Graph ADT. Lecture18: Graph II. Adjacency Matrix. Representation of Graphs. Data Vertices and edges. Methods

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1 Graph T S33: ata Structures (0) Lecture8: Graph II ohyung Han S, POSTH bhhan@postech.ac.kr ata Vertices and edges endvertices(e): an array of the two endvertices of e opposite(v,e): the vertex opposite of v on e aredjacent(v,w): true iff v and w are adjacent getdjacent(v): return a list of the vertices adjacent to vertex v insertvertex(o): insert a vertex storing element o insertdge(v,w,o): insert an edge (v,w) storing element o removevertex(v): remove vertex v (and its incident edges) removedge(e): remove edge e incidentdges(v): edges incident to v getegree(v): Returns the degree of vertex v S33: ata Structures by Prof. ohyung Han, all 0 Representation of Graphs Graphs should be able to be represented by a logical method. Information to be stored Vertices: number and labels dges: connectivity and weights djacency matrix djacency list Representation by matrix djacency Matrix Uses a matrix of size Unweighted graph: ach entry, is true if there is an edge from vertex to vertex, otherwise false. Weighted graph: ach entry stores weights. haracteristics Very simple, but large space requirement: ppropriate if the graph is dense 3 S33: ata Structures by Prof. ohyung Han, all 0 4 S33: ata Structures by Prof. ohyung Han, all 0

2 djacency Matrix Undirected and unweighted graph djacency Matrix irected and unweighted graph S33: ata Structures by Prof. ohyung Han, all 0 6 S33: ata Structures by Prof. ohyung Han, all 0 djacency Matrix djacency List irected and weighted graph Representation by a vector of lists Motivation: Maintaining adjacent matrix is inefficient if graph is sparse. vector of lists of vertices The th element of the vector is a list of vertices adjacent to. omplexity If graph is sparse: If graph is dense: 7 S33: ata Structures by Prof. ohyung Han, all 0 8 S33: ata Structures by Prof. ohyung Han, all 0

3 djacency List djacency List Undirected graph irected graph S33: ata Structures by Prof. ohyung Han, all 0 0 S33: ata Structures by Prof. ohyung Han, all 0 Performance Graph Traversal djacency List djacency Matirx Space incidentdges(v) deg aredjacent(v,w) min deg,deg insertvertex(o) insertdge(v,w,o) removevertex(v) removedge(e) max deg,deg The most common operation is to visit all the vertices in a systematic way. traversal starts at a vertex v and visits all the vertices u such that a path exists from v to u. Similar to tree traversal Unlike trees, we need to specifically guard against repeating a path from a cycle and considering the same vertices multiple times epth irst Search (S): based on stack readth irst Search (S): based on queue vertices, edges No multiple edges and no self loops S33: ata Structures by Prof. ohyung Han, all 0 S33: ata Structures by Prof. ohyung Han, all 0

4 epth irst Search xample S traversal of a graph Searches all the way down a path before backing up to explore alternatives Recursive Stack based Time complexity: S can be further extended to solve other graph problems: ind and report a path between two given vertices ind a cycle in the graph ack edge iscovery edge unexplored vertex visited vertex unexplored edge discovery edge back edge 3 S33: ata Structures by Prof. ohyung Han, all 0 4 S33: ata Structures by Prof. ohyung Han, all 0 xample Properties of S Properties S(G,v) visits all the vertices and edges in the connected component of v. The discovery edges labeled by S(G,v) form a spanning tree of the connected component of v. ack edge iscovery edge S33: ata Structures by Prof. ohyung Han, all 0 6 S33: ata Structures by Prof. ohyung Han, all 0

5 S lgorithm nalysis of S lgorithm S(G) Input graph G Output labeling of the edges of G as discovery edges and back edges for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR S(G, v) lgorithm S(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, VISIT) for all e G.incidentdges(v) if getlabel(e) = UNXPLOR if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) S(G, w) setlabel(e, K) setlabel() for a vertex and edge time ach vertex is labeled twice: UNXPLOR and VISIT ach edge is labeled twice: UNXPLOR and (ISOVRY or K) getlabel() for a vertex and edge time The label of each vertex and edge is checked time for all vertices. incidentdges() time for all vertices alled once for each vertex Time complexity of S provided the graph is represented by the adjacency list Recall that deg 7 S33: ata Structures by Prof. ohyung Han, all 0 8 S33: ata Structures by Prof. ohyung Han, all 0 Path inding ycle inding lgorithm Recursive method We use a stack S to keep track of the path between the start vertex and the current vertex. s soon as destination vertex z is encountered, we return the path as the contents of the stack. It finds all paths from v to z. lgorithm paths(g, v, z) setlabel(v, VISIT) S.push(v) if v = z return S.elements() for all e G.incidentdges(v) if getlabel(e) = UNXPLOR if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) S.push(e) paths(g, w, z) S.pop(e) setlabel(e, K) S.pop(v) lgorithm We use a stack S to keep track of the path between the start vertex and the current vertex. s 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. lgorithm cycles(g, v, z) setlabel(v, VISIT) S.push(v) for all e G.incidentdges(v) if getlabel(e) = UNXPLOR S.push(e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) paths(g, w, z) S.pop(e) T new empty stack repeat o S.pop() T.push(o) until o = w return T.elements() S.pop(v) 9 S33: ata Structures by Prof. ohyung Han, all 0 0 S33: ata Structures by Prof. ohyung Han, all 0

6 readth irst Search xample S traversal of a graph Traverse the graph level by level Queue based Iterative Time complexity: S can be further extended to solve other graph problems ind and report a path with the minimum number of edges between two given vertices ind a simple cycle, if there is one iscovery edge unexplored vertex visited vertex ross edge unexplored edge discovery edge cross edge S33: ata Structures by Prof. ohyung Han, all 0 S33: ata Structures by Prof. ohyung Han, all 0 xample xample 3 S33: ata Structures by Prof. ohyung Han, all 0 4 S33: ata Structures by Prof. ohyung Han, all 0

7 Properties S lgorithm Properties S(G,s) visits all the vertices and edges of G. The discovery edges labeled by S(G,s) form a spanning tree. Level represents the number of edges to reach the node in the level from the initial node. lgorithm S(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR S(G, v) lgorithm S(G, s) L 0 new empty sequence L 0.addLast(s) setlabel(s, VISIT) i 0 while L i.ismpty() L i + new empty sequence for all v L i.elements() for all e G.incidentdges(v) if getlabel(e) = UNXPLOR if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) setlabel(w, VISIT) L i +.addlast(w) setlabel(e, ROSS) i i + S33: ata Structures by Prof. ohyung Han, all 0 6 S33: ata Structures by Prof. ohyung Han, all 0 S lgorithm nalysis of S lgorithm S(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR S(G, v) lgorithm S(G, s) Q.enqueue(s) setlabel(s, VISIT) while Q.ismpty() for all v Q.dequeue () for all e G.incidentdges(v) if getlabel(e) = UNXPLOR if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) setlabel(w, VISIT) Q.enqueue(w) setlabel(e, ROSS) setlabel() for a vertex and edge time ach vertex is labeled twice: UNXPLOR and VISIT ach edge is labeled twice: UNXPLOR and (ISOVRY or K) getlabel() for a vertex and edge time The label of each vertex and edge is checked a constant number of time. incidentdges() time for all vertices alled once for each vertex Time complexity of S provided the graph is represented by the adjacency list Recall that deg 7 S33: ata Structures by Prof. ohyung Han, all 0 8 S33: ata Structures by Prof. ohyung Han, all 0

8 pplications ind the connected components ind a spanning forest ind a simple cycle in G, or report that G is a tree or forest Given two vertices, find a path in G between them with the minimum number of edges, or report that no such path exists S vs. S pplications S S Spanning forest, connected components, paths, cycles Shortest paths iconnected components Time complexity: S S 9 S33: ata Structures by Prof. ohyung Han, all 0 30 S33: ata Structures by Prof. ohyung Han, all 0 S vs. S ack edge (v,w) w is an ancestor of v in the tree of discovery edges ross edge (v,w) w is in the same level as w or in the next level S S 3 S33: ata Structures by Prof. ohyung Han, all 0 3