Algorithm Design and Analysis (ADA)
242-535, Semester 1 2014-2015
9. Graph Search
• Objective
o describe and compare depth-first and breadthfirst graph searching, and look at the creation of
spanning trees
242-535 ADA: 9. Graph Search
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Overview
1. Graph Searching
2. Depth First Search (DFS)
3. Uses of DFS
o
cycle detection, reachability,
topological sort
4. Breadth-first Search (BFS)
5. DFS vs. BFS
6. IP Multicasting
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1. Graph Searching
• Given: a graph G = (V, E), directed or undirected
• Goal: visit every vertex
• Often the end result is a tree built over the graph
o called a spanning tree
• it visits every vertex, but not necessarily every edge
o Pick any vertex as the root
o Choose certain edges to produce a tree
o Note: we might build a forest if the graph is not connected
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Example
search then build a spanning tree
(or trees)
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2. Depth First Search (DFS)
• 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
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Directed Graph Example
a
DFS works with directed
and undirected graphs.
b
d
e
f
c
Graph G
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Data Structures
• 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];
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The dfs() Function
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;
}
}
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Calling dfs(a,G)
• Call
d(a)
d(a)-d(b)
d(a)-d(b)-d(c)
d(a)-d(b)-d(d)
d(a)-d(b)-d(d)-d(e)
242-535 ADA: 9. Graph Search
call it d(a)
for short
Visited
{a}
{a,b}
{a,b,c}
Skip b, return to d(b)
{a,b,c,d}
Skip c
{a,b,c,d,e}
Skip c, return to d(d)
continued
9
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
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DFS Spanning Tree
• Since nodes are marked, the graph is searched
as if it were a tree:
A spanning tree is a
subgraph of a graph G
which contains all the
verticies of G.
a/1
b/2
c/3
d/4
e/5
f/6
c
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Example 2
a
a
b
b
c
d
c
d
e
f
e
f
h
g
DFS
h
g
the tree generated by
DFS is drawn with thick
lines
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dfs() Running Time
• 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)
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continued
13
• If the graph is dense, E >> V (E approaches V2) then
the O(V) term can be ignored
o in that case, the total running time = O(E) or O(V2)
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3. Uses of DFS
• Finding cycles in a graph
o e.g. for finding recursion in a call graph
• Searching complex locations, such as mazes
• Reachability detection
o i.e. can a vertex v be reached from vertex u?
o useful for e-mail routing; path finding
• Strong connectivity
• Topological sorting
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continued
15
Maze Traversal
• The DFS algorithm is similar
to a classic strategy for
exploring a maze
o 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
Graphs
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16
Reachability
• DFS tree rooted at v: what are the vertices reachable from v
via directed paths?
E
D
C
E
D
C
A
F
E
A
start at C
D
B
C
A
F
B
start at B
Strong Connectivity
• Each vertex can reach all other vertices
a
g
c
d
e
f
Graphs
b
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Strong Connectivity Algorithm
• Pick a vertex v in G.
• Perform a DFS from v in G.
a
G:
g
c
o If there’s a vertex not visited, print
“no”.
d
e
• Let G’ be G with edges reversed.
• Perform a DFS from v in G’.
o If there’s a vertex not visited, print
“no”
a
G’:
g
c
• If the algorithm gets here, print
“yes”.
d
e
f
• Running time: O(V+E).
b
f
b
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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.
a
g
c
{a,c,g}
d
e
f
b
{f,d,e,b}
20
Topological Sort
• 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)  G
o a DAG is a directed graph with no directed cycles
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Example: Getting Dressed
Underwear
Socks
Watch
Trousers
Shoes
Shirt
Belt
Tie
one topological sort
(not unique)
Jacket
Socks
Underwear
Trousers
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Shoes
Watch
Shirt
Belt
Tie
Jacket
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Topological Sort Algorithm
Topological-Sort()
{ Run DFS;
When a vertex is finished, output it;
Vertices are output in reverse topological order;
}
• Time: O(V+E)
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4. Breadth-first Search (BFS)
• Process all the verticies at a given level before
moving to the next level.
• Example graph G (again):
a
b
c
d
e
f
h
g
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Informal Algorithm
• 1) Put the verticies into an ordering
o 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 T
o i.e. (a,b), (a,c), (a,g)
T = {a, b, c, g}
a
b
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c
continued
g
25
• Repeat step 3 on the verticies just added,
these are on level 1
o i.e. b: add (b,d)
c: add (c,e)
g: nothing
T = {a,b,c,d,e}
level 1
a
b
c
d
e
g
• Repeat step 3 on the verticies just added,
these are on level 2
o i.e. d: add (d,f)
e: nothing
T = {a,b,c,d,e,f}
a
level 2
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continued
b
c
d
e
f
g
26
• Repeat step 3 on the verticies just added,
these are on level 3
o i.e. f: add (f,h)
T = {a,b,c,d,e,f,h}
level 3
a
b
c
d
e
g
f
h
• Repeat step 3 on the verticies just added,
these are on level 4
o i.e. h: nothing, so stop
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continued
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• Resulting spanning tree:
a
b
c
d
e
f
a different
spanning tree
from the earlier
solution
h
g
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Example 2
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Algorithm Graphicallypre-built
start
node
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adjency list
30
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BFS Code
boolean marked[];
int edgeTo[];
// visited this vertex?
// 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()
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5. DFS vs. BFS
Applications
Spanning forest, connected
components, paths, cycles
DFS
BFS


Shortest paths
Biconnected components


see part 11
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DFS and BFS as Maze Explorers
• DFS is like one person exploring a maze
o 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
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BFS Maze Graphically
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Also called flood filling; used in paint
software.
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Sequential / Parallel
• 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.
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6. IP Multicasting
• A network of computers and routers:
source
computer
router
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continued
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• How can a packet (message) be sent from the
source computer to every other computer?
• The inefficient way is to use broadcasting
o send a copy along every link, and have each router do
the same
o each router and computer will receive many copies of
the same packet
o loops may mean the packet never disappears!
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continued
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• IP multicasting is an efficient solution
o send a single packet to one router
o 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 paths
o i.e. BFS is a better choice
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continued
40
• One spanning tree for the network:
source
computer
the tree is
drawn with
thick lines
router
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