Adjacency Lists; Breadth-First Search & Depth-First Search Adjacency Lists For directed graphs: a b c a a b c b c c b Simple Notation { (a,a), (a,b), (a,c), (b,c), (c,b) } 1 a 1 2 b 3 3 c 2 a b c a 1 1 1 b 0 0 1 c 0 1 0 2 3 Space: Adjacency Lists vs. Matricies • Space (n vertices and m edges) – matrix: n2 + n(vertex-name size) • = matrix size + header size • matrix can be bits, but bits are not directly addressable – list: n(header-node size) + m(list-node size) • Sparse: few edges 0 in the extreme case – Matrix fixed size: so no size benefit – List variable size: as little as n(vertex-node size) • Dense: many edges n2 in the extreme case – Matrix fixed size: so no size loss – List variable size: as much as n(header-node size) + n2(list-node size) Operations: Adjacency Lists vs. Matricies • Operations depend on sparse/dense and what’s being done. • Examples (n nodes and m edges) – Is there an arc from x to y? • Matrix: O(1) check value at (x, y) • List: O(n) index to x, traverse list to y or end – Get successor nodes of a node. • Matrix: O(n) scan a row • List: O(n) traverse a linked list – Get predecessor nodes of a node. • Matrix: O(n) scan a column • List: O(n+m) traverse all linked lists, which could be as bad as O(n+n2) = O(n2). Adjacency Lists for Undirected and Weighted Graphs • Undirected Graphs: 1 a 3 5 b 7 c a a b c b a c Make each edge (except loops) go both ways. c a b • Weighted Graphs: a b c (a,1) (b,5) (c,3) (a,5) (c,7) (a,3) (b,7) - add additional field to node - node-weight pairs Breadth-First Search (BFS) Mark start node and enqueue While queue is not empty BFS Algorithm Dequeue N For each neighbor X of N Marked a If X is not marked Mark X and enqueue b c a c b d d N a a a b c c d Queue a b bc c d - Undirected edges: each edge twice Search order: a b c d = O(n+2m) = O(m) if m>>n Depth-First Search (DFS) DFS (start node) Proc DFS (N) DFS Algorithm Mark N For each neighbor X of N If X is not marked DFS (X) a Marked N a a a b b c c d c b a b c c b d d DFS a b c d Search order: a b c d = O(n+2m) = O(m) if m>>n BFS vs. DFS Marked a b c e d f N a a a a b b c c e d f a Queue a b bc bce ce ced ed edf df f - e b c f BFS Order: a b c e d f d BFS vs. DFS Marked N a a a b b c c f c b d b a e a b c f d e a DFS a b e b c d f c f d e DFS Order: a b c f d e (BFS Order: a b c e d f) BFS & DFS with Directed Graphs a b c e j a f k d g l h b i m c e n j f k d g h l i m BFS DFS a,b,c,d,e,f,g,h,i,j,k,l,m,n a,b,e,j,f,k,l,h,c,g,d,i,m,n Same as before, by chance Not same as before n

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