Lecture Notes CMSC 251 Observation For a digraph e n2 O n2 For an undirected graph e O n2 n 2 n n 1 2 A graph or digraph is allowed to have no edges at all One interesting question is what the minimum number of edges that a connected graph must have We say that a graph is sparse if e is much less than n2 For example the important class of planar graphs graphs which can be drawn on the plane so that no two edges cross over one another e O n In most application areas very large graphs tend to be sparse This is important to keep in mind when designing graph algorithms because when n is really large and O n2 running time is often unacceptably large for real time response Lecture 22 Graphs Representations and BFS Thursday April 16 1998 Read Sections 23 1 through 23 3 in CLR Representations of Graphs and Digraphs We will describe two ways of representing graphs and digraphs First we show how to represent digraphs Let G V E be a digraph with n V and let e E We will assume that the vertices of G are indexed 1 2 n Adjacency Matrix An n n matrix defined for 1 v w n 1 if v w E A v w 0 otherwise If the digraph has weights we can store the weights in the matrix For example if v w E then A v w W v w the weight on edge v w If v w E then generally W v w need not be defined but often we set it to some special value e g A v w 1 or By we mean in practice some number which is larger than any allowable weight In practice this might be some machine dependent constant like MAXINT Adjacency List An array Adj 1 n of pointers where for 1 v n Adj v points to a linked list containing the vertices which are adjacent to v i e the vertices that can be reached from v by a single edge If the edges have weights then these weights may also be stored in the linked list elements 1 2 Adj 1 1 1 2 1 3 1 1 1 2 0 0 1 2 3 3 0 1 0 3 2 2 3 3 Adjacency matrix Adjacency list Figure 23 Adjacency matrix and adjacency list for digraphs We can represent undirected graphs using exactly the same representation but we will store each edge twice In particular we representing the undirected edge v w by the two oppositely directed edges v w and w v Notice that even though we represent undirected graphs in the same way that we 65 Lecture Notes CMSC 251 represent digraphs it is important to remember that these two classes of objects are mathematically distinct from one another This can cause some complications For example suppose you write an algorithm that operates by marking edges of a graph You need to be careful when you mark edge v w in the representation that you also mark w v since they are both the same edge in reality When dealing with adjacency lists it may not be convenient to walk down the entire linked list so it is common to include cross links between corresponding edges Adj 1 4 2 3 1 1 0 2 1 3 1 4 1 2 1 0 1 0 3 1 1 0 1 4 1 0 1 0 Adjacency matrix 1 2 3 2 1 3 3 1 2 4 1 4 4 3 Adjacency list with crosslinks Figure 24 Adjacency matrix and adjacency list for graphs An adjacency matrix requires n2 storage and an adjacency list requires n e storage one entry for each vertex in Adj and each list has outdeg v entries which when summed is e For sparse graphs the adjacency list representation is more cost effective Shortest Paths To motivate our first algorithm on graphs consider the following problem You are given an undirected graph G V E by the way everything we will be saying can be extended to directed graphs with only a few small changes and a source vertex s V The length of a path in a graph without edge weights is the number of edges on the path We would like to find the shortest path from s to each other vertex in G If there are ties two shortest paths of the same length then either path may be chosen arbitrarily The final result will be represented in the following way For each vertex v V we will store d v which is the distance length of the shortest path from s to v Note that d s 0 We will also store a predecessor or parent pointer v which indicates the first vertex along the shortest path if we walk from v backwards to s We will let s NIL It may not be obvious at first but these single predecessor pointers are sufficient to reconstruct the shortest path to any vertex Why We make use of a simple fact which is an example of a more general principal of many optimization problems called the principal of optimality For a path to be a shortest path every subpath of the path must be a shortest path If not then the subpath could be replaced with a shorter subpath implying that the original path was not shortest after all Using this observation we know that the last edge on the shortest path from s to v is the edge u v then the first part of the path must consist of a shortest path from s to u Thus by following the predecessor pointers we will construct the reverse of the shortest path from s to v Obviously there is simple brute force strategy for computing shortest paths We could simply start enumerating all simple paths starting at s and keep track of the shortest path arriving at each vertex However since there can be as many as n simple paths in a graph consider a complete graph then this strategy is clearly impractical Here is a simple strategy that is more efficient Start with the source vertex s Clearly the distance to each of s s neighbors is exactly 1 Label all of them with this distance Now consider the unvisited 66 Lecture Notes CMSC 251 2 1 s 2 1 2 2 s 3 s 2 1 2 Undiscovered 3 3 Discovered 3 3 Finished Figure 25 Breadth first search for shortest paths neighbors of these neighbors They will be at distance 2 from s Next consider the unvisited neighbors of the neighbors of the neighbors and so on Repeat this until there are no more unvisited neighbors left to visit This algorithm can be visualized as simulating a wave propagating outwards from s visiting the vertices in bands at ever increasing distances from s Breadth first search Given an graph G V E breadth first search starts at some source vertex s and discovers which vertices are reachable from s Define …