Graph Graph SearchingSearching

OverviewOverview

GraphGraph– Notation and ImplementationNotation and Implementation

BFS (Breadth First Search)BFS (Breadth First Search) DFS (Depth First Search)DFS (Depth First Search) TopologyTopology SCC (Strongly Connected SCC (Strongly Connected

Component)Component) Centre, Radius, DiameterCentre, Radius, Diameter Multi-state BFSMulti-state BFS

What is What is Graph?Graph?

GraphGraph

A A graphgraph is defined as G=(V,E), is defined as G=(V,E),

i.e a set of vertices and edges, i.e a set of vertices and edges, wherewhere– V is the set of vertices (singular: V is the set of vertices (singular:

vertex)vertex)– E is the set of edges that connect E is the set of edges that connect

some of the verticessome of the vertices

1

2

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3

vertex

edge

WA

NT

SA

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NSW

V

T

GraphGraph

Directed/Undirected GraphDirected/Undirected Graph Weighted/Unweighted GraphWeighted/Unweighted Graph ConnectivityConnectivity

Representation of Representation of GraphGraph Adjacency MatrixAdjacency Matrix Adjacency listAdjacency list Edge listEdge list

Representation of Representation of GraphGraph

AdjacencAdjacency Matrixy Matrix

Adjacency Adjacency Linked ListLinked List

Edge ListEdge List

Memory Memory StorageStorage

O(VO(V22)) O(V+E)O(V+E) O(V+E)O(V+E)

Check Check whether whether ((uu,,vv) is an ) is an edgeedge

O(1)O(1) O(deg(u))O(deg(u)) O(deg(u))O(deg(u))

Find all Find all adjacent adjacent vertices of vertices of a vertex a vertex uu

O(V)O(V) O(deg(u))O(deg(u)) O(deg(u))O(deg(u))

deg(u): the number of edges connecting vertex deg(u): the number of edges connecting vertex uu

GraphGraph

root

siblings

descendents children

ancestors

parent

TreeTree

Depth-First Search Depth-First Search (DFS)(DFS) Strategy: Go as far as you can (if Strategy: Go as far as you can (if

you have not visit there), you have not visit there), otherwise, go back and try otherwise, go back and try another wayanother way

ImplementationImplementation

DFS (vertex DFS (vertex uu) {) {

mark mark uu as as visitedvisited

for each vertex for each vertex vv directly reachable from directly reachable from uu

if if vv is is unvisitedunvisited

DFS (DFS (vv))

}}

Initially all vertices are marked as Initially all vertices are marked as unvisitedunvisited

Topological SortTopological Sort

Topological order: Topological order: A numbering of the vertices of a A numbering of the vertices of a directed acyclic graph such that directed acyclic graph such that every edge from a vertex numbered every edge from a vertex numbered i to a vertex numbered j satisfies i<ji to a vertex numbered j satisfies i<j

Topological Sort: Topological Sort: Finding the topological order of a Finding the topological order of a directed acyclic graphdirected acyclic graph

Tsort AlgorithmTsort Algorithm

If the graph has more then one vertex If the graph has more then one vertex that has indegree 0, add a vertice to that has indegree 0, add a vertice to connect to all indegree-0 verticesconnect to all indegree-0 vertices

Let the indegree 0 vertice be Let the indegree 0 vertice be ss Use Use ss as start vertice, and compute the as start vertice, and compute the

DFS forestDFS forest The death time of the vertices represent The death time of the vertices represent

the reverse of topological orderthe reverse of topological order

Example: Assembly Example: Assembly LineLine1.1. In a factory, there is several process. In a factory, there is several process.

Some need to be done before others. Some need to be done before others. Can you order those processes so that Can you order those processes so that they can be done smoothly?they can be done smoothly?

2.2. Chris is now studying OI materials. Chris is now studying OI materials. There are many topics. He needs to There are many topics. He needs to master some basic topics before master some basic topics before understanding those advanced one. understanding those advanced one. Can you help him to plan a smooth Can you help him to plan a smooth study plan?study plan?

Example: SCCExample: SCC

A graph is strongly-connected ifA graph is strongly-connected if– for any pair of vertices for any pair of vertices uu and and vv, one can , one can

go from go from uu to to vv and from and from vv to to uu.. Informally speaking, an SCC of a Informally speaking, an SCC of a

graph is a subset of vertices thatgraph is a subset of vertices that– forms a strongly-connected subgraphforms a strongly-connected subgraph– does not form a strongly-connected does not form a strongly-connected

subgraph with the addition of any new subgraph with the addition of any new vertexvertex

SCC (Illustration)SCC (Illustration)

Finding Shortest PathFinding Shortest Path

How can we use DFS to find the How can we use DFS to find the shortest distance from a vertex to shortest distance from a vertex to another? The distance of the first another? The distance of the first path found? path found?

Breadth-First Search Breadth-First Search (BFS)(BFS) BFS tries to find the target from BFS tries to find the target from

nearest vertices first.nearest vertices first. BFS makes use of a queue to BFS makes use of a queue to

store visited (but not dead) store visited (but not dead) vertices, expanding the path from vertices, expanding the path from the earliest visited vertices.the earliest visited vertices.

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Simulation of BFSSimulation of BFS

Queue:Queue: 1 4 3 5 2 6

ImplementationImplementation

while queue Q not emptywhile queue Q not emptydequeue the first vertex dequeue the first vertex uu from Q from Qfor each vertex for each vertex vv directly reachable from directly reachable from uu

if if vv is is unvisitedunvisitedenqueue enqueue vv to Q to Qmark mark vv as as visitedvisited

Initially all vertices except the Initially all vertices except the start vertex are marked as start vertex are marked as unvisitedunvisited and the queue contains and the queue contains the start vertex onlythe start vertex only

AdvantagesAdvantages

Guarantee shortest paths for Guarantee shortest paths for unweighted graphsunweighted graphs

Use queue instead of recursive Use queue instead of recursive functions – Avoiding stack functions – Avoiding stack overflowoverflow

Flood FillFlood Fill

An algorithm that determines the An algorithm that determines the area connected to a given node in area connected to a given node in a multi-dimensional arraya multi-dimensional array

Start BFS/DFS from the given Start BFS/DFS from the given node, counting the total number node, counting the total number of nodes visitedof nodes visited

Example: Squareland (HKOI 2006)Example: Squareland (HKOI 2006)

Variations of BFS and Variations of BFS and DFSDFS Bidirectional Search (BDS)Bidirectional Search (BDS) Iterative Deepening Search(IDS)Iterative Deepening Search(IDS)

Bidirectional search Bidirectional search (BDS)(BDS) Searches Searches simultaneouslysimultaneously from from

both the start vertex and goal both the start vertex and goal vertexvertex

Commonly implemented as Commonly implemented as bidirectional BFSbidirectional BFS

start goal

BDS Example: Bomber Man (1 BDS Example: Bomber Man (1 Bomb)Bomb)

find the shortest path from the upper-left find the shortest path from the upper-left corner to the lower-right corner in a maze using corner to the lower-right corner in a maze using a bomb. The bomb can destroy a wall.a bomb. The bomb can destroy a wall.

SS

EE

Bomber Man (1 Bomb)Bomber Man (1 Bomb)

SS

EE

1 2 3

4

1234

5

4

Shortest Path length = 8

5

6 67 78 8

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What will happen if we stop once we find a path?

SS 11 22 33 44 55 66 77 88 99

1100

2211

1111

2200

1122

1199

1133

1188

1144

1177

1177

1166

1155

1111

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1100

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ExampleExample

Iterative deepening Iterative deepening search (IDS)search (IDS)

Iteratively performs DFS with Iteratively performs DFS with increasing depth boundincreasing depth bound

Shortest paths are guaranteedShortest paths are guaranteed

IDSIDS

IDS (pseudo code)IDS (pseudo code)

DFS (vertex DFS (vertex u, u, depth depth dd) {) {mark mark uu as as visitedvisitedif (d>0)if (d>0)

for each vertex for each vertex vv directly reachable from directly reachable from uuif if vv is is unvisitedunvisited

DFS (DFS (v,v,d-1d-1))}}

i=0i=0Do {Do {

DFS(start vertex,i)DFS(start vertex,i)Increment iIncrement i

}While (target is not found)}While (target is not found)

IDSIDS

The complexity of IDS is the same The complexity of IDS is the same as DFSas DFS

Summary of DFS, BFSSummary of DFS, BFS

We learned some variations of We learned some variations of DFS and BFS DFS and BFS – Bidirectional search (BDS) Bidirectional search (BDS) – Iterative deepening search (IDS)Iterative deepening search (IDS)

EquationEquation

Question: Find the number of solution of Question: Find the number of solution of xxii, given k, given kii,p,pii. 1<=n<=6, 1<=x. 1<=n<=6, 1<=xii<=150<=150

Vertex: possible values of Vertex: possible values of – kk11xx11

pp1 1 , k, k11xx11pp1 1 + k+ k22xx22

pp2 2 , k, k11xx11pp1 1 + k+ k22xx22

pp2 2 + k+ k33xx33pp3 3 , ,

kk44xx44pp4 4 , k, k44xx44

pp4 4 + k+ k55xx55pp5 5 , k, k44xx44

pp4 4 + k+ k55xx55pp5 5 + k+ k66xx66

pp6 6