Graphs. Terminology. Graphs & Breadth First Search (BFS) Extremely useful tool in modeling problems. Consist of: Vertices Edges

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1 COMP Spring Graphs & BS / Slide Graphs Graphs & Breadth irst Search (BS) Extremely useful tool in modeling problems. Consist of: Vertices Edges Vertex A B D C E Edge Vertices can be considered sites or locations. Edges represent connections between vertices. Graphs & BS / Slide Application Air flight system Graphs & BS / Slide Application Wireless communication problem Each vertex represents a city. Each edge represents a direct flight between two cities. A query on direct flights = a query on whether an edge exists. A query on how to get to a location = does a path exist from A to B. We can even associate costs to edges (weighted graphs), then ask what is the cheapest path from A to B. Represented by a weighted complete graph (every two vertices are connected by an edge). Each edge is associated with the Euclidean distance d ij between two stations. Each station i uses a certain power to transmit messages. Given this power, only a few nodes can be reached (bold edges). A station reachable by i then uses its own power to relay the message to other stations not reachable by i. A typical wireless communication problem is: how to broadcast between all stations such that they are all connected (i.e. a message can reach all stations) and the power consumption is minimized. Graphs & BS / Slide Definition A graph G=(V, E) consists of a set of vertices, V, and a set of edges, E. Each edge is a pair of vertices {v, w}, where v and w belong to V. If the pair is unordered, the graph is undirected; otherwise it is directed. An undirected graph {b,e} {a,b} {b,d} {e,f} {c,d} {a,c} {c,f} Graphs & BS / Slide erminology. If v and v are connected, they are said to be adjacent vertices. v and v are endpoints of the edge {v, v }.. If an edge e is connected to v, then v is said to be incident on e (or v is a vertex incident on edge e). Also, the edge e is said to be incident to v (or e is an edge incident to vertex v). If we are talking about directed graphs, where edges have direction. his means that {v,v } does not equal to {v,v }. Directed graphs are drawn with arrows. (called {v, varcs) } = between {v, edges. v } A B his means {A,B} only, not {B,A}

2 Graphs & BS / Slide Graph Representation wo popular computer representations of a graph. Both represent the vertex set and the edge set, but in different ways. Graphs & BS / Slide Adjacency Matrix rom o. Adjacency Matrix Use a D matrix to represent the graph. Use a D array of linked lists D array A[..n,..n], where n is the number of vertices in a graph. Each row and column is indexed by the vertex id e.g. a=, b=, c=, d=, e=. A[i][j]= and A[j][i]= if there is an undirected edge connecting vertices i and j; otherwise, A[i][j]= A[j][i]=. he storage requirement is Θ(n ). It is not spaceefficient if the graph has few edges. An adjacency matrix is an appropriate representation if the graph is dense: E =Θ( V ). We can detect in O() time whether two vertices are connected. Graphs & BS / Slide Graphs & BS / Slide Adjacency Matrix Example o If the graph is not dense, in other words, sparse, a better solution is an adjacency list. he adjacency list is a D array A[..n] of linked lists, where n is the number of vertices in a graph. Each array entry is indexed by the vertex id. Each list A[i] stores the ids of the vertices adjacent to vertex i. rom Graphs & BS / Slide Example Graphs & BS / Slide Storage of he array takes up Θ(n) space. Define degree of v, deg(v), to be the number of edges incident to v. hen, the total space to store the graph is proportional to: vertex v deg( v) An edge e={u,v}, where u does not equal to v, of the graph contributes a count of to deg(u) and contributes a count to deg(v). herefore, Σ vertex v deg(v) = m, where m is the total number of edges. In all, the adjacency list takes up Θ(n+m) space If m = O(n ) (i.e. dense graphs), both adjacent matrix and adjacent lists use Θ(n ) space. If m = O(n), adjacent list outperforms adjacent matrix. However, one cannot tell in O() time whether two vertices are connected.

3 Graphs & BS / Slide vs. Matrix More compact than adjacency matrix if a graph has few edges. Requires more time to find if an edge exists. Adjacency Matrix Always require Θ(n ) space. his can waste a lot of space if the number of edges are small (the matrix is sparse). Can quickly find if an edge exists. Graphs & BS / Slide Path between Vertices A path is a sequence of vertices (v, v, v, v k ) such that: or i < k, {v i, v i+ } is an edge. or i < k, v i v i+ hat is, the edge {v i, v i+ } {v i+, v i+ }. An undirected edge should not be repeated immediately. Otherwise, there are two immediately repeated undirected edges between two adjacent vertices. Note: Given that the two conditions above are satisfied, a path is allowed to go through the same vertex or the same edge any number of times! he length of a path is the number of edges on the path. Graphs & BS / Slide ypes of paths Graphs & BS / Slide Path Examples Are these paths? A path is simple if and only if it does not contain a vertex more than once. All vertices are distinct. A path is a cycle if and only if v = v k. he start and end vertices are the same vertex! A path contains a cycle as its subpath if some vertex appears twice or more. Any cycles? Yes What is the path s length?. {a,c,f,e} Simple path, L =. {a,b,d,c,f,e} Simple path, L =. {a,c,d,b,d,c,f,e} Not a path because of {d,b,d}. {a,c,d,b,a} Cycle, L =. {a,c,f,e,b,d,c,a} Cycle, L = Graphs & BS / Slide Graph raversal Application example Given a graph representation and a vertex s in the graph. ind all paths from s to other vertices, e.g. is there a path between s and v?, if so, what is the path between s and v?. wo common graph traversal algorithms Breadthirst Search (BS) ind paths and the shortest paths in an undirected graph. ind connected components. Depthirst Search (DS) ind paths in an undirected graph. ind connected components. Perform topological sort on a directed graph. Graphs & BS / Slide BS and Shortest Path Problem Given any vertex s, BS visits the other vertices at increasing distances away from s. In each step, BS visits all reachable vertices with constant distance. In doing so, BS discovers paths from s to other vertices. What does it mean by distance? he number of edges on the path (the shortest path) from s. s Example Consider s=vertex Nodes at distance?,,, Nodes at distance?,,, Nodes at distance?

4 Graphs & BS / Slide BS Algorithm Graphs & BS / Slide BS Example Visited able (/) // flag[ ]: visited table Why use queue? Need IO Initialize visited table (all alse) Q = { } Initialize Q to be empty Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) lag that has been visited Mark neighbors as visited,, Q = { } Q = {} Q = {,, } Place on the queue Dequeue. Place all unvisited neighbors of on the queue Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Mark newly visited neighbors, Mark newly visited neighbors, Q = {,, } Q = {,,, } Dequeue. Place all unvisited neighbors of on the queue. Notice that is not placed on the queue again, it has been visited! Q = {,,, } Q = {,,,, } Dequeue. Place all unvisited neighbors of on the queue. Only nodes and haven t been visited yet.

5 Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Q = {,,,, } Q = {,,, } Dequeue. has no unvisited neighbors! Q = {,,, } Q = {,, } Dequeue. has no unvisited neighbors! Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Mark newly visited neighbor Q = {,, } Q = {, } Dequeue. has no unvisited neighbors! Q = {, } Q = {, } Dequeue. place neighbor on the queue. Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Mark newly visited neighbor Q = {, } Q = {, } Dequeue. place neighbor on the queue Q = {, } Q = { } Dequeue. no unvisited neighbors of

6 Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) What did we discover? Q = { } Q = { } Dequeue. no unvisited neighbors of Q = { } SOP!!! Q is empty!!! Look at visited tables. here exists a path from vertex to all vertices in the graph Graphs & BS / Slide ime Complexity of BS (Using ) Assume adjacency list n = number of vertices, and m = number of edges. O(n + m) Each vertex will enter Q (will be enqueued and dequeued) at most once. Each iteration takes time proportional to deg(v) + (the number is to account for the work required for dequeue operation). Graphs & BS / Slide Running ime Recall: Given a graph with m edges, what is the total degree? Σ vertex v deg(v) = m he total running time of the while loop is: O( Σ vertex v (deg(v) + ) ) = O(m+n) = O(n+m) his is summing over all the iterations in the while loop! Graphs & BS / Slide ime Complexity of BS (Using Adjacency Matrix) Assume adjacency matrix n = number of vertices, m = number of edges. O(n ) inding the adjacent vertices of v requires checking all elements in the row. his takes linear time O(n). Summing over all the n iterations, the total running time is O(n ). As such, with adjacency matrix, BS is O(n ) independent of the number of edges m. With adjacency list, BS is O(n+m); if m=o(n ), similar to a dense graph, O(n+m)=O(n ). Graphs & BS / Slide Shortest Path Recording BS only tells us whether a path exists from s to other vertices v. It doesn t tell us the path! We need to modify the algorithm to record the path How can we do that? We do not know which vertices lie on this path until we reach v! But, for {w,v}, we know vertex w has been visited from vertex v. Efficient solution: Use an additional array pred[..n] to record the predecessor for each vertex. [w] = v means that vertex w has been visited from v.

7 Graphs & BS / Slide BS + Path inding Graphs & BS / Slide Example Visited able (/) initialize all pred[v] to Record where you came from. w was previously visited by pred[w]=v Q = { } Initialize Q to be empty Initialize visited table (all alse) Initialize to Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Q = { } Place on the queue. lag that has been visited. Q = {} Q = {,, } Dequeue. Place all unvisited neighbors of on the queue Mark neighbors as visited. Record in that we came from. Graphs & BS / Slide Graphs & BS / Slide Q = {,, } Q = {,,, } Visited able (/) Mark new visited. Record in Dequeue. that we came Place all unvisited neighbors of on the queue. from. Notice that is not placed on the queue again, it has been visited! Q = {,,, } Q = {,,,, } Dequeue. Place all unvisited neighbors of on the queue. Only nodes and haven t been visited yet. Visited able (/) Mark new visited. Record in that we came from.

8 Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Q = {,,,, } Q = {,,, } Dequeue. has no unvisited neighbors! Q = {,,, } Q = {,, } Dequeue. has no unvisited neighbors! Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Q = {,, } Q = {, } Dequeue. has no unvisited neighbors! Q = {, } Q = {, } Dequeue. place neighbor on the queue. Mark new visited Vertex. Record in that we came from. Graphs & BS / Slide Graphs & BS / Slide Visited able (/) Visited able (/) Q = {, } Q = {, } Dequeue. place neighbor on the queue. Mark new visited Vertex. Record in that we came from. Q = {, } Q = { } Dequeue. no unvisited neighbors of.

9 Graphs & BS / Slide Visited able (/) Graphs & BS / Slide BS inished Visited able (/) Q = { } Q = { } Dequeue. no unvisited neighbors of. Q = { } SOP!!! Q is empty!!! now can be traced backward to report the path! Graphs & BS / Slide Recursive algorithm Path Reporting nodes visited from ry some examples, report path from s to v: Path() Path() Path() he path returned is the shortest from s to v (minimum number of edges). Graphs & BS / Slide BS ree he paths found by BS is often drawn as a rooted tree (called BS tree), with the starting vertex as the root of the tree. BS tree for vertex s=. Question: What would a level order traversal tell you? Graphs & BS / Slide Recording the shortest distances between s to any connected vertices d[w]=d[v]+; d[s] = ; d[v] = ; Graphs & BS / Slide Application of BS One application concerns how to find connected components in a graph. If a graph has more than one connected components, BS builds a BSforest (not just BStree)! Each tree in the forest is a connected component.