Data Structure & Algorithm
10 – Graph & BFS & DFS
JJCAO
Graph Are Not
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Graphs
• G = (V,E)
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V[G] = {1,2,3,4,5,6} |V| = 6
E[G] = {{1,2},{1,5},{2,5},{3,6}}
Note: {u,v} = (u,v) = (v,u)
(u,v): u↔v
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Can be very complex. Excellent Tool!
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Terminology
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Adjacency List
Representation
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Adjacency Matrix
Representation
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Object-Oriented
Representation
• Node:
some structure, with all relevant information
• Edge:
name & pointers to two endpoints
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Sub-Graphs
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More Terminology
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Connectivity
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Connected Components
• Every node v is connected to itself
• if u and v are in the same connected
component then v is connected to u and
u is connected to v
• Connected components form a partition
of the nodes and so are disjoint:
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Forests & Trees
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Properties
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Directed Graph
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Adding Direction
• An undirected graph can be transformed
into a directed one
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Terminology
• endpoints of an edge = the vertices it connects
• e = (u,v) is incident from u = leaves u
incident to v = enters v
– u is the tail
– v is the head
• degree of v = indegree + outdegree
– indegree = # incoming edges
– outdegree = # outgoing edges
• Paths and cycles are now directed
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Simple Graphs
Are graphs with no parallel edges, no self
loops
Properties:
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More Properties
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Strong Connectivity
• Vertices strongly connected ↔ there is a path from each to
the other
• Graph Strongly connected↔every 2 vertices are strongly
connected
• Strongly Connected Components = maximal strongly connected
subgraphs
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Strong Connectivity
The strongly connected components of the above graph
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Graph Traversals
Given: a graph G
a source vertex s in V[G]
Goal: “visit” all the vertices of G to determine
some property:
– Is G connected?
– Does G contain a cycle?
– is there a v u path in G?
– What is the shortest path connecting v and u?
– Will G disconnect if we remove a single edge? A
vertex?
– G is the WWW - follow links to retrieve
information….
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Data Structure for BFS
• First In First Out (FIFO) Queue
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Breadth First Search
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Example
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BFS - Running Time
Adjacency list representation:
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Shortest Paths
Goal: Find a shortest path from s to every other v
Idea: - start at s
- find all vertices at distance 1
- find all vertices at distance 2
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BFS for Shortest Paths
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Shortest-Paths Claim
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BFS for Connectivity
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Depth First Search
Goal: Visit all vertices of the graph
Idea: - start at s
- keep going “deeper” into the graph
whenever possible
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Depth First Search
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Data Structure for DFS
• Last In First Out (LIFO) Stack
Rewrite the procedure DFS, using a stack to eliminate recursion
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Example
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Time Stamping
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Example
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BFS vs. DFS
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Classification of Edges in DFS Forest
Relative to the
same DFS tree
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Classification of Edges
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DFS - Running Time
Adjacency list representation
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Applications of DFS
In 𝛩(V+E) time, we can:
• Find connected components of G
• determine if G has a cycle
• determine if removing an edge / vertex
disconnects G
• determine if G is planar
• …
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Cycles in Directed Graphs
Theorem: DiGraph G has a cycle
 DFS forest has a back edge
Lemma: A directed graph G is acyclic if and only if a depth-first search of
G yields no back edges.
Proof:
• back edge => cycle: obvious
• cycle => back edge:
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Topological Sort
an ordering "<" of V[G] such that if (u, v)
E[G] => u < v
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Application of Topological Sort
Many applications use directed acyclic graphs to
indicate precedence among events, such as
Professor Bumstead gets dressed in the morning.
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Topological Sort
Lemma:
G can be topologically sorted <=> G is a DAG (directed
acyclic graph)
Proof:
cycle => G can't be sorted - obvious:
no cycle => G can be sorted:
we will show an algorithm
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A Simple TopSort Algorithm
Source = a vertex with indegree = 0
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A Simple TopSort Algorithm
Source = a vertex with indegree = 0
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Adjacency Matrix
Implementation
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Adjacency List Implementation
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Topological-Sort
1 call DFS.G/ to compute
finishing times v.f for each
vertex
2 as each vertex is finished,
insert it onto the front of a
linked list
3 return the linked list of
vertices
We can perform a topological sort in time Θ(V + E), since depth-first
search takes Θ(V + E) time and it takes O(1) time to insert each of the
|V| vertices onto the front of the linked list.
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1 call DFS.G/ to compute finishing times v.f for each
vertex
2 as each vertex is finished, insert it onto the front of a
linked list
3 return the linked list of vertices
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Graph in C++
• Boost/graph
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Over
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Strongly
Connected Components
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Simple SCC Algorithms
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