bbm 202 - algorithmsbbm202/spring2017/... · mar. 31, 2016 bbm 202 - algorithms directed graphs...

Mar. 31, 2016

BBM 202 - ALGORITHMS

DIRECTED GRAPHS

DEPT. OF COMPUTER ENGINEERING

Acknowledgement:ThecourseslidesareadaptedfromtheslidespreparedbyR.SedgewickandK.WayneofPrincetonUniversity.

TODAY ‣ Directed Graphs ‣ Digraph API ‣ Digraph search ‣ Topological sort ‣ Strong components

Digraph. Set of vertices connected pairwise by directed edges.

3

Directed graphs

directedcycle

directed pathfrom 0 to 2

vertex of outdegree 4

and indegree 2

4

Road network

Vertex = intersection; edge = one-way street.Address Holland Tunnel

New York, NY 10013

©2008 Google - Map data ©2008 Sanborn, NAVTEQ™ - Terms of Use

To see all the details that are visible on the screen,use the"Print" link next to the map.

Vertex = political blog; edge = link.

5

Political blogosphere graph

The Political Blogosphere and the 2004 U.S. Election: Divided They Blog, Adamic and Glance, 2005Figure 1: Community structure of political blogs (expanded set), shown using utilizing a GEMlayout [11] in the GUESS[3] visualization and analysis tool. The colors reflect political orientation,red for conservative, and blue for liberal. Orange links go from liberal to conservative, and purpleones from conservative to liberal. The size of each blog reflects the number of other blogs that linkto it.

longer existed, or had moved to a different location. When looking at the front page of a blog we didnot make a distinction between blog references made in blogrolls (blogroll links) from those madein posts (post citations). This had the disadvantage of not differentiating between blogs that wereactively mentioned in a post on that day, from blogroll links that remain static over many weeks [10].Since posts usually contain sparse references to other blogs, and blogrolls usually contain dozens ofblogs, we assumed that the network obtained by crawling the front page of each blog would stronglyreflect blogroll links. 479 blogs had blogrolls through blogrolling.com, while many others simplymaintained a list of links to their favorite blogs. We did not include blogrolls placed on a secondarypage.

We constructed a citation network by identifying whether a URL present on the page of one blogreferences another political blog. We called a link found anywhere on a blog’s page, a “page link” todistinguish it from a “post citation”, a link to another blog that occurs strictly within a post. Figure 1shows the unmistakable division between the liberal and conservative political (blogo)spheres. Infact, 91% of the links originating within either the conservative or liberal communities stay withinthat community. An effect that may not be as apparent from the visualization is that even thoughwe started with a balanced set of blogs, conservative blogs show a greater tendency to link. 84%of conservative blogs link to at least one other blog, and 82% receive a link. In contrast, 74% ofliberal blogs link to another blog, while only 67% are linked to by another blog. So overall, we see aslightly higher tendency for conservative blogs to link. Liberal blogs linked to 13.6 blogs on average,while conservative blogs linked to an average of 15.1, and this difference is almost entirely due tothe higher proportion of liberal blogs with no links at all.

Although liberal blogs may not link as generously on average, the most popular liberal blogs,Daily Kos and Eschaton (atrios.blogspot.com), had 338 and 264 links from our single-day snapshot

4

Vertex = bank; edge = overnight loan.

6

Overnight interbank loan graph

The Topology of the Federal Funds Market, Bech and Atalay, 2008

GSCC

GWCC

Tendril

DC

GOUTGIN

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S7

Vertex = variable; edge = logical implication.

7

Implication graph

~x0

~x3

~x1~x5

x6

x5

~x6

~x4

~x2

x2

x4

x1

x3

x0

if x5 is true,

then x0 is true

Vertex = logical gate; edge = wire.

8

Combinational circuit

Vertex = synset; edge = hypernym relationship.

9

WordNet graph

http://wordnet.princeton.edu

event

happening occurrence occurrent natural_event

change alteration modification

damage harm ..impairment transition

leap jump saltation jump leap

act human_action human_activity

group_action

forfeit forfeiture sacrifice action

change

resistance opposition transgression

demotion variation

motion movement move

locomotion travel

run running

dash sprint

descent

jump parachuting

increase

miracle

miracle

10

Digraph applications

digraph vertex directed edge

transportation street intersection one-way street

web web page hyperlink

food web species predator-prey relationship

WordNet synset hypernym

scheduling task precedence constraint

financial bank transaction

cell phone person placed call

infectious disease person infection

game board position legal move

citation journal article citation

object graph object pointer

inheritance hierarchy class inherits from

control flow code block jump

11

Some digraph problems

Path. Is there a directed path from s to t ?Shortest path. What is the shortest directed path from s to t ?Topological sort. Can you draw the digraph so that all edges point upwards?Strong connectivity. Is there a directed path between all pairs of vertices?Transitive closure. For which vertices v and w is there a path from v to w ?PageRank. What is the importance of a web page?

s

t

DIRECTED GRAPHS

‣ Digraph API ‣ Digraph search ‣ Topological sort ‣ Strong components

13

Digraph API

public class Digraph

Digraph(int V) create an empty digraph with V vertices

Digraph(In in) create a digraph from input stream

void addEdge(int v, int w) add a directed edge v→w

Iterable<Integer> adj(int v) vertices pointing from v

int V() number of vertices

int E() number of edges

Digraph reverse() reverse of this digraph

String toString() string representation

In in = new In(args[0]); Digraph G = new Digraph(in);

for (int v = 0; v < G.V(); v++) for (int w : G.adj(v)) StdOut.println(v + "->" + w);

read digraph from

input stream

print out each

edge (once)

14

Digraph API

In in = new In(args[0]); Digraph G = new Digraph(in);

for (int v = 0; v < G.V(); v++) for (int w : G.adj(v)) StdOut.println(v + "->" + w);

% java Digraph tinyDG.txt 0->5 0->1 2->0 2->3 3->5 3->2 4->3 4->2 5->4 ⋮ 11->4 11->12 12-9

read digraph from

input stream

print out each

edge (once)

adj[]0

1

2

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12

5 2

3 2

0 3

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9 4 8

1322 4 2 2 3 3 2 6 0 0 1 2 011 1212 9 9 10 9 11 7 910 1211 4 4 3 3 5 6 8 8 6 5 4 0 5 6 4 6 9 7 6

tinyDG.txtV

E

⋮

Maintain vertex-indexed array of lists.

adj[]0

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1322 4 2 2 3 3 2 6 0 0 1 2 011 1212 9 9 10 9 11 7 910 1211 4 4 3 3 5 6 8 8 6 5 4 0 5 6 4 6 9 7 6

tinyDG.txtV

E

15

Adjacency-lists digraph representation

adj[]0

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5 2

3 2

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5 1

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0

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6 9

4 12

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12

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9 4 8

1322 4 2 2 3 3 2 6 0 0 1 2 011 1212 9 9 10 9 11 7 910 1211 4 4 3 3 5 6 8 8 6 5 4 0 5 6 4 6 9 7 6

tinyDG.txtV

E

16

Adjacency-lists graph representation: Java implementation

public class Graph { private final int V; private final Bag<Integer>[] adj;

public Graph(int V) { this.V = V; adj = (Bag<Integer>[]) new Bag[V]; for (int v = 0; v < V; v++) adj[v] = new Bag<Integer>(); }

public void addEdge(int v, int w) { adj[v].add(w); adj[w].add(v); }

public Iterable<Integer> adj(int v) { return adj[v]; } }

adjacency lists

create empty graphwith V vertices

iterator for vertices

adjacent to v

add edge v–w

17

Adjacency-lists digraph representation: Java implementation

public class Digraph { private final int V; private final Bag<Integer>[] adj;

public Digraph(int V) { this.V = V; adj = (Bag<Integer>[]) new Bag[V]; for (int v = 0; v < V; v++) adj[v] = new Bag<Integer>(); }

public void addEdge(int v, int w) { adj[v].add(w);

}

public Iterable<Integer> adj(int v) { return adj[v]; } }

adjacency lists

create empty digraphwith V vertices

add edge v→w

iterator for vertices

pointing from v

In practice. Use adjacency-lists representation.• Algorithms based on iterating over vertices pointing from v.

• Real-world digraphs tend to be sparse.

18

Digraph representations

representation spaceinsert edge from v to w

edge from

v to w?

iterate over vertices

pointing from v?

list of edges E 1 E E

adjacency matrix V 2 1 † 1 V

adjacency lists E + V 1 outdegree(v) outdegree(v)

huge number of vertices,small average vertex degree

† disallows parallel edges

DIRECTED GRAPHS


20

Reachability

Problem. Find all vertices reachable from s along a directed path.

s

Same method as for undirected graphs.• Every undirected graph is a digraph (with edges in both directions).

• DFS is a digraph algorithm.

21

Depth-first search in digraphs

Mark v as visited.

Recursively visit all unmarked

vertices w pointing from v.

DFS (to visit a vertex v)

adj[]0

1

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5 2

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6 9

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9 4 8

1322 4 2 2 3 3 2 6 0 0 1 2 011 1212 9 9 10 9 11 7 910 1211 4 4 3 3 5 6 8 8 6 5 4 0 5 6 4 6 9 7 6

tinyDG.txtV

E

To visit a vertex v :• Mark vertex v as visited.

• Recursively visit all unmarked vertices pointing from v.

1

4

9

2

5

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1211

10

Depth-first search

22

a directed graph

4→2

2→3

3→2

6→0

0→1

2→0

11→12

12→9

9→10

9→11

8→9

10→12

11→4

4→3

3→5

6→8

8→6

5→4

0→5

6→4

6→9

7→6

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a directed graph

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edgeTo[]



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Depth-first search

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visit 0: check 5 and check 1

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8 76 –

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Depth-first search

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visit 5: check 4

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Depth-first search

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Depth-first search

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Depth-first search

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done 2

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done 3

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done 4

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done 5

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visit 1

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done 1

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Depth-first search

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done 0

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marked[]



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Depth-first search

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done

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v edgeTo[]



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marked[]

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Depth-first search

41

reachable from 0

reachable

from vertex 0

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v edgeTo[]

4

Recall code for undirected graphs.

public class DepthFirstSearch { private boolean[] marked;

public DepthFirstSearch(Graph G, int s) { marked = new boolean[G.V()]; dfs(G, s); }

private void dfs(Graph G, int v) { marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(G, w); }

public boolean visited(int v) { return marked[v]; } }

42

Depth-first search (in undirected graphs)

true if path to s

constructor marks

vertices connected to s

recursive DFS does the work

client can ask whether any

vertex is connected to s

Code for directed graphs identical to undirected one. [substitute Digraph for Graph]

public class DirectedDFS { private boolean[] marked;

public DirectedDFS(Digraph G, int s) { marked = new boolean[G.V()]; dfs(G, s); }

private void dfs(Digraph G, int v) { marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(G, w); }

public boolean visited(int v) { return marked[v]; } }

43

Depth-first search (in directed graphs)

true if path from s

constructor marks vertices

reachable from s

recursive DFS does the work

client can ask whether any

vertex is reachable from s

44

Reachability application: program control-flow analysis

Every program is a digraph.• Vertex = basic block of instructions (straight-line program).

• Edge = jump.

Dead-code elimination. Find (and remove) unreachable code. Infinite-loop detection. Determine whether exit is unreachable.

Every data structure is a digraph.• Vertex = object.

• Edge = reference.

Roots. Objects known to be directly accessible by program (e.g., stack).Reachable objects. Objects indirectly accessible by program(starting at a root and following a chain of pointers).

45

Reachability application: mark-sweep garbage collector

roo

ts

46

Reachability application: mark-sweep garbage collector

Mark-sweep algorithm. [McCarthy, 1960]• Mark: mark all reachable objects.

• Sweep: if object is unmarked, it is garbage (so add to free list).

Memory cost. Uses 1 extra mark bit per object (plus DFS stack).

roo

ts

DFS enables direct solution of simple digraph problems.• Reachability.

• Path finding.

• Topological sort.

• Directed cycle detection.

Basis for solving difficult digraph problems.• 2-satisfiability.

• Directed Euler path.

• Strongly-connected components.

47

Depth-first search in digraphs summary

✓

SIAM J. COMPUT.Vol. 1, No. 2, June 1972

DEPTH-FIRST SEARCH AND LINEAR GRAPH ALGORITHMS*

ROBERT TARJAN"

Abstract. The value of depth-first search or "bacltracking" as a technique for solving problems isillustrated by two examples. An improved version of an algorithm for finding the strongly connectedcomponents of a directed graph and ar algorithm for finding the biconnected components of an un-direct graph are presented. The space and time requirements of both algorithms are bounded byk1V + k2E d- k for some constants kl, k2, and ka, where Vis the number of vertices and E is the numberof edges of the graph being examined.

Key words. Algorithm, backtracking, biconnectivity, connectivity, depth-first, graph, search,spanning tree, strong-connectivity.

1. Introduction. Consider a graph G, consisting of a set of vertices U and aset of edges g. The graph may either be directed (the edges are ordered pairs (v, w)of vertices; v is the tail and w is the head of the edge) or undirected (the edges areunordered pairs of vertices, also represented as (v, w)). Graphs form a suitableabstraction for problems in many areas; chemistry, electrical engineering, andsociology, for example. Thus it is important to have the most economical algo-rithms for answering graph-theoretical questions.

In studying graph algorithms we cannot avoid at least a few definitions.These definitions are more-or-less standard in the literature. (See Harary [3],for instance.) If G (, g) is a graph, a path p’v w in G is a sequence of verticesand edges leading from v to w. A path is simple if all its vertices are distinct. A pathp’v v is called a closed path. A closed path p’v v is a cycle if all its edges aredistinct and the only vertex to occur twice in p is v, which occurs exactly twice.Two cycles which are cyclic permutations of each other are considered to be thesame cycle. The undirected version of a directed graph is the graph formed byconverting each edge of the directed graph into an undirected edge and removingduplicate edges. An undirected graph is connected if there is a path between everypair of vertices.

A (directed rooted) tree T is a directed graph whose undirected version isconnected, having one vertex which is the head of no edges (called the root),and such that all vertices except the root are the head of exactly one edge. Therelation "(v, w) is an edge of T" is denoted by v- w. The relation "There is apath from v to w in T" is denoted by v w. If v - w, v is the father ofw and w is ason of v. If v w, v is an ancestor ofw and w is a descendant of v. Every vertex is anancestor and a descendant of itself. If v is a vertex in a tree T, T is the subtree of Thaving as vertices all the descendants of v in T. If G is a directed graph, a tree Tis a spanning tree of G if T is a subgraph of G and T contains all the vertices of G.

If R and S are binary relations, R* is the transitive closure of R, R-1 is theinverse of R, and

RS {(u, w)lZlv((u, v) R & (v, w) e S)}.

* Received by the editors August 30, 1971, and in revised form March 9, 1972.

" Department of Computer Science, Cornell University, Ithaca, New York 14850. This researchwas supported by the Hertz Foundation and the National Science Foundation under Grant GJ-992.

146

Same method as for undirected graphs.• Every undirected graph is a digraph (with edges in both directions).

• BFS is a digraph algorithm.

Proposition. BFS computes shortest paths (fewest number of edges) from s to all other vertices n a digraph in time proportional to E+V.

48

Breadth-first search in digraphs

Is w reachable from v in this digraph?

v

w

s

Put s onto a FIFO queue, and mark s as visited.Repeat until the queue is empty:

- remove the least recently added vertex v

- for each unmarked vertex pointing from v: add to queue and mark as visited.

BFS (from source vertex s)

Multiple-source shortest paths. Given a digraph and a set of source vertices, find shortest path from any vertex in the set to each other vertex.Ex. S={ 1, 7, 10 }. • Shortest path to 4 is 7→6→4.

• Shortest path to 5 is 7→6→0→5.

• Shortest path to 12 is 10→12.

Q. How to implement multi-source constructor?A. Use BFS, but initialize by enqueuing all source vertices.

49

Multiple-source shortest paths

adj[]0

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1322 4 2 2 3 3 2 6 0 0 1 2 011 1212 9 9 10 9 11 7 910 1211 4 4 3 3 5 6 8 8 6 5 4 0 5 6 4 6 9 7 6

tinyDG.txtV

E

50

Breadth-first search in digraphs application: web crawler

Goal. Crawl web, starting from some root web page, say www.princeton.edu.Solution. BFS with implicit graph.BFS.• Choose root web page as source s.

• Maintain a Queue of websites to explore.

• Maintain a SET of discovered websites.

• Dequeue the next website and enqueue websites to which it links (provided you haven't done so before).

Q. Why not use DFS?

18

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How many strong components are there in this digraph?

51

Bare-bones web crawler: Java implementation Queue<String> queue = new Queue<String>(); SET<String> discovered = new SET<String>();

String root = "http://www.princeton.edu"; queue.enqueue(root); discovered.add(root);

while (!queue.isEmpty()) { String v = queue.dequeue(); StdOut.println(v); In in = new In(v); String input = in.readAll();

String regexp = "http://(\\w+\\.)*(\\w+)"; Pattern pattern = Pattern.compile(regexp); Matcher matcher = pattern.matcher(input); while (matcher.find()) { String w = matcher.group(); if (!discovered.contains(w)) { discovered.add(w); queue.enqueue(w); } } }

read in raw html from next

website in queue

use regular expression to find all URLsin website of form http://xxx.yyy.zzz

[crude pattern misses relative URLs]

if undiscovered, mark it as discovered and put on queue

start crawling from root website

queue of websites to crawlset of discovered websites

DIRECTED GRAPHS


Goal. Given a set of tasks to be completed with precedence constraints, in which order should we schedule the tasks?

Digraph model. vertex = task; edge = precedence constraint.

53

Precedence scheduling

tasks precedence constraint graph

0

1

4

52

6

3

feasible schedule

0. Algorithms

1. Complexity Theory

2. Artificial Intelligence

3. Intro to CS

4. Cryptography

5. Scientific Computing

6. Advanced Programming

54

Topological sort

DAG. Directed acyclic graph.

Topological sort. Redraw DAG so all edges point upwards.

Solution. DFS. What else?topological order

directed edges DAG

0→5 0→2 0→1 3→6 3→5 3→4 5→4 6→4 6→0 3→2 1→4

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• Run depth-first search.

• Return vertices in reverse postorder.

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Topological sort algorithm

55

a directed acyclic graph

0→5 0→2 0→1 3→6 3→5 3→4 5→4 6→4 6→0 3→2 1→4

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postorder




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4 done

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visit 1

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5 done

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visit 3: check 2, check 4, check 5, and check 6

postorder




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postorder




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6 done

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visit 3

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81

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3 done

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check 4

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check 5

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4 1 2 5 0 6 3

postorder

done

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topological order

86

Depth-first search order

public class DepthFirstOrder { private boolean[] marked; private Stack<Integer> reversePost;

public DepthFirstOrder(Digraph G) { reversePost = new Stack<Integer>(); marked = new boolean[G.V()]; for (int v = 0; v < G.V(); v++) if (!marked[v]) dfs(G, v); }

private void dfs(Digraph G, int v) { marked[v] = true; for (int w : G.adj(v)) if (!marked[w]) dfs(G, w); reversePost.push(v); } public Iterable<Integer> reversePost() { return reversePost; } }

returns all vertices in

“reverse DFS postorder”

Proposition. Reverse DFS postorder of a DAG is a topological order.Pf. Consider any edge v→w. When dfs(v) is called:

• Case 1: dfs(w) has already been called and returned. Thus, w was done before v.

• Case 2: dfs(w) has not yet been called. dfs(w) will get called directly or indirectly by dfs(v) and will finish before dfs(v). Thus, w will be done before v.

• Case 3: dfs(w) has already been called, but has not yet returned. Can’t happen in a DAG: function call stack containspath from w to v, so v→w would complete a cycle.

dfs(0) dfs(1) dfs(4) 4 done 1 done dfs(2) 2 done dfs(5) check 2 5 done 0 done check 1 check 2 dfs(3) check 2 check 4 check 5 dfs(6) 6 done 3 done check 4 check 5 check 6 done

87

Topological sort in a DAG: correctness proof

all vertices pointing from 3 are done before 3 is done,

so they appear after 3 in topological order

Ex:

case 1

case 2

Proposition. A digraph has a topological order iff no directed cycle.Pf.• If directed cycle, topological order impossible.

• If no directed cycle, DFS-based algorithm finds a topological order.

Goal. Given a digraph, find a directed cycle.Solution. DFS. What else? See textbook.

88

Directed cycle detection

Finding a directed cycle in a digraph

dfs(0) dfs(5) dfs(4) dfs(3) check 5

marked[] edgeTo[] onStack[]0 1 2 3 4 5 ... 0 1 2 3 4 5 ... 0 1 2 3 4 5 ...

1 0 0 0 0 0 - - - - - 0 1 0 0 0 0 0 1 0 0 0 0 1 - - - - 5 0 1 0 0 0 0 1 1 0 0 0 1 1 - - - 4 5 0 1 0 0 0 1 1 1 0 0 1 1 1 - - - 4 5 0 1 0 0 1 1 1

a digraph with a directed cycle

Scheduling. Given a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks?

Remark. A directed cycle implies scheduling problem is infeasible.

89

Directed cycle detection application: precedence scheduling

http://xkcd.com/754

90

Directed cycle detection application: cyclic inheritance

The Java compiler does cycle detection.

public class A extends B { ... }

public class B extends C { ... }

public class C extends A { ... }

% javac A.java A.java:1: cyclic inheritance involving A public class A extends B { } ^ 1 error

Microsoft Excel does cycle detection (and has a circular reference toolbar!)

91

Directed cycle detection application: spreadsheet recalculation

92

Directed cycle detection applications

• Causalities.

• Email loops.

• Compilation units.

• Class inheritance.

• Course prerequisites.

• Deadlocking detection.

• Precedence scheduling.

• Temporal dependencies.

• Pipeline of computing jobs.

• Check for symbolic link loop.

• Evaluate formula in spreadsheet.

DIRECTED GRAPHS


Def. Vertices v and w are strongly connected if there is a directed path from v to w and a directed path from w to v.

Key property. Strong connectivity is an equivalence relation:• v is strongly connected to v.

• If v is strongly connected to w, then w is strongly connected to v.

• If v is strongly connected to w and w to x, then v is strongly connected to x.

Def. A strong component is a maximal subset of strongly-connected vertices.

Strongly-connected components

94

A digraph and its strong components

Examples of strongly-connected digraphs

95

public int connected(int v, int w) { return cc[v] == cc[w]; }

Connected components vs. strongly-connected components

96

0 1 2 3 4 5 6 7 8 9 10 11 12 cc[] 0 0 0 0 0 0 1 1 1 2 2 2 2

• v and w are connected if there isa path between v and w

•v and w are strongly connected if there is a directed

path from v to w and a directed path from w to v

0 1 2 3 4 5 6 7 8 9 10 11 12 scc[] 1 0 1 1 1 1 3 4 3 2 2 2 2

constant-time client connectivity query constant-time client strong-connectivity query

•3 connected components •5 strongly-connected components

connected component id (easy to compute with DFS) strongly-connected component id (how to compute?)

A digraph and its strong componentsA graph and its connected components

public int stronglyConnected(int v, int w) { return scc[v] == scc[w]; }


97

Strong component application: ecological food webs

Food web graph. Vertex = species; edge = from producer to consumer.

Strong component. Subset of species with common energy flow.

http://www.twingroves.district96.k12.il.us/Wetlands/Salamander/SalGraphics/salfoodweb.gif

98

Strong component application: software modules

Software module dependency graph.• Vertex = software module.

• Edge: from module to dependency.

Strong component. Subset of mutually interacting modules. Approach 1. Package strong components together.Approach 2. Use to improve design!

Internet ExplorerFirefox

Strong components algorithms: brief history

1960s: Core OR problem.• Widely studied; some practical algorithms.

• Complexity not understood.

1972: linear-time DFS algorithm (Tarjan).• Classic algorithm.

• Level of difficulty: Algs4++.

• Demonstrated broad applicability and importance of DFS.

1980s: easy two-pass linear-time algorithm (Kosaraju-Sharir).• Forgot notes for lecture; developed algorithm in order to teach it!

• Later found in Russian scientific literature (1972).

1990s: more easy linear-time algorithms.• Gabow: fixed old OR algorithm.

• Cheriyan-Mehlhorn: needed one-pass algorithm for LEDA.

99


Reverse graph. Strong components in G are same as in GR.Kernel DAG. Contract each strong component into a single vertex.Idea.• Compute topological order (reverse postorder) in kernel DAG.

• Run DFS, considering vertices in reverse topological order.

100

Kosaraju's algorithm: intuition

digraph G and its strong components kernel DAG of G (in reverse topological order)

how to compute?

Kernel DAG in reverse topological order

first vertex is a sink (has no edges pointing from it)

KOSARAJU'S ALGORITHM

‣ DFS in reverse graph ‣ DFS in original graph

Phase 1. Compute reverse postorder in GR.

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102

digraph G

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Kosaraju-Sharir

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reverse digraph GR

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7

8

9

10

11

12

0marked[v]v

6 7 8

T

F

F

F

F

F

T

T

T

F

F

F

F


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

113


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

F

F

T

T

T

F

F

F

F

0marked[v]v

6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

114


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

F

F

F

F

0

6 7 8

marked[v]v


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

115

visit 11: check 9

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

F

F

T

F

0marked[v]v

6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

116


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

F

T

F

0marked[v]v

6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

117


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

F

T

T

0marked[v]v

6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

118


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

F

T

T

0marked[v]v

6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

119

visit 10: check 9

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0marked[v]v

6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

1211

10

1

Kosaraju-Sharir

120

10 done

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0marked[v]v

10 6 7 8

12


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

9

2

5

3

11

1

Kosaraju-Sharir

121

12 done

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

12

marked[v]v

12 10 6 7 8

9


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

2

5

3

11

1

Kosaraju-Sharir

122


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

9

marked[v]v

12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

2

5

3

11

1

Kosaraju-Sharir

123


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

9

marked[v]v

12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

2

5

3

11

1

Kosaraju-Sharir

124

9 done

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

9

marked[v]v

9 12 10 6 7 8

11


1

4

9

2

5

3

0

1211

10

1

8 76

1

4

2

5

3

1

Kosaraju-Sharir

125

11 done

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

11

marked[v]v

11 9 12 10 6 7 8

4

Phase 1. Compute reverse postorder in GR.11 9 12 10 6 7 8

1

4

9

2

5

3

0

1211

10

1

8 76

1 2

5

3

1

Kosaraju-Sharir

126


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

4

marked[v]v


1

4

9

2

5

3

0

1211

10

1

8 76

1 2

5

3

1

Kosaraju-Sharir

127


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

F

T

T

T

T

T

T

T

0

4

marked[v]v

11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1 2

5

3

1

Kosaraju-Sharir

128


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

F

T

T

T

T

T

T

T

T

T

0

4

marked[v]v

11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1 2

5

3

1

Kosaraju-Sharir

129


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

T

T

T

T

T

T

T

T

T

T

0

4

marked[v]v

11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1 2

5

3

1

Kosaraju-Sharir

130


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

T

T

T

T

T

T

T

T

T

T

0

4

marked[v]v

11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1 2

5

3

1

Kosaraju-Sharir

131

3 done

0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

T

T

T

T

T

T

T

T

T

T

0

4

marked[v]v

3 11 9 12 10 6 7 8

5


1

4

9

2

5

3

0

1211

10

1

8 76

1 21

Kosaraju-Sharir

132


0

1

2

3

4

5

6

7

8

9

10

11

12

T

F

T

T

T

T

T

T

T

T

T

T

T

0

4

5

marked[v]v

3 11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

1 21

Kosaraju-Sharir

133

5 done

0

1

2

3

4

5

6

7

8

9

10

11

12

0

4

5

marked[v]v

5 3 11 9 12 10 6 7 8

T

F

T

T

T

T

T

T

T

T

T

T

T

4


1

4

9

2

5

3

0

1211

10

1

8 76

1 21

Kosaraju-Sharir

134

4 done

0

1

2

3

4

5

6

7

8

9

10

11

12

0

4

marked[v]v

4 5 3 11 9 12 10 6 7 8

T

F

T

T

T

T

T

T

T

T

T

T

T

2


1

4

9

2

5

3

0

1211

10

1

8 76

11

Kosaraju-Sharir

135


0

1

2

3

4

5

6

7

8

9

10

11

12

0

2

marked[v]v

4 5 3 11 9 12 10 6 7 8

T

F

T

T

T

T

T

T

T

T

T

T

T


1

4

9

2

5

3

0

1211

10

1

8 76

11

Kosaraju-Sharir

136

2 done

0

1

2

3

4

5

6

7

8

9

10

11

12

0

2

marked[v]v

2 4 5 3 11 9 12 10 6 7 8

T

F

T

T

T

T

T

T

T

T

T

T

T

0


1

4

9

2

5

3

0

1211

10

1

8 76

11

Kosaraju-Sharir

137

0 done

0

1

2

3

4

5

6

7

8

9

10

11

12

0marked[v]v

0 2 4 5 3 11 9 12 10 6 7 8

T

F

T

T

T

T

T

T

T

T

T

T

T


1

4

9

2

5

3

0

1211

10

1

8 76

11

Kosaraju-Sharir

138

visit 1: check 0

0

1

2

3

4

5

6

7

8

9

10

11

12

T

T

T

T

T

T

T

T

T

T

T

T

T

marked[v]v

0 2 4 5 3 11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

Kosaraju-Sharir

139

1 done

0

1

2

3

4

5

6

7

8

9

10

11

12

T

T

T

T

T

T

T

T

T

T

T

T

T

marked[v]v

1 0 2 4 5 3 11 9 12 10 6 7 8

1


1

4

9

2

5

3

0

1211

10

1

8 76

Kosaraju-Sharir

140

check 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

9

10

11

12

T

T

T

T

T

T

T

T

T

T

T

T

T

marked[v]v

1 0 2 4 5 3 11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

1

8 76

Kosaraju-Sharir

141

1 0 2 4 5 3 11 9 12 10 6 7 8

reverse digraph GR

Simple (but mysterious) algorithm for computing strong components.• Run DFS on GR to compute reverse postorder.

• Run DFS on G, considering vertices in order given by first DFS.

142

Kosaraju's algorithm

...

check unmarked vertices in the order 0 1 2 3 4 5 6 7 8 9 10 11 12

dfs(0) dfs(6) dfs(8) check 6 8 done dfs(7) 7 done 6 done dfs(2) dfs(4) dfs(11) dfs(9) dfs(12) check 11 dfs(10) check 9 10 done 12 done check 7 check 6 9 done 11 done check 6 dfs(5) dfs(3) check 4 check 2 3 done check 0 5 done 4 done check 3 2 done0 donedfs(1) check 01 donecheck 2check 3check 4check 5check 6check 7check 8check 9check 10check 11check 12

DFS in reverse digraph GR

reverse postorder for use in second dfs()1 0 2 4 5 3 11 9 12 10 6 7 8

KOSARAJU'S ALGORITHM

‣ DFS in reverse graph ‣ DFS in original graph

Phase 2. Run DFS in G, visiting unmarked vertices in reverse postorder of GR.

1

4

9

2

5

3

0

1211

10

Kosaraju-Sharir

144

1

4

9

2

5

3

0

1211

10

8 76 8 76 0

1

2

3

4

5

6

7

8

9

10

11

12

scc[v]v

–

–

–

–

–

–

–

–

–

–

–

–

–

original digraph G

1 0 2 4 5 3 11 9 12 10 6 7 8


1

4

9

2

5

3

0

1211

10

Kosaraju-Sharir

145

1

4

9

2

5

3

0

1211

10

8 76 8 76

visit 1

0

1

2

3

4

5

6

7

8

9

10

11

12

–

0

–

–

–

–

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


11

4

9

2

5

3

0

1211

10

Kosaraju-Sharir

146

4

9

2

5

3

0

1211

10

8 76 8 76

1 done

0

1

2

3

4

5

6

7

8

9

10

11

12

–

0

–

–

–

–

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

Kosaraju-Sharir

147

1

4

9

2

5

3

0

1211

10

8 76 8 76

strong component: 1

0

1

2

3

4

5

6

7

8

9

10

11

12

–

0

–

–

–

–

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

1

0

–

–

–

–

–

–

–

–

–

–

–


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

148


4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

1

0

–

–

–

1

–

–

–

–

–

–

–


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

149

visit 5: check 4

4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

1

0

–

–

1

1

–

–

–

–

–

–

–


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

150


4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

1

0

–

1

1

1

–

–

–

–

–

–

–


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

151


4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

152


4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

–

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

1

0

1

1

1

1

–

–

–

–

–

–

–


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

153


4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

154


4

9

2

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


22

4

9

5

3

0

1211

10

Kosaraju-Sharir

155

2 done

4

9

5

3

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

3

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


3

4

9

2

5

0

1211

10

Kosaraju-Sharir

156

3 done

4

9

5

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

3

4

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

157


4

9

5

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


9

2

5

3

0

1211

10

Kosaraju-Sharir

158

4 done

9

5

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

44

5

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


5

4

9

2

3

0

1211

10

Kosaraju-Sharir

159

5 done

9

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

5

0

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


4

9

2

5

3

0

1211

10

Kosaraju-Sharir

160


9

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

8 76

1

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

00


1

4

9

2

5

3

1211

10

8 76

Kosaraju-Sharir

161

0 done

9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

3

0

1211

10

8 76

Kosaraju-Sharir

162

strong component: 0 2 3 4 5

9

0

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

–

–

5

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

163

check 2

9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

164

check 4

9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

165

check 5

9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

166

check 3

9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

–

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

167


9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

2

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

168


9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

2

–

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

169

visit 12: check 9

9

1211

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

–

–

2

2

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

170


9

12

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

–

2

2

11

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

171


9

12

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

–

2

2

11

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

8 76

Kosaraju-Sharir

172

visit 10: check 12

9

12

10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

11

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


10

1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

173

10 done

9

12

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

11

9 10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


9

1

4

2

5

3

0

1211

8 76

Kosaraju-Sharir

174

9 done

12

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

11

109

12

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


12

1

4

9

2

5

3

0

11

8 76

Kosaraju-Sharir

175

12 done

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

11

10

11 12

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

1111


1

4

9

2

5

3

0

12

8 76

Kosaraju-Sharir

176

11 done

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

177

strong component: 9 10 11 12

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

11

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

178

check 9

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

179

check 12

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

180

check 10

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

–

–

–

2

2

2

2

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

181


0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

–

2

2

2

2

86 7

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

182


0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

–

2

2

2

2

8 76

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

183


0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

–

2

2

2

2

8 76

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

184

visit 8: check 6

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

3

2

2

2

2

8 76

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


8

1

4

9

2

5

3

0

1211

7

Kosaraju-Sharir

185

8 done

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

3

2

2

2

2

7

10

6 866

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

186


0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

3

2

2

2

2

76

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

187

6 done

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

3

2

2

2

2

7

10

6

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 7

Kosaraju-Sharir

188

strong component: 6 8

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

–

3

2

2

2

2

7

10

6

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

189


0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

4

3

2

2

2

2

7

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

190


0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

4

3

2

2

2

2

7

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

191

7 done

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

4

3

2

2

2

2

7

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

8 76

Kosaraju-Sharir

192

strong component: 7

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

4

3

2

2

2

2

7

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

86 7

Kosaraju-Sharir

193

check 8

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

4

3

2

2

2

2

10

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v


1

4

9

2

5

3

0

1211

10

86 7

Kosaraju-Sharir

194

done

0

1

2

3

4

5

6

7

8

9

10

11

12

1

0

1

1

1

1

3

4

3

2

2

2

2

1 0 2 4 5 3 11 9 12 10 6 7 8

scc[v]v

Simple (but mysterious) algorithm for computing strong components.• Run DFS on GR to compute reverse postorder.

• Run DFS on G, considering vertices in order given by first DFS.

Proposition. Second DFS gives strong components. (!!)

195

Kosaraju's algorithm

dfs(7) check 6 check 97 donecheck 8

check unmarked vertices in the order 1 0 2 4 5 3 11 9 12 10 6 7 8

DFS in original digraph G

dfs(1)1 done

dfs(0) dfs(5) dfs(4) dfs(3) check 5 dfs(2) check 0 check 3 2 done 3 done check 2 4 done 5 done check 10 donecheck 2check 4check 5check 3

dfs(11) check 4 dfs(12) dfs(9) check 11 dfs(10) check 12 10 done 9 done 12 done11 donecheck 9check 12check 10

dfs(6) check 9 check 4 dfs(8) check 6 8 done check 06 done

196

Connected components in an undirected graph (with DFS)

public class CC { private boolean marked[]; private int[] id; private int count;

public CC(Graph G) { marked = new boolean[G.V()]; id = new int[G.V()];

for (int v = 0; v < G.V(); v++) { if (!marked[v]) { dfs(G, v); count++; } } }

private void dfs(Graph G, int v) { marked[v] = true; id[v] = count; for (int w : G.adj(v)) if (!marked[w]) dfs(G, w); }

public boolean connected(int v, int w) { return id[v] == id[w]; } }

197

Strong components in a digraph (with two DFSs)

public class KosarajuSCC { private boolean marked[]; private int[] id; private int count;

public KosarajuSCC(Digraph G) { marked = new boolean[G.V()]; id = new int[G.V()]; DepthFirstOrder dfs = new DepthFirstOrder(G.reverse()); for (int v : dfs.reversePost()) { if (!marked[v]) { dfs(G, v); count++; } } }

private void dfs(Digraph G, int v) { marked[v] = true; id[v] = count; for (int w : G.adj(v)) if (!marked[w]) dfs(G, w); }

public boolean stronglyConnected(int v, int w) { return id[v] == id[w]; } }

Digraph-processing summary: algorithms of the day

198

single-source reachability

DFS

topological sort (DAG)

DFS

strong components

Kosaraju DFS (twice)

06

4

21

5

3

7

12

109

11

8

bbm 202 - algorithmsbbm202/spring2017/... · mar. 31, 2016 bbm 202 - algorithms directed graphs...

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