Subgraphs, Connected Components, Spanning Trees Depth-First Search 2/4/2019 8:01 PM Subgraphs, Connected Components, Spanning Trees Depth-First Search

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 Depth-First Search

Non connected graph with two connected components 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 Depth-First Search

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 Depth-First Search

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 Depth-First Search

Cycles How to detect that a graph has a cycle? Hint: BFS (or DFS) Depth-First Search

Connectivity Determines whether G is connected How do we know G is not connected? Hint: BFS or DFS Depth-First Search

Connected Components Computes the connected components of G How do we find all the connected components? Depth-First Search

BFS for an entire graph Algorithm BFS(G, s) Queue  new empty queue enqueue(Queue, s) while Queue.isEmpty() v  dequeue(Queue) if getLabel(v) != VISITED visit v setLabel(v, VISITED) neighbors  getAdjacentVertices(G, v) for each w  neighbors if getLabel(w) != VISITED // “children” enqueue(Queue, w) The algorithm uses a mechanism for setting and getting “labels” of vertices and edges Algorithm BFS(G) Input graph G Output labeling of the edges and partition of the vertices of G for each u  G.vertices() setLabel(u, UNEXPLORED) for each v  G.vertices() if getLabel(v) = UNEXPLORED BFS(G, v) Breadth-First Search

DFS for an Entire Graph The algorithm uses a mechanism for setting and getting “labels” of vertices and edges Algorithm DFS(G, v) visit v setLabel(v, VISITED) neighbors  getAdjacentVertices(G, v) for each w  neighbors if getLabel(w) != VISITED // “children” DFS(G, w) Algorithm DFS(G) Input graph G Output labeling of the edges of G as discovery edges and back edges for each u  G.vertices() setLabel(u, UNEXPLORED) for each v  G.vertices() if getLabel(v) = UNEXPLORED DFS(G, v) Depth-First Search

Spanning Tree Computes a spanning tree of G How do we find a spanning tree? Depth-First Search

BFS Example unexplored vertex visited vertex unexplored edge 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 Breadth-First Search

BFS Example (cont.) L0 L1 L0 L1 L2 L0 L1 L2 L0 L1 L2 C B A E D F C B A Breadth-First Search

BFS Example (cont.) L0 L1 L2 L0 L1 L2 L0 L1 L2 C B A E D F A B C D E F Breadth-First Search

DFS Example unexplored vertex visited vertex unexplored edge B A C E A A visited vertex unexplored edge discovery edge back edge D B A C E D B A C E Depth-First Search

DFS Example (cont.) D B A C E D B A C E D B A C E D B A C E Depth-First Search

Spanning Forest Computes a spanning forest of G How do we find a spanning forest? Depth-First Search

Path finding Given a start vertex s and a destination vertex z Find a path from s to z Depth-First Search

Path finding in a maze Vertex intersection, corner and dead end Edge Corridor Depth-First Search

BFS and path finding Hint: use the spanning tree Depth-First Search

BFS and path finding Use the spanning tree Start from destination Go to its parent Until source is reached Depth-First Search

BFS and path finding Use the spanning tree Start from destination Go to its parent Until source is reached The found path is the shortest in terms of number of edges --- Why? Depth-First Search

DFS and path finding Use the spanning tree Start from destination Go to its parent Until source is reached The spanning tree could be large Use DFS but faster, ideas? Depth-First Search

Path Finding find a path between two given vertices v and z Algorithm pathDFS(G, v, z, S) setLabel(v, VISITED) push(S, v) if v = z return S.elements() path = null neighbors  getAdjacentVertices(G, v) for each w  neighbors if getLabel(w) != VISITED // “child” path = pathDFS(G, w, z, S) if path != null // has path via a child return path pop(S, v) //no path via a child or no children return null find a path between two given vertices v and z call DFS(G, v) with v as the start vertex use a stack S to keep track of the path When destination vertex z is encountered, return the path as the contents of the stack Depth-First Search

DFS vs. BFS Applications DFS BFS DFS BFS Spanning forest, connected components, paths, cycles  Shortest paths Biconnected components (still connected after removing one vertex, algorithm not discussed) C B A E D L0 L1 F L2 A B C D E F DFS BFS Breadth-First Search

DFS vs. BFS (cont.) Back edge (v,w) Cross edge (v,w) DFS BFS 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 C B A E D L0 L1 F L2 A B C D E F DFS BFS Breadth-First Search