Lecture 22 Tuesday, April 10
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1 CIS Spring 2018 (instructor Val Tannen) Lecture 22 Tuesday, April 10 GRAPH THEORY Directed Graphs Directed graphs (a.k.a. digraphs) are an important mathematical modeling tool in Computer Science, Statistics, and most disciplines of physical and social science. I will follow closely the terminology in Appendix B.4 of CLRS, the textbook for CIS More about digraphs in CIS 121. Definition 22.1 A directed graph (digraph) G = (V, E) consists of a non-empty set V of vertices (nodes) and a set E V V of (directed) edges (some authors call them arcs, others call them arrows). Therefore, an edge in a digraph is an ordered pair of vertices. Note that E is a binary relation on V. This definition, like CLRS, allows for self-loops (we often drop the self ): (v, v) where v V. (Some authors disallow these in what they call simple digraphs; for computer scientists self-loops are natural because of the matrix representation.) We say that the edge (u, v) goes from u to v and, in our course, we will use the notation u v whenever (u, v) E. CLRS calls an edge u v incident from u and incident to v but I will rather say that u v is an outgoing edge from u and an incoming edge to v. The definition allows for at most one edge from u to v, so no parallel edges. Note however that the edges u v and v u are distinct and we can have both (in undirected graphs these would be indistiguishable). CLRS gives an asymmetric definition for adjacency: if u v then v is adjacent to u, but u is not necessarily adjacent to v, unless also v u. I prefer the following terminology: Definition 22.2 If u v then v is a successor of u and u is a predecessor of v. Moreover, u and v are neighbors, when u v or v u, a symmetric relationship. Since we allow self-loops, a vertex can be its own neighbor in which case it is also its own predecessor and its own succcessor. Isolated vertices: vertices with no neighbors. 1 Introduction to Algorithms, 3rd Edition, by Cormen, Leiserson, Rivest, and Stein, The MIT Press (2009). Online text available through the Penn Libraries website. 1
2 Examples: edgeless graph, directed path graph P n for n 1: ([1..n], {1 n}), directed cycle graph C n for n 1: ([1..n], {1 n 1}). (What is C 1? What is C 2?) There is no standard definition for the directed complete graph. However, the maximum number of edges of a digraph with n vertices is n 2. Definition 22.3 The outdegree of a vertex u is the number of successors of u, same as the number of outgoing edges from u. We will denote it by out(u). Similarly, the indegree of a vertex u is the number of predecessors of u, same as the number of incoming edges to u. We will denote it by in(u). The degree of a node is d(u) = out(u)+in(u). In any digraph, a node of indegree 0 is called a source and a node of outdegree 0 is called a sink. Proposition 22.4 The sum of the outdegrees of all vertices equals the number of edges. The sum of the indegrees of all vertices also equals the number of edges. Proof: Just count the number of edges in three ways: by the definition, by the number of vertices they go out of and by the number of vertices they come into. Representing a digraph as a matrix (Not required for the 3rd midterm or the final exam.) Let G be a digraph with n nodes and let v 1,..., v n be some ordering of its vertices. the n n matrix in which the element at row i and column j is 1 when v i v j and 0 otherwise is called the adjacency matrix of G. Directed walks, paths and cycles Definition 22.5 A directed walk is a non-empty sequence of vertices with edges between them: u 0, u 1,..., u k such that u 0 u 1 u k. We call this a directed walk from u 0 to u k of length k. For every vertex v there is a directed walk of length 0: v. A directed path is a walk with no repeated vertices. Do not confuse the directed walk of length 0, u with the directed walk of length 1 given by the existence of a self-loop: v v. Walks of length 0 are paths but walks of length 1 given by self-loops are not paths. The theorem where there is a walk, there is a path holds in digraphs also, with a similar proof. We often skip the directed qualifier if it s clear from the context that we work in a digraph. Definition 22.6 A directed cycle is a closed walk u 0 u k u 0, with u 0,..., u k all distinct. The length of the cycle is k. 2
3 There are no cycles of length 0. A self-loop gives a cycle of length 1. A cycle of length 2 consists of two vertices and edges between them in opposite directions. Reachability, strong connectivity, reduced graph Definition 22.7 A vertex v is reachable from a vertex u when there is a walk (and therefore a path) from u to v. We write u v. Let s define E = {(u, v) V V u v} Reachability is a binary relation on V, with as the infix notation for reachability and E as its notation as a set of pairs. Proposition 22.8 The reachability relation is reflexive, i.e., u u and transitive, i.e., u v and v w imply u w. Moreover, E E and for any binary relation R that is reflexive and transitive and such that E R we have E R. Proof:(sketch) For reflexivity consider walks of length 0. For transitivity we concatenate walks. To prove E R we show by induction on the length of the walk that if u v then (u, v) R. We use the reflexivity of R in the base case and the transitivity of R together with E R in the induction step. Because of this property, E is called the reflexive-transitive closure of E. Note that this discussion is valid for any binary relation E on a any set V, even if we do not use graph terminology. (Not required for the midterm or the final exam.) (V, E ) is also a digraph. Its adjacency matrix can be obtained by multiplying the adjacency matrix of G with itself until no more new entries appear, adding 1 s on the diagonal if not already there and replacing every element > 0 with 1. Definition 22.9 Two vertices u and v are strongly connected, denoted u v when v is reachable from u and u is reachable from v, i.e., u v and v u. Proposition Strong connectivity is an equivalence relation. Proof: (sketch) For reflexivity we consider walks of length 0 and for transitivity we concatenate walks (both ways). 3
4 Definition Therefore strong connectivity determines a partition of the vertices. The blocks of this partition are called the strongly connected components (SCCs) of the graph. As you will see in CIS 121 and CIS 320, determining the strongly connected components of a digraph is more complicated than determining the connected components of an undirected graph. Note also that there may exist edges between nodes in distinct SCCs, but they cannot go in opposite directions. Definition Given a digraph G = (V, E) its reduced graph has as vertices the SCCs of G and as edges the pairs (S 1, S 2 ) where S 1 and S 2 are distinct SCCs and there exist u 1 S 1 and u 2 S 2 such that u 1 u 2 is an edge in G. Proposition The reduced graph has no directed cycles. Proof: (sketch) Because the reduced graph has edges only between distinct vertices we cannot have cycles of length 1. Now suppose, toward a contradiction, that the reduced graph has a directed cycle of length 2. This cycle has at least two distinct vertices, i.e., two distinct SCCs, S 1 and S 2. Let v 1, v 2 V such that v 1 S 1 and v 2 S 2. We then show (omitted) that the cycle in the reduced graph implies that there exists a cycle C in G such that v 1, v 2 belong to C. It then follows that v 1 v 2, hence S 1 = S 2, which is a contradiction. DAGs Definition A directed acyclic graphs is a digraph without directed cycles (not even of length 1). Most authors use the acronym DAG. DAGs occur quite often, for example the course prerequisites graphs. Lemma Every DAG has at least one source and at least one sink. Proof: Isolated vertices are both sources and sinks. This takes care of the case when the DAG consists of just one vertex. For two or more vertices, recall how we proved that every tree with two or more vertices has a leaf (actually two!). Consider a directed path of maximum length, p, from u to v. Then u is a source. Indeed, suppose, toward a contradiction, that in(u) 1 hence there exists an edge w u. We have two cases: 4
5 Case 1: w p. Then w u v is a directed path that is strictly longer than p, contradiction. Case 2: w p. Then w u w is a directed cycle, contradiction. Similarly we show that v is a sink. Definition A topological sort of a digraph is a sequence σ in which every vertex appears exactly once (i.e., a permutation of its vertices) such that for any edge u v in the graph, the vertex u appears in σ before ( not necessarily immediately before) the vertex v. Lemma If a digraph has a topological sort σ then it has no self-loops and if u v for u v, i.e., the vertex v is reachable from another vertex u then the vertex u appears in σ before ( not necessarily immediately before) the vertex v. Proof: Omitted, but one hint is to use induction on the length of the path from u to v. Corollary If a digraph has a directed cycle then it cannot have a topological sort. Proposition Every DAG has a topological sort. Proof: By induction on n, the number of vertices. Base case: n = 1. We cannot have any edges because they would form a cycle of length 1. With just one vertex v and zero edges the topological sort is the sequence v. Induction step: Let k 1 arbitrary. Assume (IH) that any DAG with k vertices has a topological sort. Now take any DAG G with k + 1 vertices. By Lemma G has a source u (and a sink v, but we don t use it in this proof). Delete u from G. This deletes also any outgoing edge from u. There are no incoming edges to u because u is a source. The resulting graph G must also be a DAG because no directed cycle can pass through a source vertex. G is a DAG with k vertices so we can apply the IH and obtain a topological sort, call it σ. Now we claim that the concatenated sequence u σ is a topological sort of G. Clearly it is a permutation of the vertices of G. For any edge x y of G, if x y G then x appears before y in σ, hence in u σ also. If x y is not in G then x u and u occurs before y in u σ. Note that topological sorts are not unique. Consider G = ({1, 2, 3}, {1 3, 2 3}). It has two topological sorts: and
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