Graph Traversal. 1 Breadth First Search. Correctness. find all nodes reachable from some source node s

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Tyrone Briggs
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1 1 Graph Traversal 1 Breadth First Search visit all nodes and edges in a graph systematically gathering global information find all nodes reachable from some source node s Prove this by giving a minimum length path Applications: Minimize number of train changes; Mr. Euler decides that he hates bridges; low radius spanning tree; subroutines; Lemma 1. If Ô ¼ Ú Ú ¼ is a shortest path to Ú ¼ then Ô Ú is a shortest path to Ú 3 // Find a BFS tree rooted at Correctness parent : NodeArray of Node Ò : NodeArray of ¼ Ò depth Ò parent := depth := ¼ Õ : FIFOQueue of Node while Õ do Ù:= Õ.popFront µ foreach Ù Úµ do if parent Úµ then Õ pushback Úµ parent Úµ:= Ù depth Ú := ÔØ Ù ½ // Ò ½ // selfloop signals root // Note that the Ô Ö ÒØ pointers are reverse to the corresponding edges in The execution of BFS can be subdivided in phases such that in phase Õ, contains only nodes with depth and (after it) ½. At the beginning of phase, Õ contains all nodes of depth. Proof. Induction over. True for ¼. ½ : When phase ½ ends, only nodes of depth are in Õ. Suppose some depth node Ú is missing in Õ. Let Ô Ú ¼ Ú denote a length path to Ú. Ú ¼ was in Õ at the beginning of phase ½. We have explored the edges leaving Ú ¼. Contradiction

2 5 Analysis 1.1 Generalization Theorem 2. BFS runs in time Ç Ñ Òµ // Find a Q-tree rooted at Use (e.g.) adjacency arrays We look at each of Ñ edges once (navigation) parent : NodeArray of Node parent := Õ : Q of Node // selfloop signals root // any queue type Q Ò queue insertions and removals all operations are constant time while Õ do Õ.removeSomeElement µ Ù:= Ù Úµ foreach do if parent Úµ then Õ pushback Úµ parent Úµ:= Ù else do sth with Ù Úµ Possibly restart at any nonvisited node 7 Preview 2 Depth First Search What different queue types give us: FIFO: BFS Idea: Nodes on stack; descend without looking left and right Presentation is close to Ulrik Brandes Dagstuhl 2003 talk Stack: perhaps fastest way (sequential) to get some spanning trees of the CCs of an undirected graph (not quite DFS) Priority Queue: depends on key type: edge weights: Minimum spanning trees paths: Shortest paths

3 9 2.1 DFS Template 2.2 DFS Tree Ë : Stack foreach Î do parent : NodeArray of Node if is not marked then mark ; Ë ÔÙ µ // root of a DFS tree // node marks could be implemented Ô Ö ÒØ using root µ Procedure parent µ:= backtrack Ú Ûµ Procedure Û parent Úµ:= root µ while Ë do Ú:= Ë ØÓÔ µ if Ú Ûµ is unmarked then Ûµ traverse Ú mark e if Û is not marked then // does NOT mark reverse edge mark w; Ë ÔÙ Ûµ else Û:= Ë ÔÓÔ µ; backtrack Ë ØÓÔ µ Ûµ DFS Numbering 2.4 Topological Sorting Procedure root µ Sort nodes such that all edges go from left to right ¼ dfsnum := ½ := toporder : Sequence of Node traverse Ú Ûµ Procedure Ù Ûµ if is a back edge then throw exception the graph contains a cycle if Û is not marked then dfsnum Û := // One cycle consists of Ú Ûµ and // a path of tree edges from Û to Ú. Def: Ù Ú ÆÙÑ Ù ÆÙÑ Ú toporder.pushfront Ûµ tree edge forward edge Classification of edges: back edge Û not marked Û marked, Ú Û Û marked, Û Ú, Û Ë cross edge Û marked, Û Ú, Û Ë

4 13 Correctness 2.5 Strongly Connected Components (SCC) A node Û is finished, when ØÖ Ù Ûµ is called. To prove correcntess it suffices to show that when Û is inserted into ØÓÔÇÖ Ö, there are no edges Ù Ûµ with Ù already in ØÓÔÇÖ Ö. Assume the contrary. What kind of edge could Ù Ûµ be? Lemma 3. two vertices are in the same strongly connected component, if and only if they lie on a cycle. Ù ÓÑÔÓÒ ÒØ Ù output: is a unique representative node. Every graph is a DAG of SCCs tree edge forward edge back edge cross edge would not be finished Ù would not be finished Ù alg. would signalled a cycle would have finisheded before Û Applications: comm. network, equivalence classes, lifeness, 2SAT SCC solution DAG solution general solution? 15 Idea for a single pass algorithm Algorithm Maintain SCCs of of nodes and edges seen so far. Procedure root µ ÓÔ Ò ÔÙ µ; ÓÆÓ ÔÙ µ components are open, if dfs has visited some of its inducing elements. closed backtracked over all if Ú Ûµ is a tree edge then ÓÔ Ò ÔÙ Ûµ; elsif Û ÓÆÓ then ÓÆÓ ÔÙ Ûµ // collapse components on cycle Maintain two additional stacks: open: unfinished: order of discovery components nodes while Û ÓÔ Ò ØÓÔ µ do ÓÔ Ò.ÔÓÔ µ if Û ÓÔ Ò ØÓÔ µ then Ù:= ÓÆÓ ÔÓÔ µ component Ù := Û until Ù Û open: unfinished: order of discovery

5 Biconnected and 2-Edge-Connected Components Generic Algorithm: Components Based on Cycles nodes / edges of an undirected graph are in the same Procedure root µ new component 2-edge connected / biconnected component if they lie on a cycle / simple cycle if Ú Ûµ is a tree edge then create new open component elsif Ú Ûµ closes a cycle then merge components if leaving component then close component Undirected graphs are trees of such components. An application: networks that tolerate link / node failures 19 2-Edge-Connected Components Biconnected Components Procedure root µ ÓÔ Ò ÔÙ µ; ÓÆÓ ÔÙ µ if Ú Ûµ is a tree edge then ÓÔ Ò ÔÙ Ûµ; elsif Û ÓÆÓ then while Û ÓÔ Ò ØÓÔ µ do ÓÔ Ò.ÔÓÔ µ ÓÆÓ ÔÙ Ûµ // collapse components on cycle Ó ÔÙ Ú Ûµµ if Ú Û is a tree edge then ÓÔ Ò ÔÙ Ú Û µ elsif Ú Û is a back edge then // no cross edges! while ÆÙÑ Ûµ Ñ Ò ÆÙÑ ÓÔ Ò ØÓÔ µµ do ÓÔ Ò.ÔÓÔ µ if Û ÓÔ Ò ØÓÔ µ then if Ú Û ÓÔ Ò ØÓÔ µ then Ù:= ÓÆÓ ÔÓÔ µ component Ù := Û until Ù Û := Ó ÔÓÔ µ component := Ú Û until Ú Û

Strongly Connected Components

Strongly Connected Components Let G = (V, E) be a directed graph Write if there is a path from to in G Write if and is an equivalence relation: implies and implies s equivalence classes are called the