Graph Searching
Overview

Graph
– Notation and Implementation






BFS (Breadth First Search)
DFS (Depth First Search)
Topology
SCC (Strongly Connected Component)
Centre, Radius, Diameter
Multi-state BFS
What is Graph?
Graph

A graph is defined as G=(V,E),
i.e a set of vertices and edges, where
– V is the set of vertices (singular: vertex)
– E is the set of edges that connect some
of the vertices
NT
1
4
2
vertex
edge
3
WA
Q
SA
NSW
V
T
Graph



Directed/Undirected Graph
Weighted/Unweighted Graph
Connectivity
Representation of Graph



Adjacency Matrix
Adjacency list
Edge list
Representation of Graph
Adjacency Adjacency
Matrix
Linked List
Edge List
Memory
Storage
O(V2)
O(V+E)
O(V+E)
Check
whether
(u,v) is an
edge
O(1)
O(deg(u))
O(deg(u))
Find all
adjacent
vertices of a
vertex u
O(V)
O(deg(u))
O(deg(u))
deg(u): the number of edges connecting vertex u
Graph

Tree
ancestors
root
parent
siblings
descendents
children
Depth-First Search (DFS)

Strategy: Go as far as you can (if you
have not visit there), otherwise, go
back and try another way
Implementation
DFS (vertex u) {
mark u as visited
for each vertex v directly reachable from u
if v is unvisited
DFS (v)
}

Initially all vertices are marked as
unvisited
Topological Sort


Topological order:
A numbering of the vertices of a
directed acyclic graph such that every
edge from a vertex numbered i to a
vertex numbered j satisfies i<j
Topological Sort:
Finding the topological order of a
directed acyclic graph
Tsort Algorithm




If the graph has more then one vertex that
has indegree 0, add a vertice to connect to all
indegree-0 vertices
Let the indegree 0 vertice be s
Use s as start vertice, and compute the DFS
forest
The death time of the vertices represent the
reverse of topological order
Example: Assembly Line
1.
2.
In a factory, there is several process. Some
need to be done before others. Can you
order those processes so that they can be
done smoothly?
Chris is now studying OI materials. There
are many topics. He needs to master some
basic topics before understanding those
advanced one. Can you help him to plan a
smooth study plan?
Example: SCC

A graph is strongly-connected if
– for any pair of vertices u and v, one can
go from u to v and from v to u.

Informally speaking, an SCC of a graph
is a subset of vertices that
– forms a strongly-connected subgraph
– does not form a strongly-connected
subgraph with the addition of any new
vertex
SCC (Illustration)
Finding Shortest Path

How can we use DFS to find the
shortest distance from a vertex to
another? The distance of the first path
found?
Breadth-First Search
(BFS)


BFS tries to find the target from
nearest vertices first.
BFS makes use of a queue to store
visited (but not dead) vertices,
expanding the path from the earliest
visited vertices.
Simulation of BFS

Queue:
1
4
3
5
2
4
1
3
2
5
6
6
Implementation
while queue Q not empty
dequeue the first vertex u from Q
for each vertex v directly reachable from u
if v is unvisited
enqueue v to Q
mark v as visited

Initially all vertices except the start
vertex are marked as unvisited and
the queue contains the start vertex
only
Advantages


Guarantee shortest paths for
unweighted graphs
Use queue instead of recursive
functions – Avoiding stack overflow
Flood Fill



An algorithm that determines the area
connected to a given node in a multidimensional array
Start BFS/DFS from the given node,
counting the total number of nodes
visited
Example: Squareland (HKOI 2006)
Variations of BFS and DFS


Bidirectional Search (BDS)
Iterative Deepening Search(IDS)
Bidirectional search (BDS)


Searches simultaneously from both the
start vertex and goal vertex
Commonly implemented as
bidirectional BFS
start
goal
BDS Example: Bomber Man (1
Bomb)

find the shortest path from the upper-left corner
to the lower-right corner in a maze using a
bomb. The bomb can destroy a wall.
S
E
Bomber Man (1 Bomb)
S
1
2
3 4
13 12 11 12
4
8
7
6
5
10
6
9
5
10 11 12 13
4
7
8
9
3
2
1 E
Shortest Path length = 8
What will happen if we stop once we find a
path?
Example
S
1
2
3
4
5
6
21
20
19
18
17
11 12 13 14 15 16
10
9 8 7 6 5 4 3
7
8
9
10
11
12
13
14
17 16 15
2
1
E
Iterative deepening search
(IDS)


Iteratively performs DFS with
increasing depth bound
Shortest paths are guaranteed
IDS
IDS (pseudo code)
DFS (vertex u, depth d) {
mark u as visited
if (d>0)
for each vertex v directly reachable from u
if v is unvisited
DFS (v,d-1)
}
i=0
Do {
DFS(start vertex,i)
Increment i
}While (target is not found)
IDS

The complexity of IDS is the same as
DFS
Summary of DFS, BFS

We learned some variations of DFS
and BFS
– Bidirectional search (BDS)
– Iterative deepening search (IDS)
Equation


Question: Find the number of solution of
xi, given ki,pi. 1<=n<=6, 1<=xi<=150
Vertex: possible values of
– k1x1p1 , k1x1p1 + k2x2p2 , k1x1p1 + k2x2p2 + k3x3p3 ,
k4x4p4 , k4x4p4 + k5x5p5 , k4x4p4 + k5x5p5 + k6x6p6