Adjacency Matrix Undirected Graph. » Matrix [ p, q ] is 1 (True) if < p, q > is an edge in the graph. Graphs Representation & Paths

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1 djacency Matrix Undirected Graph Row and column indices are vertex numbers Graphs Representation & Paths Mixture of Chapters 9 & 0» Matrix [ p, q ] is (True) if < p, q > is an edge in the graph if < p, q > is not an edge in the graph Space is O ( (#V) ) from 3 to GR- GR- djacency Matrix Directed Graph Incidence Matrix» Matrix [ p, q ] is (True) if < p, q > is an edge in the graph if < p, q > is not an edge in the graph In general the matrix entries can be any data representing the edge. irline flight number for example True, False are the minimum to say an edge exists or does not exist from 3 to Row indices are edges numbers Column indices are vertex numbers» Matrix [ p, q ] is (True) if edge p is incident on vertex q if edge p is not incident on vertex q Space is O ( #V * # ) edge 3 3 vertex GR-3 GR- Path xistence? Path existence is a fundamental question djacency Matrix corresponds to the problem Consider matrix multiplication C = B = B + B B j,, k j,, k jn, nk, What does it mean if = B and we have an adjacency matrix? C = Path xistence? = j,, k j,, k jn, nk, ssume we store integer zeros and ones» [ j, p ] = means an edge goes from j to p [ p, k ] = means an edge goes from p to k [ j, p ] x [ p, k ] = means a path exists between j and k > In particular the path is of length and goes through vertex p GR- GR-6

2 Path xistence? 3 Path xistence? C = = j,, k j,, k jn, nk,» [ j, p ] x [ p, k ] 0 means a path exists between j and k > In particular the path is of length and goes through vertex p» So C[ j, k ] 0 if and only if a path of length exists between j and k» p :.. [ j, p ] x [ p, k ] 0 > In fact the value of C[ j, k ] is the number of such paths GR-7 = The above expression, using matrix multiplication gives us the number of paths of length two between any pair of matrices By extension p gives the number of paths of length p between any pair of matrices The following expression gives the number of paths from lengths to (the longest interesting path, if one considers a Hamiltonian cycle as a path ) between any pair of vertices P= GR-8 Computing the Path xistence Matrix Matrix Multiplication We could naively compute the expression as written O ( * max ( O(matrix mult), O(matrix add) ) P // paths of length T // for computing longer paths // LI: for j :.. k- P includes paths of length j // and T gives paths of length k- for k :.. do // Compute paths of length k and add to P T T x // paths one edge longer P P + T // Include the longer paths GR-9 O ( ^ 3 ) Loop invariants C[..j, * ] C[..j,..k ] t contains Σ[j,p-]xB[p-,k] C = B for j = :.. do // Over rows of for k = :.. do // Over columns of B t 0 // Initial result for p = :.. do t t + [ j, p ] x B[ p, k ] C [ j, k ] t GR-0 O ( ^ ) Loop invariants C[..j, * ] C[..j,..k ] Matrix ddition C = + B jk, jk, jk, for j = :.. do // Rows of & B for k = :.. do // Columns of & B C[ j, k ] [ j, k ] + B[ j, k ] Big O aive Path Calculation O ( * max ( O(matrix mult), O(matrix add) ) From previous slides we have» O ( * 3 ) = O ( ) Using Warshall's algorithm we can reduce this to» O ( 3 ) GR- GR-

3 Warshall's lgorithm Other lgorithms P // Start with the adjacency matrix for j = :.. do // Over the rows of the result for k = :.. do // Over the columns of the result if P [ j, k ] 0 then // Have something to add for p = :.. do // xtend paths from j to p P [ j, p ] P [ j, p ] + ( P [ j, k ] x P [ k, p ] ) existing paths j to p plus paths from j to k joined with paths from k to p Warshall's algorithm as given results in a matrix that gives the number of paths of all lengths between every pair of vertices > It will include paths with self-loops and cycles» Suppose we want a boolean matrix that has True if and only if a path of some length exists» Suppose we want the list of edges for each path in all possible paths» Suppose we want the shortest path The above plus other algorithms are variations on Warshall's algorithm GR-3 GR- Generalizing Warshall's lgorithm Matrix multiplication is a special case of operating on matrices with two different operators» For matrix multiplication we use < +, * >» But other pairs of operators are possible > < or, and > > < or, string_concatenation > > < minimum, + > > nd many others depending upon the functions you want to provide and the result wanted We can write a generalized algorithm to take all possible variations into account GR- Generalizing Warshall's lgorithm P for j = :.. do for k = :.. do suitable condition is satisfied if P [ j, k ] satisfies COD then for p = :.. do P [ j, p ] P [ j, p ] ( P [ j, k ] P [ k, p ] ) path info j to p combine with general plus join with general times path info j to k path info k to p GR-6 Boolean Path Matrix djacency matrix is boolean» OR for D for if P [ j, k ] then P [ j, p ] P [ j, p ] OR ( P [ j, k ] D P [ k, p ] ) Result» P [ j, k ] is true iff a path exists from j to k GR-7 dge ames of ll Paths in the Path Matrix Path matrix contains list of strings representing a path adjacent matrix contains name of edge» dd to list (function L) for create all possible list pairs (function CLP) for if P [ j, k ] not empty list then P [ j, p ] L (P [ j, p ], CLP ( ( P [ j, k ], P [ k, p ] ) ) ) Result» P [ j, k ] is a list of edge names that give a path from j to k GR-8

4 xample Path ames Length of Minimum Path ssume single character names save slide space Suppose» P[ j, p ] contains < BC, DF, HIJK > 3 paths P[ j, k ] contains < B, XY> paths P[ k, p ] contains < L, M, PQR > 3 paths Then P [ j, p ] L (P [ j, p ], CLP ( ( P [ j, k ], P [ k, p ] ) ) Gives the result» P[ j, p ] = < BC, DF, HIJK original paths BL, BM, BPQR, joined paths XYL, XYM, XYPQR > Path matrix contains length of minimum path djacency matrix contains edge lengths, no edge is ( too large a value for the result )» minimum (min) for + for if P [ j, k ] then P [ j, p ] min (P [ j, p ], ( P [ j, k ] + P [ k, p ] ) ) P [ j, k ] = minimumum length of a path from j to k GR-9 GR-0 dge ames for Minimum Path Keep a record containing» Path length» ame of path Substitution» to both keep minimum length and the name of the corresponding path to add path lengths and concatenate the names of the subpaths nd so on» Limited only by your imagination Transitive Closure The transitive closure graph T of a graph G» V (T) = V (G)» u, v : V path ( u, v ) in G < u, v > (T) > If a path exists in G between vertices u and v, then edge < u, v > is in T The transitive closure is computed using OR and D with Warshall's algorithm GR- GR Transitive Closure xample Blocks of 's down the main diagonal give the connected components note 0 on the main diagonalis a single vertex component with no self-loop GR-3 Weighted Graph The last few variations with Warshall s algorithm were done using weighted graphs weighted graph is a triple» G = < V,, W >» W is a set of weights or attributes associated with the edge > n edge is more than a pair of vertices Some meanings of weight» length of edge (distance between cities)» flow capacity (oil pipeline, network connection speed)» ame of edge airline flight number, highway number GR-

5 djacency List The adjacency matrix is very useful but is wasteful of space due to many 'zero' entries Instead of a matrix use sequence representations based on lists and not matrices or vectors (arrays) Pointers to all the vertices are kept in one sequence» Data is the name of vertex plus other attributes Diagram General djacency List ll vertices Sequence of adjacent vertices ach vertex is itself a sequence of adjacent vertices» ach adjacent element points to the vertex in the list of all vertices > For a weighted graph ach adjacent vertex element has a pointer to the edge information dge attributes GR- GR-6 ll vertices 3 xample djacency List List of adjacent vertices B B C C D dge attributes B C 3 D djacency list is itself a graph with vertices of different types GR-7 lgorithms Can use Warshall's algorithm» Modify to use sequence operations instead of array operations for j :.. do... end becomes for j : seq.first.. seq.last do... end on the appropriate sequence object seq» Results are maintained in appropriate sequence structures GR-8 Warshall's lgorithm Using Sequences P for pj = : allvertices.first.. allvertices.last do for pk = : pj.first.. pj.last do // if P [ j, k ] satisfies COD then // no need must exist for pp = : allvertices.first.. allvertices.last do P < j, p > P < j, p > ( P < j, k > P < k, p > ) Use appropriate data structure for the Path matrix sparse matrix? GR-9