Design and Analysis of Algorithms

Transcription

1 S 101, Winter 2018 esign and nalysis of lgorithms Lecture 3: onnected omponents lass URL:

2 nd of Lecture 2: s, Topological Order irected cyclic raph.g., Vertices = tasks; irected dges = dependencies cyclic: no solution if there s a cycle of dependencies Terminology: sources and sinks The vertex with largest post number has no incoming edges (we call this vertex a source ) vertex with smallest post number has no outgoing edges (we call this vertex a sink ) act: n a, every edge leads to a vertex with lower post number rom last time: edge (u,v) with post(v) > post(u) is a back edge. ut a has no cycles no back edges

3 Topological Ordering of a TOP_ORR PROLM: iven a = (V,) with V =n, assign labels 1,...,n to v i V such that every edge is from a lower label to a higher label nductive thinking ny always has a vertex with indegree = 0 ( source ) ive this vertex the next label, delete vertex and its edges, keep doing this to get a topological ordering S-based algorithm: Run S, then perform tasks (label vertices) in decreasing order of post numbers ecause in a, every edge leads to a vertex with lower post number

4 onnectivity and onnected omponents 1,10 11,22 23,24 K L 2,3 1 4,9 5,8 6,7 14,17 K 15, ,21 13,20 L 18,19 3 Undirected ase: onnected components found by S

5 What is onnectivity in a irected raph? s this digraph (= another term for directed graph ) connected?.g., can t reach from L Only one sensible definition efinition: u is connected to w iff there is a path from u to w and a path from w to u With this definition, we partition V into strongly connected components

6 Strongly onnected omponents s this digraph (= another term for directed graph ) connected?.g., can t reach from L Only one sensible definition efinition: u is connected to w iff there is a path from u to w and a path from w to u With this definition, we partition V into strongly connected components xample shown (igure 3.9(a)): 5 S s

7 What s onnectivity etween S s? Meta-graph Shrink each S to a meta-node Put a directed edge from one metanode to another if there is an edge in that direction between any of their respective vertices The Meta-raph

8 igraph as a Over ts S s Meta-graph Shrink each S to a meta-node Put a directed edge from one metanode to another if there is an edge in that direction between any of their respective vertices The meta-graph is a WY? very directed graph is the of its strongly connected components We have a two-tiered connectivity structure of directed graph: oarse = Top-level ; ine = nside a meta-node / S

9 inding S s in irected raphs oal: n algorithm for finding S s in directed graphs. act 1: f the explore subroutine is started at node u, it will terminate when all nodes reachable from u have been visited. onsequence: f we start in a sink S, then we will identify precisely that S : (1) ind a node that is guaranteed to be in a sink S (2) fter identifying a sink S, somehow continue

10 What Node is uaranteed to e in a Sink S? act 2: Run S on. The node with highest post number lies in a source S. Why? act 3: f, are S s and there is an edge from to, then the highest post number in is larger than the highest post number in. act 3 implies act 2

11 What Node is uaranteed to e in a Sink S? act 2: Run S on. The node with highest post number lies in a source S. Why? act 3: f, are S s and there is an edge from to, then the highest post number in is larger than the highest post number in. ase 1: S sees before. The first node visited in will have higher post number than any node in.

12 What Node is uaranteed to e in a Sink S? act 2: Run S on. The node with highest post number lies in a source S. Why? act 3: f, are S s and there is an edge from to, then the highest post number in is larger than the highest post number in. ase 2: S sees before. S will get stuck after seeing all of but before seeing any node of.

13 Topological Ordering of S s act 2: Run S on. The node with highest post number lies in a source S. Why? act 3: f, are S s and there is an edge from to, then the highest post number in is larger than the highest post number in. S s can be topologically sorted by arranging them in decreasing order of their highest post numbers = eneralization of earlier algorithm for topological ordering. (n, each S is a singleton node.)

14 (Recall) : ack To inding S s (1) ind a node that is guaranteed to be in a sink S (2) fter identifying a sink S, somehow continue efinition: reverse graph R = same as, with edges reversed Source S in R = sink S in o S on R, pick node with highest post number This node is in a sink S of

15 (Recall) : ack To inding S s (1) ind a node that is guaranteed to be in a sink S (2) fter identifying a sink S, somehow continue dentify sink S, delete it from graph Of remaining nodes, the one with highest post number will be in a sink S of whatever is left of

16 lgorithm for inding S s in igraph (Recall) : (1) ind a node that is guaranteed to be in a sink S (2) fter identifying a sink S, somehow continue lgorithm Run S on R for v V in decreasing order of post numbers if not visited[v] explore(,v) output nodes seen as a S

17 S xample Meta-graph: 2 source S s, 1 sink S

18 S xample R onstruct reverse graph R

19 S xample 1,2 3,4 5,20 12,13 9,10 6,17 18,19 11,14 8,15 7,16 R Run S on R (lexicographic tie-breaking)

20 S xample 1,2 3,4 5,20 12,13 9,10 6,17 18,19 11,14 8,15 R 7,16 Run S on in order of decreasing POST numbers from R

21 S xample 1,2 3,4 5,20 12,13 9,10 6,17 18,19 11,14 8,15 R 7,16 Run S on in order of decreasing POST numbers from R

22 S xample 1,2 3,4 5,20 12,13 9,10 6,17 18,19 11,14 8,15 R 7,16 Run S on in order of decreasing POST numbers from R

23 S xample 1,2 3,4 5,20 12,13 9,10 6,17 18,19 11,14 8,15 R 7,16 xercise: What happens when you run this algorithm on a? Run S on in order of decreasing POST numbers from R

24 xamples 1

25 xamples 2

26 xamples 3