working with graphs: What is the cheapest way to fly to NY What route involves the least flying time? If one city is closed due to bad weather, does a route to NY exist?

Edge Types flight AA 1206 ORD PVD 849 miles ORD PVD Directed edge Graphs 11/16/2018 2:34 AM Edge Types Directed edge ordered pair of vertices (u,v) first vertex u is the origin second vertex v is the destination e.g., a flight Undirected edge unordered pair of vertices (u,v) e.g., a flight route Directed graph all the edges are directed e.g., flight network Undirected graph all the edges are undirected e.g., route network flight AA 1206 ORD PVD 849 miles ORD PVD

Applications Electronic circuits Transportation networks Graphs 11/16/2018 2:34 AM Applications Electronic circuits Printed circuit board Integrated circuit Transportation networks Highway network Flight network Computer networks Local area network Internet Web Databases Entity-relationship diagram

Terminology X U V W Z Y a c b e d f g h i j Graphs 11/16/2018 2:34 AM Terminology End vertices (or endpoints) of an edge U and V are the endpoints of a Edges incident on a vertex a, d, and b are incident on V Adjacent vertices U and V are adjacent Degree of a vertex X has degree 5 Parallel edges h and i are parallel edges Self-loop j is a self-loop X U V W Z Y a c b e d f g h i j

Terminology (cont.) V a b P1 d U X Z P2 h c e W g f Y Path Graphs 11/16/2018 2:34 AM Terminology (cont.) Path sequence of alternating vertices and edges begins with a vertex ends with a vertex each edge is preceded and followed by its endpoints Simple path path such that all its vertices and edges are distinct Examples P1=(V,b,X,h,Z) is a simple path P2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple V a b P1 d U X Z P2 h c e W g f Y

Terminology (cont.) V a b d U X Z C2 h e C1 c W g f Y Cycle Graphs 11/16/2018 2:34 AM Terminology (cont.) Cycle circular sequence of alternating vertices and edges each edge is preceded and followed by its endpoints Simple cycle cycle such that all its vertices and edges are distinct Examples C1=(V,b,X,g,Y,f,W,c,U,a,) is a simple cycle C2=(U,c,W,e,X,g,Y,f,W,d,V,a,) is a cycle that is not simple V a b d U X Z C2 h e C1 c W g f Y

Properties Sv deg(v) = 2m Property 1 Property 2 Notation Graphs 11/16/2018 2:34 AM Properties Property 1 Sv deg(v) = 2m Proof: each edge is counted twice Property 2 In an undirected graph with no self-loops and no multiple edges (i.e. a simple graph) m  n (n - 1)/2 Proof: No 2 edges can have the same endpoints. Thus, each vertex has degree at most (n -1). Thus, by above, 2m≤n(n-1) What is the bound for a directed graph? m≤n(n-1) Proof: No edge can have the same origin and destination. Notation n number of vertices m number of edges deg(v) degree of vertex v Example n = 4 m = 6 deg(v) = 3

Iterator ADT- Recall from Chpt 2 Graphs 11/16/2018 2:34 AM An iterator abstracts the process of scanning through a collection of elements Methods of the ObjectIterator ADT: object object() boolean hasNext() object nextObject() reset() Extends the concept of Position by adding a traversal capability Implement with an array or singly linked list An iterator is typically associated with an another data structure We can augment the Stack, Queue, Vector, List and Sequence ADTs with method: ObjectIterator elements() Two notions of iterator: snapshot: freezes the contents of the data structure at a given time dynamic: follows changes to the data structure

Main Methods of the Graph ADT Graphs 11/16/2018 2:34 AM Main Methods of the Graph ADT v=vertex & e=edge are positions store elements (o) Update methods insertVertex(o) insertEdge(v, w, o) insertDirectedEdge(v, w, o) removeVertex(v) removeEdge(e) makeUndirected(e) reverseDirection(e) setDirectionFrom(e,v) setDirectionTo(e,v) Generic methods numVertices() numEdges() vertices() edges() See pages 294 and 295 for definitions, although most are obvious. Color key: Returns an iterator Boolean

Main Methods of the Graph ADT Graphs 11/16/2018 2:34 AM Main Methods of the Graph ADT Accessor methods aVertex() - returns any vertex as a graph may not have a special vertex incidentEdges(v) adjacentEdges(v) endVertices(e) opposite(v, e) areAdjacent(v, w) degree(v) See pages 294 and 295 for definitions, although most are obvious. Color key: Returns an iterator Boolean Needed for graphs with directed edges directedEdges() undirectedEdges() origin(e) destination(e) isDirected(e) Relating methods inDegree(v) outDegree(v) inIncidentEdges(v) outIncidentEdges(v) inAdjacentVertices(v) outAdjacentVertices(v)

Why are there so many methods for graphs? 11/16/2018 2:34 AM Why are there so many methods for graphs? Graphs are very rich and versatile structures. Therefore, the number of methods is unavoidable when describing the needed methods. Realize that graphs support two kinds of positions - vertices and edges. Moreover, edges can be directed or undirected. Consequently, we need different methods for accessing and updating all these different positions as well as handling relationships between the different positions.

Data Structures for Graphs 11/16/2018 2:34 AM Data Structures for Graphs Most commonly used Edge list structure Adjacency list structure Adjacency matrix structure Each has its own strengths and weaknesses in terms of what methods are easier to use and the cost of storage space. The edge list and the adjacency list actually store representations of the edges. The adjacency matrix format only uses a placeholder for every pair of vertices. This implies, as we will see, that the list structures use O(n+m) space and matrix format uses O(n^2) where n is number of vertices and m is number of edges.

Sequence ADT Recall from Chpt 2 Graphs 11/16/2018 2:34 AM Sequence ADT Recall from Chpt 2 The Sequence ADT is the union of the Vector and List ADTs Elements accessed by Rank, or Position Generic methods: size(), isEmpty() Vector-based methods: elemAtRank(r), replaceAtRank(r, o), insertAtRank(r, o), removeAtRank(r) List-based methods: first(), last(), before(p), after(p), replaceElement(p, o), swapElements(p, q), insertBefore(p, o), insertAfter(p, o), insertFirst(o), insertLast(o), remove(p) Bridge methods: atRank(r), rankOf(p)

Applications of Sequences Graphs 11/16/2018 2:34 AM Applications of Sequences The Sequence ADT is a basic, general-purpose, data structure for storing an ordered collection of elements Direct applications: Generic replacement for stack, queue, vector, or list small database (e.g., address book) Indirect applications: Building block of more complex data structures

Edge List Structure Vertex sequence sequence of vertex objects Graphs 11/16/2018 2:34 AM Edge List Structure Vertex sequence sequence of vertex objects Vertex object element reference to position in vertex sequence Edge object origin vertex object destination vertex object reference to position in edge sequence Edge sequence sequence of edge objects Note: Works for both directed and undirected graphs. u a c b d v w z u v w z a b c d

Edge List Structure - Notes Graphs 11/16/2018 2:34 AM Edge List Structure - Notes This is the simplest and most used data structure for a graph. The vertex and edge sequence could be a list, vector, or dictionary. If we use a vector representation, we could think of the vertices as being numbered. Vertex and edge objects may contain additional information, especially for graphs with some directed edges i,e, an edge vertex could have a Boolean variable indicating whether or not it was directed. Main feature: Provides direct access from edges to the vertices to which they are incident. But, accessing the edges that are incident on a vertex requires that all edges be examined. Obviously, performance of the various methods is very dependent upon which representation is chosen.

Adjacency List Structure Graphs 11/16/2018 2:34 AM Adjacency List Structure Vertex sequence sequence of vertex objects Vertex object element reference to position in vertex sequence reference to incident object Incidence sequence for each vertex sequence of references to edge objects of incident edges Augmented edge objects references to associated positions in incidence sequences of end vertices a v b u w u v w a b

Adjacency List Structure - Notes Graphs 11/16/2018 2:34 AM Adjacency List Structure - Notes Typically the incident sequence is implemented as a list, but could use a dictionary or a priority queue if their characteristics are useful. Structure provides direct access from both the edges to the vertices and from the vertices to their incident edges. Structure matches the performance of the edge list representation, but has improved running time when the above characteristic can be exploited - i.e. given a vertex, return an iterator of incident edges.

Adjacency Matrix Structure Graphs 11/16/2018 2:34 AM Adjacency Matrix Structure Vertex list structure Augmented vertex objects Integer key (index) associated with vertex 2D adjacency array Reference to edge object for adjacent vertices Null for non nonadjacent vertices The “old fashioned” version just has 0 for no edge and 1 for edge Edge objects Edge list structure a v b u w u 1 v 2 w 1 2  a b

Notes for Adjacency Matrix Graphs 11/16/2018 2:34 AM Notes for Adjacency Matrix Historically, the adjacency matrix was the first represented as a Boolean matrix with A[i,j] = 1 if an edge existed between vertex i and j and 0 otherwise. The representation shown here updates this to a more object-oriented environment where you use an interface to access the structure. In all of these structures, additional data may be stored in vertex or edge objects. For example, if edges are being used in a problem about distances between to cities, those distances could be stored in the edge objects.

Asymptotic Performance Graphs 11/16/2018 2:34 AM Asymptotic Performance n vertices, m edges no parallel edges no self-loops Bounds are “big-Oh” Edge List Adjacency List Adjacency Matrix Space n + m n2 incidentEdges(v) m deg(v) n areAdjacent (v, w) min(deg(v), deg(w)) 1 insertVertex(o) insertEdge(v, w, o) removeVertex(v) removeEdge(e)

Graphs 11/16/2018 2:34 AM Depth-First Search D B A C E

Outline and Reading Definitions (§6.1) Subgraph Connectivity Graphs 11/16/2018 2:34 AM Outline and Reading Definitions (§6.1) Subgraph Connectivity Spanning trees and forests Depth-first search (§6.3.1) Algorithm Example Properties Analysis Applications of DFS (§6.5) Path finding Cycle finding

Subgraphs A subgraph S of a graph G is a graph such that 11/16/2018 2:34 AM Subgraphs A subgraph S of a graph G is a graph such that The vertices of S are a subset of the vertices of G The edges of S are a subset of the edges of G A spanning subgraph of G is a subgraph that contains all the vertices of G Subgraph Spanning subgraph

Non connected graph with two connected components Graphs 11/16/2018 2:34 AM Connectivity A graph is connected if there is a path between every pair of vertices A connected component of a graph G is a maximal connected subgraph of G Connected graph Non connected graph with two connected components

Trees and Forests A (free) tree is an undirected graph T such that Graphs 11/16/2018 2:34 AM Trees and Forests A (free) tree is an undirected graph T such that T is connected T has no cycles This definition of tree is different from the one of a rooted tree A forest is an undirected graph without cycles The connected components of a forest are trees Tree Forest

Spanning Trees and Forests Graphs 11/16/2018 2:34 AM Spanning Trees and Forests A spanning tree of a connected graph is a spanning subgraph that is a tree A spanning tree is not unique unless the graph is a tree Spanning trees have applications to the design of communication networks A spanning forest of a graph is a spanning subgraph that is a forest Graph Spanning tree

Graphs 11/16/2018 2:34 AM Traversing a Graph A traversal of a graph is a systematic procedure for exploring a graph by examining all of its vertices and edges. Example: Web spider or crawler which is the data collecting part of a search engine that visits all the hypertext documents on the web. (The documents are vertices and the hyperlinks are the edges).

Graphs 11/16/2018 2:34 AM Depth-First Search Depth-first search (DFS) is a general technique for traversing a graph A DFS traversal of a graph G does the following: Visits all the vertices and edges of G Determines whether G is connected Computes the connected components of G Computes a spanning forest of G DFS on a graph with n vertices and m edges takes O(n + m ) time DFS can be further extended to solve other graph problems Find and report a path between two given vertices Find a cycle in the graph Depth-first search is to graphs what Euler tour is to binary trees

Graphs 11/16/2018 2:34 AM Depth-First Search DFS utilizes backtracking as you would if you were wandering through a maze. Start at a vertex in G with string; attached the string at the vertex; mark the vertex as "visited". Explore G by moving out along the incident edges according to some predetermined scheme for ordering and unroll the string. If you can move to a new vertex, mark it as "visited". If you hit a "visited" node, backtrack to the last vertex rolling up the string as you go and see if we can move out from there. The process terminates when the backtracking returns to the place where the string was attached - i.e. the start vertex.

Depth-First Search It's useful to distinguish edges by their use: Graphs 11/16/2018 2:34 AM Depth-First Search It's useful to distinguish edges by their use: Discovery (or tree) edges - those that are followed to discover new vertices. Back edges - those that lead to already visited vertices. For back edges, you need to backtrack to the last visited node and decide if you can move out from it without hitting a visited node. The discovery edges form a spanning tree of the connected components of the starting vertex - called a DFS tree. The back edges are also useful - assuming the DFS tree is rooted at the starting vertex, each back edge leads back from a vertex in the tree to one of its ancestors in the tree.

Example unexplored vertex visited vertex unexplored edge Graphs 11/16/2018 2:34 AM Example unexplored vertex D B A C E A A visited vertex unexplored edge discovery edge back edge D B A C E D B A C E

Example (cont.) D B A C E D B A C E D B A C E D B A C E Graphs

Graphs 11/16/2018 2:34 AM DFS and Maze Traversal The DFS algorithm is similar to a classic strategy for exploring a maze We mark each intersection, corner and dead end (vertex) visited We mark each corridor (edge ) traversed We keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack)

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S

Often Posed as Programming Contest Problem Graphs 11/16/2018 2:34 AM Often Posed as Programming Contest Problem Given a maze represented as follows: -1 1 X 1=wall , -1=exit. Start at the bottom. To walk systematically, we'll try to go N, W, E, S The exact path taken depends on this ordering.

DFS Algorithm Algorithm DFS(G, v) Graphs 11/16/2018 2:34 AM DFS Algorithm The algorithm uses a mechanism for setting and getting “labels” of vertices and edges Algorithm DFS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G in the connected component of v as discovery edges and back edges setLabel(v, VISITED) for all e  G.incidentEdges(v) if getLabel(e) = UNEXPLORED w  G.opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) DFS(G, w) else setLabel(e, BACK) Algorithm DFS(G) Input graph G Output labeling of the edges of G as discovery edges and back edges for all u  G.vertices() setLabel(u, UNEXPLORED) for all e  G.edges() setLabel(e, UNEXPLORED) for all v  G.vertices() if getLabel(v) = UNEXPLORED DFS(G, v)

Properties of DFS Property 1 Graphs 11/16/2018 2:34 AM Properties of DFS Property 1 DFS(G, v) visits all the vertices and edges in the connected component of v Property 2 The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v D B A C E

Analysis of DFS Setting/getting a vertex/edge label takes O(1) time Graphs 11/16/2018 2:34 AM Analysis of DFS Setting/getting a vertex/edge label takes O(1) time Each vertex is labeled twice once as UNEXPLORED once as VISITED Each edge is labeled twice once as DISCOVERY or BACK Method incidentEdges is called once for each vertex DFS runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that Sv deg(v) = 2m

Graphs 11/16/2018 2:34 AM Path Finding We can specialize the DFS algorithm to find a path between two given vertices u and z We call DFS(G, u) with u as the start vertex We use a stack S to keep track of the path between the start vertex and the current vertex As soon as destination vertex z is encountered, we return the path as the contents of the stack Algorithm pathDFS(G, v, z) setLabel(v, VISITED) S.push(v) if v = z return S.elements() for all e  G.incidentEdges(v) if getLabel(e) = UNEXPLORED w  opposite(v, e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) S.push(e) pathDFS(G, w, z) S.pop() { e gets popped } else setLabel(e, BACK) S.pop() { v gets popped }

Graphs 11/16/2018 2:34 AM Cycle Finding We can specialize the DFS algorithm to find a simple cycle We use a stack S to keep track of the path between the start vertex and the current vertex As soon as a back edge (v, w) is encountered, we return the cycle as the portion of the stack from the top to vertex w Algorithm cycleDFS(G, v, z) setLabel(v, VISITED) S.push(v) for all e  G.incidentEdges(v) if getLabel(e) = UNEXPLORED w  opposite(v,e) S.push(e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) pathDFS(G, w, z) S.pop() else C  new empty stack repeat o  S.pop() C.push(o) until o = w return C.elements()

Graphs 11/16/2018 2:34 AM Biconnectivity SEA PVD ORD FCO SNA MIA

Outline and Reading Definitions (§6.3.2) Algorithms (§6.3.2) Graphs 11/16/2018 2:34 AM Outline and Reading Definitions (§6.3.2) Separation vertices and edges Biconnected graph Biconnected components Equivalence classes Linked edges and link components Algorithms (§6.3.2) Auxiliary graph Proxy graph

Separation Edges and Vertices Graphs 11/16/2018 2:34 AM Separation Edges and Vertices Definitions Let G be a connected graph A separation edge of G is an edge whose removal disconnects G A separation vertex of G is a vertex whose removal disconnects G Applications Separation edges and vertices represent single points of failure in a network and are critical to the operation of the network Example DFW, LGA and LAX are separation vertices (DFW,LAX) is a separation edge ORD PVD SFO LGA HNL LAX DFW MIA

Biconnected Graph Equivalent definitions of a biconnected graph G Graphs 11/16/2018 2:34 AM Biconnected Graph Equivalent definitions of a biconnected graph G Graph G has no separation edges and no separation vertices For any two vertices u and v of G, there are two disjoint simple paths between u and v (i.e., two simple paths between u and v that share no other vertices or edges) For any two vertices u and v of G, there is a simple cycle containing u and v Example PVD ORD SFO LGA HNL LAX DFW MIA

Biconnected Components Graphs 11/16/2018 2:34 AM Biconnected Components Biconnected component of a graph G A maximal biconnected subgraph of G, or A subgraph consisting of a separation edge of G and its end vertices Interaction of biconnected components An edge belongs to exactly one biconnected component A nonseparation vertex belongs to exactly one biconnected component A separation vertex belongs to two or more biconnected components Example of a graph with four biconnected components ORD PVD SFO LGA RDU HNL LAX DFW MIA

Graphs 11/16/2018 2:34 AM Equivalence Classes Given a set S, a relation R on S is a set of ordered pairs of elements of S, i.e., R is a subset of SS An equivalence relation R on S satisfies the following properties Reflexive: (x,x)  R Symmetric: (x,y)  R  (y,x)  R Transitive: (x,y)  R  (y,z)  R  (x,z)  R An equivalence relation R on S induces a partition of the elements of S into equivalence classes (i.e. the classes are pairwise disjoint and their union is S. Example (connectivity relation among the vertices of a graph): Let V be the set of vertices of a graph G Define the relation C = {(v,w)  VV such that G has a path from v to w} Relation C is an equivalence relation The equivalence classes of relation C are the vertices in each connected component of graph G

Equivalence Relations Graphs 11/16/2018 2:34 AM Equivalence Relations These abstract the notion of equality for numbers. Obviously, equality for numbers is an equivalence relation. But, there are many such equivalence relations. Example: For positive integers a,b,c,d, define for fractions, a/b = c/d if and only if ad=bc. Proof: reflexive: a/b = a/b because ab=ba symmetric: a/b = c/d implies c/d = a/b because ad=bc implies cb=da. So, c/d=a/b transitive: a/b = c/d and c/d = e/f implies a/b = e/f (complete this proof)

Equivalence Relations Graphs 11/16/2018 2:34 AM Equivalence Relations What are the equivalence classes of the last equivalence relation? [1/2] = {x/y | 1/2= x/y} i.e. all the positive fractions we view as having 1/2 as its reduced form. i.e. [1/2] = {1/2, 2/4, 3/6, 4/8,...} Note: There is really nothing special about 1/2, [2/4], [3/6], [4/8] ... are all [1/2].

Equivalence classes of linked edges: Graphs 11/16/2018 2:34 AM Link Relation Edges e and f of connected graph G are linked if e = f, or G has a simple cycle containing e and f Theorem: The link relation on the edges of a graph is an equivalence relation Proof Sketch: The reflexive and symmetric properties follow from the definition For the transitive property, consider two simple cycles sharing an edge a b g c j d e f i Equivalence classes of linked edges: {a} {b, c, d, e, f} {g, i, j} i g e b a d j c f

Graphs 11/16/2018 2:34 AM Link Components The link components of a connected graph G are the equivalence classes of edges with respect to the link relation A biconnected component of G is the subgraph of G induced by an equivalence class of linked edges A separation edge is a single-element equivalence class of linked edges A separation vertex has incident edges in at least two distinct equivalence classes of linked edge (Note this does not violate the property that equivalence classes are pairwise disjoint.) ORD PVD SFO LGA RDU HNL LAX DFW MIA

Auxiliary Graph Auxiliary graph B for a connected graph G Graphs 11/16/2018 2:34 AM Auxiliary Graph Auxiliary graph B for a connected graph G Associated with a DFS traversal of G The vertices of B are the edges of G For each back edge e of G, B has edges (e,f1), (e,f2) , …, (e,fk), where f1, f2, …, fk are the discovery edges of G that form a simple cycle with e Its connected components correspond to the the link components of G h g i b e j i d c f a DFS on graph G g e i h b f j d c a Auxiliary graph B

Auxiliary Graph (cont.) Graphs 11/16/2018 2:34 AM Auxiliary Graph (cont.) In the worst case, the number of edges of the auxiliary graph is proportional to nm DFS on graph G Auxiliary graph B

Proxy Graph Proxy graph F for a connected graph G Graphs 11/16/2018 2:34 AM Proxy Graph Proxy graph F for a connected graph G Spanning forest of the auxiliary graph B Has m vertices and O(m) edges Can be constructed in O(n + m) time Its connected components (trees) correspond to the the link components of G Given a graph G with n vertices and m edges, we can compute the following in O(n + m) time The biconnected components of G The separation vertices of G The separation edges of G h g i b e j i d c f a DFS on graph G g e i h b f j d c a Proxy graph F

Proxy Graph Constructed Using DFS Graphs 11/16/2018 2:34 AM Proxy Graph Constructed Using DFS Algorithm proxyGraph(G) Input connected graph G Output proxy graph F for G F  empty graph DFS(G, s) { s is any vertex of G} for all discovery edges e of G F.insertVertex(e) setLabel(e, UNLINKED) for all vertices v of G in DFS visit order for all back edges e = (u,v) repeat f  discovery edge with dest. u F.insertEdge(e,f,) if f getLabel(f) = UNLINKED setLabel(f, LINKED) u  origin of edge f else u  v { ends the loop } until u = v return F h g i b e j i d c f a DFS on graph G g e i h b f j d c a Proxy graph F

Biconnected Components Graphs 11/16/2018 2:34 AM Biconnected Components The bottleneck in the auxiliary graph approach is the construction of the auxiliary graph B, which take O(nm) time. But we do not need all of the auxiliary graph B in order to find the biconnected components of G. We only need to identify the connected components of B. We only need a spanning forest of B. Thus we can reduce the time required to compute the link components of G by using the spanning forest of B (Proxy graph, F), which can be constructed in time O(n+m)

Graphs 11/16/2018 2:34 AM Breadth-First Search C B A E D L0 L1 F L2

Outline and Reading Breadth-first search (§6.3.3) DFS vs. BFS (§6.3.3) Graphs 11/16/2018 2:34 AM Outline and Reading Breadth-first search (§6.3.3) Algorithm Example Properties Analysis Applications DFS vs. BFS (§6.3.3) Comparison of applications Comparison of edge labels

Graphs 11/16/2018 2:34 AM Breadth-First Search Breadth-first search (BFS) is a general technique for traversing a graph A BFS traversal of a graph G Visits all the vertices and edges of G Determines whether G is connected Computes the connected components of G Computes a spanning forest of G BFS on a graph with n vertices and m edges takes O(n + m ) time BFS can be further extended to solve other graph problems Find and report a path with the minimum number of edges between two given vertices Find a simple cycle, if there is one

Graphs 11/16/2018 2:34 AM BFS Algorithm The algorithm uses a mechanism for setting and getting “labels” of vertices and edges Algorithm BFS(G, s) L0  new empty sequence L0.insertLast(s) setLabel(s, VISITED) i  0 while Li.isEmpty() Li +1  new empty sequence for all v  Li.elements() for all e  G.incidentEdges(v) if getLabel(e) = UNEXPLORED w  opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) setLabel(w, VISITED) Li +1.insertLast(w) else setLabel(e, CROSS) i  i +1 Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u  G.vertices() setLabel(u, UNEXPLORED) for all e  G.edges() setLabel(e, UNEXPLORED) for all v  G.vertices() if getLabel(v) = UNEXPLORED BFS(G, v)

Example unexplored vertex visited vertex unexplored edge Graphs 11/16/2018 2:34 AM Example C B A E D L0 L1 F A unexplored vertex A visited vertex unexplored edge discovery edge cross edge L0 L0 A A L1 L1 B C D B C D E F E F

Example (cont.) L0 L1 L0 L1 L2 L0 L1 L2 L0 L1 L2 C B A E D F C B A E D Graphs 11/16/2018 2:34 AM Example (cont.) C B A E D L0 L1 F C B A E D L0 L1 F L2 C B A E D L0 L1 F L2 C B A E D L0 L1 F L2

Example (cont.) L0 L1 L2 L0 L1 L2 L0 L1 L2 C B A E D F A B C D E F C B Graphs 11/16/2018 2:34 AM Example (cont.) C B A E D L0 L1 F L2 L0 A L1 B C D L2 E F C B A E D L0 L1 F L2

Properties Notation Gs: connected component of s Property 1 Graphs 11/16/2018 2:34 AM Properties Notation Gs: connected component of s Property 1 BFS(G, s) visits all the vertices and edges of Gs Property 2 The discovery edges labeled by BFS(G, s) form a spanning tree Ts of Gs Property 3 For each vertex v in Li The path of Ts from s to v has i edges Every path from s to v in Gs has at least i edges A B C D E F L0 A L1 B C D L2 E F

Analysis Setting/getting a vertex/edge label takes O(1) time Graphs 11/16/2018 2:34 AM Analysis Setting/getting a vertex/edge label takes O(1) time Each vertex is labeled twice once as UNEXPLORED once as VISITED Each edge is labeled twice once as DISCOVERY or CROSS Each vertex is inserted once into a sequence Li Method incidentEdges is called once for each vertex BFS runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that Sv deg(v) = 2m

Graphs 11/16/2018 2:34 AM Applications We can specialize the BFS traversal of a graph G to solve the following problems in O(n + m) time Compute the connected components of G Compute a spanning forest of G Find a simple cycle in G, or report that G is a forest Given two vertices of G, find a path in G between them with the minimum number of edges, or report that no such path exists

DFS vs. BFS DFS BFS Applications DFS BFS L0 L1 L2 Graphs 11/16/2018 2:34 AM DFS vs. BFS Applications DFS BFS Spanning forest, connected components, paths, cycles  Shortest paths Biconnected components C B A E D L0 L1 F L2 A B C D E F DFS BFS

DFS vs. BFS (cont.) DFS BFS Back edge (v,w) Cross edge (v,w) L0 L1 L2 Graphs 11/16/2018 2:34 AM DFS vs. BFS (cont.) Back edge (v,w) w is an ancestor of v in the tree of discovery edges Cross edge (v,w) w is in the same level as v or in the next level in the tree of discovery edges C B A E D L0 L1 F L2 A B C D E F DFS BFS

Graphs 11/16/2018 2:34 AM Directed Graphs BOS ORD JFK SFO DFW LAX MIA

Outline and Reading (§6.4) Graphs 11/16/2018 2:34 AM Outline and Reading (§6.4) Reachability (§6.4.1) Directed DFS Strong connectivity Transitive closure (§6.4.2) The Floyd-Warshall Algorithm Directed Acyclic Graphs (DAG’s) (§6.4.4) Topological Sorting

Digraphs A digraph is a graph whose edges are all directed E 11/16/2018 2:34 AM Digraphs A digraph is a graph whose edges are all directed Short for “directed graph” Applications one-way streets flights task scheduling A C E B D

Digraph Properties E D C B A graph G=(V,E) such that Graphs 11/16/2018 2:34 AM A C E B D Digraph Properties A graph G=(V,E) such that Each edge goes in one direction: Edge (a,b) goes from a to b, but not b to a. If G is simple, m < n(n-1). If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of of the sets of in-edges and out-edges in time proportional to their size.

Graphs 11/16/2018 2:34 AM Digraph Application Scheduling: edge (a,b) means task a must be completed before b can be started ics21 ics22 ics23 ics51 ics53 ics52 ics161 ics131 ics141 ics121 ics171 The good life ics151

Graphs 11/16/2018 2:34 AM Directed DFS We can specialize the traversal algorithms (DFS and BFS) to digraphs by traversing edges only along their direction In the directed DFS algorithm, we have four types of edges discovery edges back edges forward edges cross edges A directed DFS starting at a vertex s determines the vertices reachable from s E D C B A

Graphs 11/16/2018 2:34 AM Reachability DFS tree rooted at v: vertices reachable from v via directed paths E D E D C A C F E D A B C F A B

Strong Connectivity Each vertex can reach all other vertices a g c d e Graphs 11/16/2018 2:34 AM Strong Connectivity Each vertex can reach all other vertices a d c b e f g

Strong Connectivity Algorithm Graphs 11/16/2018 2:34 AM Strong Connectivity Algorithm Pick a vertex v in G. Perform a DFS from v in G. If there’s a w not visited, print “no”. Let G’ be G with edges reversed. Perform a DFS from v in G’. Else, print “yes”. Running time: O(n+m). a G: g c d e b f a G’: g c d e b f

Strongly Connected Components Graphs 11/16/2018 2:34 AM Strongly Connected Components Maximal subgraphs such that each vertex can reach all other vertices in the subgraph Can also be done in O(n+m) time using DFS, but is more complicated (similar to biconnectivity). a d c b e f g { a , c , g } { f , d , e , b }

Graphs 11/16/2018 2:34 AM Transitive Closure Given a digraph G, the transitive closure of G is the digraph G* such that G* has the same vertices as G if G has a directed path from u to v (u  v), G* has a directed edge from u to v The transitive closure provides reachability information about a digraph D E B G C A D E B C A G*

Computing the Transitive Closure Graphs 11/16/2018 2:34 AM Computing the Transitive Closure If there's a way to get from A to B and from B to C, then there's a way to get from A to C. We can perform DFS starting at each vertex O(n(n+m)) Alternatively ... Use dynamic programming: the Floyd-Warshall Algorithm

Floyd-Warshall Transitive Closure Graphs 11/16/2018 2:34 AM Floyd-Warshall Transitive Closure Idea #1: Number the vertices 1, 2, …, n. Idea #2: Consider paths that use only vertices numbered 1, 2, …, k, as intermediate vertices: Uses only vertices numbered 1,…,k (add this edge if it’s not already in) i j Uses only vertices numbered 1,…,k-1 Uses only vertices numbered 1,…,k-1 k

Floyd-Warshall’s Algorithm Graphs 11/16/2018 2:34 AM Floyd-Warshall’s Algorithm Algorithm FloydWarshall(G) Input digraph G Output transitive closure G* of G i  1 for all v  G.vertices() denote v as vi i  i + 1 G0  G for k  1 to n do Gk  Gk - 1 for i  1 to n (i  k) do for j  1 to n (j  i, k) do if Gk - 1.areAdjacent(vi, vk)  Gk - 1.areAdjacent(vk, vj) if Gk.areAdjacent(vi, vj) Gk.insertDirectedEdge(vi, vj , k) return Gn Floyd-Warshall’s algorithm numbers the vertices of G as v1 , …, vn and computes a series of digraphs G0, …, Gn G0=G Gk has a directed edge (vi, vj) if G has a directed path from vi to vj with intermediate vertices in the set {v1 , …, vk} We have that Gn = G* In phase k, digraph Gk is computed from Gk - 1 Running time: O(n3), assuming areAdjacent is O(1) (e.g., adjacency matrix)

Floyd-Warshall Example Graphs 11/16/2018 2:34 AM Floyd-Warshall Example BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Iteration 1 Graphs 11/16/2018 2:34 AM Floyd-Warshall, Iteration 1 BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Iteration 2 Graphs 11/16/2018 2:34 AM Floyd-Warshall, Iteration 2 BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Iteration 3 Graphs 11/16/2018 2:34 AM Floyd-Warshall, Iteration 3 BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Iteration 4 Graphs 11/16/2018 2:34 AM Floyd-Warshall, Iteration 4 BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Iteration 5 Graphs 11/16/2018 2:34 AM Floyd-Warshall, Iteration 5 BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Iteration 6 Graphs 11/16/2018 2:34 AM Floyd-Warshall, Iteration 6 BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

Floyd-Warshall, Conclusion Graphs 11/16/2018 2:34 AM Floyd-Warshall, Conclusion BOS v ORD 4 JFK v 2 v 6 SFO DFW LAX v 3 v 1 MIA v 5

DAGs and Topological Ordering Graphs 11/16/2018 2:34 AM DAGs and Topological Ordering A directed acyclic graph (DAG) is a digraph that has no directed cycles A topological ordering of a digraph is a numbering v1 , …, vn of the vertices such that for every edge (vi , vj), we have i < j Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints Theorem A digraph admits a topological ordering if and only if it is a DAG D E B C A DAG G v4 v5 D E v2 B v3 C v1 Topological ordering of G A

Graphs 11/16/2018 2:34 AM Topological Sorting Number vertices, so that (u,v) in E implies u < v 1 A typical student day wake up 2 3 eat study computer sci. 4 5 nap more c.s. 7 play 8 write c.s. program 6 9 work out make cookies for professors 10 sleep 11 dream about graphs

Algorithm for Topological Sorting Graphs 11/16/2018 2:34 AM Algorithm for Topological Sorting Note: This algorithm is different than the one in Goodrich-Tamassia Running time: O(n + m). How…? Method TopologicalSort(G) H  G // Temporary copy of G n  G.numVertices() while H is not empty do Let v be a vertex with no outgoing edges Label v  n n  n - 1 Remove v from H

Topological Sorting Algorithm using DFS Graphs Topological Sorting Algorithm using DFS 11/16/2018 2:34 AM Simulate the algorithm by using depth-first search O(n+m) time. Algorithm topologicalDFS(G, v) Input graph G and a start vertex v of G Output labeling of the vertices of G in the connected component of v setLabel(v, VISITED) for all e  G.incidentEdges(v) if getLabel(e) = UNEXPLORED w  opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) topologicalDFS(G, w) else {e is a forward or cross edge} Label v with topological number n n  n - 1 Algorithm topologicalDFS(G) Input dag G Output topological ordering of G n  G.numVertices() for all u  G.vertices() setLabel(u, UNEXPLORED) for all e  G.edges() setLabel(e, UNEXPLORED) for all v  G.vertices() if getLabel(v) = UNEXPLORED topologicalDFS(G, v)

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 7 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 6 7 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 6 5 7 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 4 6 5 7 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 3 4 6 5 7 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 2 3 4 6 5 7 8 9

Topological Sorting Example Graphs 11/16/2018 2:34 AM Topological Sorting Example 2 1 3 4 6 5 7 8 9

Graphs 11/16/2018 2:34 AM Campus Tour

Outline and Reading Review Adjacency matrix structure (§6.2.3) Graphs 11/16/2018 2:34 AM Outline and Reading Review Adjacency matrix structure (§6.2.3) Kruskal’s MST algorithm (§7.3.1) Partition ADT and implementation (§4.2.2) The decorator pattern (§6.5.1) The traveling salesperson problem Definition Approximation algorithm (§13.4.3)

Graph Assignment Goals Graphs 11/16/2018 2:34 AM Graph Assignment Goals Learn and implement the adjacency matrix structure an Kruskal’s minimum spanning tree algorithm Understand and use the decorator pattern and various JDSL classes and interfaces Our task Implement the adjacency matrix structure for representing a graph Implement Kruskal’s MST algorithm Frontend Computation and visualization of an approximate traveling salesperson tour

Adjacency Matrix Structure Graphs 11/16/2018 2:34 AM Adjacency Matrix Structure Edge list structure Augmented vertex objects Integer key (index) associated with vertex 2D-array adjacency array Reference to edge object for adjacent vertices Null for non nonadjacent vertices a v b u w u 1 v 2 w 1 2  a b

Kruskal’s Algorithm The vertices are partitioned into clouds Graphs 11/16/2018 2:34 AM Kruskal’s Algorithm The vertices are partitioned into clouds We start with one cloud per vertex Clouds are merged during the execution of the algorithm Partition ADT: makeSet(o): create set {o} and return a locator for object o find(l): return the set of the object with locator l union(A,B): merge sets A and B Algorithm KruskalMSF(G) Input weighted graph G Output labeling of the edges of a minimum spanning forest of G Q  new heap-based priority queue for all v  G.vertices() do l  makeSet(v) { elementary cloud } setLocator(v,l) for all e  G.edges() do Q.insert(weight(e), e) while Q.isEmpty() e  Q.removeMin() [u,v]  G.endVertices(e) A  find(getLocator(u)) B  find(getLocator(v)) if A  B setMSFedge(e) { merge clouds } union(A, B)

Graphs 11/16/2018 2:34 AM Example G G 8 8 B 4 B 4 E 9 E 6 9 6 5 F 5 1 1 F C 3 3 11 C 11 2 2 7 H 7 H D D A 10 A 10 G G 8 8 B 4 B 4 9 E E 6 9 6 5 F 5 1 1 F C 3 11 C 3 11 2 2 7 H 7 H D D A A 10 10

Example (contd.) four steps two steps G G 8 8 B 4 B 4 E 9 E 6 9 6 5 F Graphs 11/16/2018 2:34 AM Example (contd.) G G 8 8 B 4 B 4 E 9 E 6 9 6 5 F 5 1 1 F C 3 C 3 11 11 2 2 7 H 7 H D D A 10 A 10 four steps two steps G G 8 8 B 4 B 4 E E 9 6 9 6 5 5 1 F 1 F C 3 11 C 3 11 2 2 7 H 7 H D D A A 10 10

Partition Implementation Graphs 11/16/2018 2:34 AM Partition Implementation Partition implementation A set is represented the sequence of its elements A position stores a reference back to the sequence itself (for operation find) The position of an element in the sequence serves as locator for the element in the set In operation union, we move the elements of the smaller sequence into to the larger sequence Worst-case running times makeSet, find: O(1) union: O(min(nA, nB)) Amortized analysis Consider a series of k Partiton ADT operations that includes n makeSet operations Each time we move an element into a new sequence, the size of its set at least doubles An element is moved at most log2 n times Moving an element takes O(1) time The total time for the series of operations is O(k  n log n)

Analysis of Kruskal’s Algorithm Graphs 11/16/2018 2:34 AM Analysis of Kruskal’s Algorithm Graph operations Methods vertices and edges are called once Method endVertices is called m times Priority queue operations We perform m insert operations and m removeMin operations Partition operations We perform n makeSet operations, 2m find operations and no more than n - 1 union operations Label operations We set vertex labels n times and get them 2m times Kruskal’s algorithm runs in time O((n + m) log n) time provided the graph has no parallel edges and is represented by the adjacency list structure

Graphs 11/16/2018 2:34 AM Decorator Pattern The decorator pattern extends the methods of the Position ADT to support the handling of attributes (labels) has(a): tests whether the position has attribute a get(a): returns the value of attribute a set(a, x): sets to x the value of attribute a destroy(a): removes attribute a and its associated value (for cleanup purposes) The decorator pattern can be implemented by storing a dictionary of (attribute, value) items at each position Labels are commonly used in graph algorithms Auxiliary data Output Examples DFS: unexplored/visited label for vertices and unexplored/ forward/back labels for edges Dijkstra and Prim-Jarnik: distance, locator, and parent labels for vertices Kruskal: locator label for vertices and MSF label for edges

Traveling Salesperson Problem Graphs 11/16/2018 2:34 AM Traveling Salesperson Problem A tour of a graph is a spanning cycle (e.g., a cycle that goes through all the vertices) A traveling salesperson tour of a weighted graph is a tour that is simple (i.e., no repeated vertices or edges) and has has minimum weight No polynomial-time algorithms are known for computing traveling salesperson tours The traveling salesperson problem (TSP) is a major open problem in computer science Find a polynomial-time algorithm computing a traveling salesperson tour or prove that none exists 7 D B 4 2 5 2 F C 8 3 6 E A 1 Example of traveling salesperson tour (with weight 17)

Graphs 11/16/2018 2:34 AM TSP Approximation We can approximate a TSP tour with a tour of at most twice the weight for the case of Euclidean graphs Vertices are points in the plane Every pair of vertices is connected by an edge The weight of an edge is the length of the segment joining the points Approximation algorithm Compute a minimum spanning tree Form an Eulerian circuit around the MST Transform the circuit into a tour