242-535 ADA: 9. Graph Search1 Objective o describe and compare depth-first and breadth- first graph...

242-535 ADA: 9. Graph Search 1

• Objectiveo describe and compare depth-first and breadth-

first graph searching, and look at the creation of spanning trees

Algorithm Design and Analysis

(ADA)242-535, Semester 1 2014-2015

9. Graph Search


1. Graph Searching2. Depth First Search (DFS)3. Uses of DFS

o cycle detection, reachability, topological sort

4. Breadth-first Search (BFS)5. DFS vs. BFS6. IP Multicasting

Overview


• Given: a graph G = (V, E), directed or undirected

• Goal: visit every vertex

• Often the end result is a tree built over the grapho called a spanning tree

• it visits every vertex, but not necessarily every edge

o Pick any vertex as the rooto Choose certain edges to produce a treeo Note: we might build a forest if the graph is not

connected

1. Graph Searching


Example

search then build a spanning tree(or trees)


• DFS is “depth first” because it always fully explores down a path away from a vertex v before it looks at other paths leaving v.

• Crucial DFS properties:• uses recursion: essential for graph structures• choice: at a vertex there may be a choice of several

edges to follow to the next vertex• backtracking: "return to where you came from"• avoid cycles by grouping vertices into visited and

unvisited

2. Depth First Search (DFS)


Directed Graph Examplea

d

fec

b

Graph G

DFS works with directedand undirected graphs.


• enum MARKTYPE {VISITED, UNVISITED};

struct cell { /* adj. list */ NODE nodeName; struct cell *next;};typedef struct cell *LIST;

struct graph { enum MARKTYPE mark; LIST successors;};typedef struct graph GRAPH[NUMNODES];

Data Structures


void dfs(NODE u, GRAPH G)// recursively search G, starting from u{ LIST p; // runs down adj. list of u NODE v; // node in cell that p points at

G[u].mark = VISITED; // visited u p = G[u].successors; while (p != NULL) { // visit u’s succ’s v = p->nodeName; if (G[v].mark == UNVISITED) dfs(v, G); // visit v p = p->next; }}

The dfs() Function


• Call Visitedd(a) {a}d(a)-d(b) {a,b}d(a)-d(b)-d(c) {a,b,c}

Skip b, return to d(b)d(a)-d(b)-d(d) {a,b,c,d}

Skip cd(a)-d(b)-d(d)-d(e) {a,b,c,d,e}

Skip c, return to d(d)

Calling dfs(a,G)

continued

call it d(a)for short


d(a)-d(b)-d(d)-d(f){a,b,c,d,e,f}Skip c, return to

d(d)

d(a)-d(b)-d(d) {a,b,c,d,e,f}Return to d(b)

d(a)-d(b) {a,b,c,d,e,f}Return to d(a)

d(a) {a,b,c,d,e,f}Skip d, return


• Since nodes are marked, the graph is searched as if it were a tree:

DFS Spanning Tree

d/4

f/6e/5c

b/2

a/1

c/3

A spanning tree is a subgraph of a graph G which contains all the verticies of G.


Example 2

a b

dc

e f

hg

a b

dc

e f

hg

the tree generated by DFS is drawn with thick lines

DFS


• The time taken to search from a node is proportional to the no. of successors of that node.

• Total search time for all nodes = O(|V|)Total search time for all successors = time to search all edges = O(|E|)

• Total running time is O(V + E)

dfs() Running Time

continued


• If the graph is dense, E >> V (E approaches V2) then the O(V) term can be ignoredo in that case, the total running time = O(E) or O(V2)


• Finding cycles in a grapho e.g. for finding recursion in a call graph

• Searching complex locations, such as mazes

• Reachability detectiono i.e. can a vertex v be reached from vertex u?o useful for e-mail routing; path finding

• Strong connectivity

• Topological sorting

3. Uses of DFS

continued


• The DFS algorithm is similar to a classic strategy for exploring a mazeo mark each intersection,

corner and dead end (vertex) as visited

o mark each corridor (edge ) traversed

o keep track of the path back to the previous branch points

Maze Traversal

Graphs 16

Reachability• DFS tree rooted at v: what are the vertices reachable from v

via directed paths?

A

C

E

B

D

F

A

C

E D

A

C

E

B

D

F

start at C

start at B

Strong Connectivity• Each vertex can reach all other vertices

Graphs 18

a

d

c

b

e

f

g

Strong Connectivity Algorithm

• Pick a vertex v in G.• Perform a DFS from v in G.

o If there’s a vertex not visited, print “no”.

• Let G’ be G with edges reversed.• Perform a DFS from v in G’.

o If there’s a vertex not visited, print “no”

• If the algorithm gets here, print “yes”.

• Running time: O(V+E).19

G:

G’:

a

d

c

b

e

f

g

a

d

c

b

e

f

g

Strongly Connected Components

• List all the subgraphs where each vertex can reach all the other vertices in that subgraph.

• Can also be done in O(V+E) time using DFS.

20

{ a , c , g }

{ f , d , e , b }

a

d

c

b

e

f

g


• Topological sort of a directed acyclic graph (DAG):o linearly order all the vertices in a graph G such that

vertex u comes before vertex v if edge (u, v) Go a DAG is a directed graph with no directed cycles

Topological Sort


Example: Getting Dressed

Underwear Socks

ShoesTrousers

Belt

Shirt

Watch

Tie

Jacket

Socks Underwear Trousers Shoes Watch Shirt Belt Tie Jacket

one topological sort(not unique)


Topological-Sort(){ Run DFS;

When a vertex is finished, output it;Vertices are output in reverse topological order;

}

• Time: O(V+E)

Topological Sort Algorithm


• Process all the verticies at a given level before moving to the next level.

• Example graph G (again):

4. Breadth-first Search (BFS)

a b

dc

e f

hg


• 1) Put the verticies into an orderingo e.g. {a, b, c, d, e, f, g, h}

• 2) Select a vertex, add it to the spanning tree T: e.g. a

• 3) Add to T all edges (a,X) and X verticies that do not create a cycle in To i.e. (a,b), (a,c), (a,g)

T = {a, b, c, g}

Informal Algorithm

continued

a

b c g


• Repeat step 3 on the verticies just added, these are on level 1o i.e. b: add (b,d)

c: add (c,e) g: nothing

T = {a,b,c,d,e}

• Repeat step 3 on the verticies just added, these are on level 2o i.e. d: add (d,f)

e: nothingT = {a,b,c,d,e,f}

a

b c g

d e

a

b c g

d e

fcontinued

level 1

level 2


• Repeat step 3 on the verticies just added, these are on level 3o i.e. f: add (f,h)

T = {a,b,c,d,e,f,h}

• Repeat step 3 on the verticies just added, these are on level 4o i.e. h: nothing, so stop

a

b c g

d e

f

h

continued

level 3


• Resulting spanning tree:

a b

dc

e f

hg

a differentspanning treefrom the earliersolution


Example 2


Algorithm Graphicallystart node

pre-builtadjency list


boolean marked[]; // visited this vertex?int edgeTo[]; // vertex number going to this vertex

void bfs(Graph graph, int start){ Queue q = new Queue(); marked[start] = true; q.add(start); // add to end of queue while (!q.isEmpty()) { int v = q.remove(); // get from start of queue for (int w : graph.adjacentTo(v)) // v --> w if (!marked[w]) { edgeTo[w] = v; // save last edge on a shortest path marked[w] = true; q.add(w); // add to end of queue } }} // end of bfs()

BFS Code


5. DFS vs. BFS

Applications DFS BFSSpanning forest, connected components, paths, cycles

Shortest paths

Biconnected components

see part 11


• DFS is like one person exploring a mazeo do down a path to the end, get to a dead-end,

backtrack, and try a different path

• BFS is like a group of searchers fanning out in all directions, each unrolling a ball of string. o at a branch point, the searchers split up to explore all

the branches at once o if two groups meet up, they join forces (using the ball of

string of the group that got there first)o the group that gets to the exit first has found the

shortest path

DFS and BFS as Maze Explorers


BFS Maze Graphically

Also called flood filling; used in paintsoftware.


• The BFS "fanning out" algorithm is best implemented in a parallel language, where each "group of explorers" is a separate thread of execution.o e.g. use fork and join in Java

• The earlier implementation uses a queue to implement the fanning out as a sequential algorithm.

• DFS is inherently a sequential algorithm.

Sequential / Parallel


• A network of computers and routers:

6. IP Multicasting

sourcecomputer

router

continued


• How can a packet (message) be sent from the source computer to every other computer?

• The inefficient way is to use broadcastingo send a copy along every link, and have each router

do the sameo each router and computer will receive many copies

of the same packeto loops may mean the packet never disappears!

continued


• IP multicasting is an efficient solutiono send a single packet to one routero have the router send it to 1 or more routers in such a

way that a computer never receives the packet more than once

• This behaviour can be represented by a spanning tree.

• Can use either BFS or DFS, but BFS will usually produce shorter pathso i.e. BFS is a better choice

continued


• One spanning tree for the network:

sourcecomputer

router

the tree isdrawn withthick lines

242-535 ADA: 9. Graph Search1 Objective o describe and compare depth-first and breadth- first graph...

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