Slides credited from Hsueh-I Lu, Hsu-Chun Hsiao, & Michael Tsai

Slides credited from Hsueh-I Lu, Hsu-Chun Hsiao, & Michael Tsai

 Mini-HW

 NTU COOL

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 Course recordings

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 Classroom (crowded, sleepy, etc.)

 Homework due time

 Pseudo code

 Mini-HW 7 released

 Due on 11/29 (Thur) 14:20

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 Due on 12/13 (Thur) 14:20 (three weeks)

3

Frequently check the website for the updated information!



Graph Basics



Graph Theory



Graph Representations



Graph Traversal



Breadth-First Search (BFS)



Depth-First Search (DFS)



DFS Applications



Connected Components



Strongly Connected Components



Topological Sorting

5



A graph G is defined as

 V: a finite, nonempty set of vertices

 E: a set of edges / pairs of vertices

3 5

1

4

2



Graph type

 Undirected: edge 𝑢, 𝑣 = 𝑣, 𝑢

 Directed: edge 𝑢, 𝑣 goes from vertex 𝑢 to vertex 𝑣; 𝑢, 𝑣 ≠ 𝑣, 𝑢

 Weighted: edges associate with weights

7

3 5

1 4

2

3 5

1 4

2

How many edges at most can a undirected (or directed) graph have?



Adjacent (相鄰)



If there is an edge 𝑢, 𝑣 , then 𝑢 and 𝑣 are adjacent.



Incident (作用)



If there is an edge 𝑢, 𝑣 , the edge

is incident from 𝑢 and is incident to 𝑣.



Subgraph (子圖)



If a graph 𝐺

= 𝑉

, 𝐸′ is a subgraph of 𝐺 = 𝑉, 𝐸 , then 𝑉

⊆

𝑉 and 𝐸

⊆ 𝐸



Degree



The degree of a vertex

is the number of edges incident on

 In-degree of 𝑢: #edges 𝑥, 𝑢 in a directed graph

 Out-degree of 𝑢: #edges 𝑢, 𝑥 in a directed graph

 Degree = in-degree + out-degree

 Isolated vertex: degree = 0

9

𝐸 = σ

𝑑

2



Path



a sequence of edges that connect a sequence of vertices



If there is a path from 𝑢 (source) to 𝑣 (target), there is a sequence of edges 𝑢, 𝑖

, 𝑖

, 𝑖

, … , 𝑖

, 𝑖

, (𝑖

, 𝑣)

 Reachable: 𝑣 is reachable from 𝑢 if there exists a path from 𝑢 to 𝑣



Simple Path

 All vertices except for 𝑢 and 𝑣 are all distinct



Cycle



Connected



Two vertices are connected if there is a path between them



A connected graph has a path from every vertex to every other



Tree



a connected, acyclic, undirected graph



Forest



an acyclic, undirected but possibly disconnected graph

11

3 5

1 4

2

3 5

1 4

2

3 5

1 4

2



Theorem. Let 𝐺 be an undirected graph. The following statements are equivalent:



𝐺 is a tree



Any two vertices in 𝐺 are connected by a unique simple path



𝐺 is connected, but if any edge is removed from 𝐸, the resulting graph is disconnected.



𝐺 is connected and 𝐸 = 𝑉 − 1



𝐺 is acyclic, and 𝐸 = 𝑉 − 1



𝐺 is acyclic, but if any edge is added to 𝐸, the resulting graph

13



How to traverse all bridges where each one can only be passed through once

A

B C C

A

B



Euler path

 Can you traverse each edge in a connected graph exactly once without lifting the pen from the paper?



Euler tour

 Can you finish where you started?

15

C A

B D

C A

B D

C A

B D

Euler path Euler tour

Euler path Euler tour

Euler path Euler tour



Solved by Leonhard Euler in 1736



𝐺 has an Euler path iff 𝐺 has exactly 0 or 2 odd vertices



𝐺 has an Euler tour iff all vertices must be even vertices

Is it possible to determine whether a graph has an Euler path or an Euler tour, without necessarily having to find one explicitly?

Even vertices = vertices with even degrees Odd vertices = vertices with odd degrees



Hamiltonian Path



A path that visits each vertex exactly once



Hamiltonian Cycle



A Hamiltonian path where the start and destination are the same



Both are NP-complete

17



Modeling applications using graph theory

 What do the vertices represent?

 What do the edges represent?

 Undirected or directed?

19



How to represent a graph in computer programs?



Two standard ways to represent a graph 𝐺 = 𝑉, 𝐸



Adjacency matrix



Adjacency list

Matrix

21



Adjacency matrix = 𝑉 × 𝑉 matrix 𝐴 with 𝐴[𝑢][𝑣] = 1 if (𝑢, 𝑣) is an edge

1 2 3 4 5 6

1 1 1

2 1 1 1

3 1 1 1

4 1 1 1

5 1

6 1 1

1

2

3

5 4

6

• For undirected graphs, 𝐴 is symmetric; i.e., 𝐴 = 𝐴^𝑇

• If weighted, store weights instead of bits in 𝐴

Matrix



Space:



Time for querying an edge:



Time for inserting an edge:



Time for deleting an edge:



Time for listing all neighbors of a vertex:



Time for identifying all edges:



Time for finding in-degree and out-degree of a vertex?

List



Adjacency lists = vertex indexed array of lists



One list per vertex, where for 𝑢 ∈ 𝑉, 𝐴[𝑢] consists of all vertices adjacent to 𝑢

23

1 2 3 4 5 6

1 4

3

2 3

2

3 2

5

1 4 6

4

6 1

2

3

5 4

6

If weighted, store weights also in adjacency lists

List



Space:



Time for querying an edge:



Time for inserting an edge:



Time for deleting an edge:



Time for listing all neighbors of a vertex:



Time for identifying all edges:



Time for finding in-degree and out-degree of a vertex?



Matrix representation is suitable for dense graphs



List representation is suitable for sparse graphs



Besides graph density, you may also choose a data structure based on the performance of other operations

25

Space Query an edge

Insert an edge

Delete an edge

List a vertex’s neighbors

Identify all edges

Adjacency Matrix Adjacency List

Textbook Chapter 22 – Elementary Graph Algorithms

26



From a source vertex, systematically follow the edges of a graph to visit all reachable vertices of the graph



Useful to discover the structure of a graph



Standard graph-searching algorithms

 Breadth-First Search (BFS, 廣度優先搜尋)

 Depth-First Search (DFS, 深度優先搜尋)

27

Textbook Chapter 22.2 – Breadth-first search

28

29

Source 𝒔

Layer 1

Layer 2



Input: directed/undirected graph 𝐺 = (𝑉, 𝐸) and source 𝑠



Output: a breadth-first tree with root 𝑠 (𝑇

) that contains all reachable vertices



𝑣. 𝑑: distance from 𝑠 to 𝑣, for all 𝑣 ∈ 𝑉

 Distance is the length of a shortest path in G

 𝑣. 𝑑 = ∞ if 𝑣 is not reachable from 𝑠

 𝑣. 𝑑 is also the depth of 𝑣 in 𝑇_BFS



𝑣. 𝜋 = 𝑢 if (𝑢, 𝑣) is the last edge on shortest path to 𝑣



Initially 𝑇

contains only 𝑠



As 𝑣 is discovered from 𝑢, 𝑣 and (𝑢, 𝑣) are added to 𝑇

 𝑇_BFS is not explicitly stored; can be reconstructed from 𝑣. 𝜋



Implemented via a FIFO queue



Color the vertices to keep track of progress:

 GRAY: discovered (first time encountered)

 BLACK: finished (all adjacent vertices discovered)

 WHITE: undiscovered

31

BFS(G, s)

for each vertex u in G.V-{s}

u.color = WHITE u.d = ∞

u.pi = NIL s.color = GRAY s.d = 0

s.pi = NIL Q = {}

ENQUEUE(Q, s) while Q! = {}

u = DEQUEUE(Q)

for each v in G.Adj[u]

if v.color == WHITE v.color = GRAY v.d = u.d + 1 v.pi = u

ENQUEUE(Q,v) u.color = BLACK

𝑠 0

𝑤 𝑟

1 1

𝑟 𝑡 𝑥

1 2 2

𝑡 𝑥 𝑣

2 2 2

33

𝑢 𝑦

3 3

𝑦 3

 Definition of 𝛿(𝑠, 𝑣): the shortest-path distance from 𝑠 to 𝑣 = the minimum number of edges in any path from 𝑠 to 𝑣

 If there is no path from 𝑠 to 𝑣, then 𝛿 𝑠, 𝑣 = ∞

 The BFS algorithm finds the shortest-path distance to each reachable vertex in a graph 𝐺 from a given source vertex 𝑠 ∈ 𝑉.



Proof

 Case 1: 𝑢 is reachable from 𝑠

 𝑠- 𝑢- 𝑣 is a path from 𝑠 to 𝑣 with length 𝛿 𝑠, 𝑢 + 1

 Hence, 𝛿 𝑠, 𝑣 ≤ 𝛿 𝑠, 𝑢 + 1

 Case 2: 𝑢 is unreachable from 𝑠

 Then 𝑣 must be unreachable too.

 Hence, the inequality still holds.

35

Lemma 22.1

Let 𝐺 = 𝑉, 𝐸 be a directed or undirected graph, and let 𝑠 ∈ 𝑉 be an arbitrary vertex. Then, for any edge 𝑢, 𝑣 ∈ 𝐸, 𝛿 𝑠, 𝑣 ≤ 𝛿 𝑠, 𝑢 + 1.

𝑠-𝑣的最短路徑一定會小於等於𝑠-𝑢的最短路徑距離+1

s

v

𝛿 𝑠, 𝑢 u



Proof by induction

 Holds when 𝑛 = 1: 𝑠 is in the queue and 𝑣. 𝑑 = ∞ for all 𝑣 ∈ 𝑉 𝑠

 After 𝑛 + 1 ENQUEUE ops, consider a white vertex 𝑣 that is discovered during the search from a vertex 𝑢

Lemma 22.2

Let 𝐺 = 𝑉, 𝐸 be a directed or undirected graph, and suppose BFS is run on 𝐺 from a given source vertex 𝑠 ∈ 𝑉. Then upon termination, for each vertex 𝑣 ∈ 𝑉, the value 𝑣. 𝑑 computed by BFS satisfies 𝑣. 𝑑 ≥ 𝛿 𝑠, 𝑣 .

BFS算出的d值必定大於等於真正距離

Inductive hypothesis: 𝑣. 𝑑 ≥ 𝛿 𝑠, 𝑣 after 𝑛 ENQUEUE ops



Proof by induction

 Holds when 𝑄 = 𝑠 .

 Consider two operations for inductive step:

 Dequeue op: when 𝑄 = 𝑣₁, 𝑣₂, … , 𝑣_𝑟 and dequeue 𝑣₁

 Enqueue op: when 𝑄 = 𝑣₁, 𝑣₂, … , 𝑣_𝑟 and enqueue 𝑣_𝑟+1

37

Lemma 22.3

Suppose that during the execution of BFS on a graph 𝐺 = 𝑉, 𝐸 , the queue 𝑄 contains the vertices 𝑣₁, 𝑣₂, … , 𝑣_𝑟 , where 𝑣₁ is the head of 𝑄 and 𝑣_𝑟 is the tail. Then, 𝑣_𝑟. 𝑑 ≤ 𝑣₁. 𝑑 + 1 and 𝑣_𝑖. 𝑑 ≤ 𝑣_𝑖+1. 𝑑 for 1 ≤ 𝑖 < 𝑟.

• Q中最後一個點的d值 ≤ Q中第一個點的d值+1

• Q中第i個點的d值 ≤ Q中第i+1點的d值

Inductive hypothesis:𝑣_𝑟. 𝑑 ≤ 𝑣₁. 𝑑 + 1 and 𝑣_𝑖. 𝑑 ≤ 𝑣_𝑖+1. 𝑑 after 𝑛 queue ops



Dequeue op



Enqueue op

Inductive hypothesis:

𝑣₁ 𝑣₂ … 𝑣_𝑟−1 𝑣_𝑟

𝑣₂ … 𝑣_𝑟−1 𝑣_𝑟 (induction hypothesis H2)

𝑣₁ 𝑣₂ … 𝑣_𝑟−1 𝑣_𝑟

(induction hypothesis H2)

𝑣 𝑣 … 𝑣 𝑣 𝑣

𝑢

Let 𝑢 be 𝑣_𝑟+1’s predecessor,

Since 𝑢 has been removed from 𝑄, the new head 𝑣₁ satisfies

(induction hypothesis H1) H1

H2

 H1 holds

 H2 holds

𝑢

(Q中最後一個點的d值 ≤ Q中第一個點的d值+1)

(Q中第i個點的d值 ≤ Q中第i+1點的d值)



Proof

 Lemma 22.3 proves that 𝑣_𝑖. 𝑑 ≤ 𝑣_𝑖+1. 𝑑 for 1 ≤ 𝑖 < 𝑟

 Each vertex receives a finite 𝑑 value at most once during the course of BFS

 Hence, this is proved.

39

Corollary 22.4

Suppose that vertices 𝑣_𝑖 and 𝑣_𝑗 are enqueued during the execution of BFS, and that 𝑣_𝑖 is enqueued before 𝑣_𝑗. Then 𝑣_𝑖. 𝑑 ≤ 𝑣_𝑗. 𝑑 at the time that 𝑣_𝑗 is enqueued.

若𝑣_𝑖比𝑣_𝑗早加入queue  𝑣_𝑖. 𝑑 ≤ 𝑣_𝑗. 𝑑



Proof of (1)

 All vertices 𝑣 reachable from 𝑠 must be discovered; otherwise they would have 𝑣. 𝑑 = ∞ > 𝛿 𝑠, 𝑣 . (contradicting with Lemma 22.2) Theorem 22.5 – BFS Correctness

Let 𝐺 = 𝑉, 𝐸 be a directed or undirected graph, and and suppose that BFS is run on 𝐺 from a given source vertex 𝑠 ∈ 𝑉.

1) BFS discovers every vertex 𝑣 ∈ 𝑉 that is reachable from the source 𝑠 2) Upon termination, 𝑣. 𝑑 = 𝛿 𝑠, 𝑣 for all 𝑣 ∈ 𝑉

3) For any vertex 𝑣 ≠ 𝑠 that is reachable from 𝑠, one of the shortest paths from 𝑠 to 𝑣 is a shortest path from 𝑠 to 𝑣. 𝜋 followed by the edge 𝑣. 𝜋, 𝑣

(2)



Proof of (2) by contradiction

 Assume some vertices receive 𝑑 values not equal to its shortest-path distance

 Let 𝑣 be the vertex with minimum 𝛿 𝑠, 𝑣 that receives such an incorrect 𝑑 value; clearly 𝑣 ≠ 𝑠

 By Lemma 22.2, 𝑣. 𝑑 ≥ 𝛿 𝑠, 𝑣 , thus 𝑣. 𝑑 > 𝛿 𝑠, 𝑣 (𝑣 must be reachable)

 Let 𝑢 be the vertex immediately preceding 𝑣 on a shortest path from 𝑠 to 𝑣, so 𝛿 𝑠, 𝑣 = 𝛿 𝑠, 𝑢 + 1

 Because 𝛿 𝑠, 𝑢 < 𝛿 𝑠, 𝑣 and 𝑣 is the minimum 𝛿 𝑠, 𝑣 , we have 𝑢. 𝑑 = 𝛿 𝑠, 𝑢

 𝑣. 𝑑 > 𝛿 𝑠, 𝑣 = 𝛿 𝑠, 𝑢 + 1 = 𝑢. 𝑑 + 1

41

(2)



Proof of (2) by contradiction (cont.)

 𝑣. 𝑑 > 𝛿 𝑠, 𝑣 = 𝛿 𝑠, 𝑢 + 1 = 𝑢. 𝑑 + 1

 When dequeuing 𝑢 from 𝑄, vertex 𝑣 is either WHITE, GRAY, or BLACK

 WHITE: 𝑣. 𝑑 = 𝑢. 𝑑 + 1, contradiction

 BLACK: it was already removed from the queue

 By Corollary 22.4, we have 𝑣. 𝑑 ≤ 𝑢. 𝑑, contradiction

 GRAY: it was painted GRAY upon dequeuing some vertex 𝑤

 Thus 𝑣. 𝑑 = 𝑤. 𝑑 + 1 (by construction)

(3) For any vertex 𝑣 ≠ 𝑠 that is reachable from 𝑠, one of the shortest paths from 𝑠 to 𝑣 is a shortest path from 𝑠 to 𝑣. 𝜋 followed by the edge 𝑣. 𝜋, 𝑣



Proof of (3)

 If 𝑣. 𝜋 = 𝑢, then 𝑣. 𝑑 = 𝑢. 𝑑 + 1. Thus, we can obtain a shortest path from 𝑠 to 𝑣 by taking a shortest path from 𝑠 to 𝑣. 𝜋 and then traversing the edge 𝑣. 𝜋, 𝑣 .

43

 BFS(G, s) forms a BFS tree with all reachable 𝑣 from 𝑠

 We can extend the algorithm to find a BFS forest that contains every vertex in 𝐺

BFS-Visit(G, s) s.color = GRAY s.d = 0

s.π = NIL Q = empty ENQUEUE(Q, s) while Q ≠ empty

u = DEQUEUE(Q) for v in G.adj[u]

if v.color == WHITE //explore full graph and builds up

a collection of BFS trees BFS(G)

for u in G.V

u.color = WHITE u.d = ∞

u.π = NIL for s in G.V

if(s.color == WHITE)

Textbook Chapter 22.3 – Depth-first search

45

 Search as deep as possible and then backtrack until finding a new path

1

2

3 4

8

9 12 13

14

5 6

7

10 11



Implemented via recursion (stack)



Color the vertices to keep track of progress:

 GRAY: discovered (first time encountered)

 BLACK: finished (all adjacent vertices discovered)

 WHITE: undiscovered ⁴⁷

// Explore full graph and builds up a collection of DFS trees

DFS(G)

for each vertex u in G.V u.color = WHITE

u.pi = NIL

time = 0 // global timestamp for each vertex u in G.V

if u.color == WHITE DFS-VISIT(G, u)

DFS-Visit(G, u) time = time + 1

u.d = time // discover time u.color = GRAY

for each v in G.Adj[u]

if v.color == WHITE v.pi = u

DFS-VISIT(G, v) u.color = BLACK

time = time + 1

u.f = time // finish time



Parenthesis Theorem

 Parenthesis structure: represent the discovery of vertex 𝑢 with a left

parenthesis “(𝑢” and represent its finishing by a right parenthesis “𝑢)”. In DFS, the parentheses are properly nested.



White Path Theorem

 In a DFS forest of a directed or undirected graph 𝐺 = 𝑉, 𝐸 ,

 vertex 𝑣 is a descendant of vertex 𝑢 in the forest  at the time 𝑢. 𝑑 that the search discovers 𝑢, there is a path from 𝑢 to 𝑣 in 𝐺 consisting entirely of WHITE vertices



Classification of Edges in 𝐺

 Tree Edge



Parenthesis Theorem

 Parenthesis structure: represent the discovery of vertex 𝑢 with a left parenthesis “(𝑢” and represent its finishing by a right parenthesis

“𝑢)”. In DFS, the parentheses are properly nested.

49

Properly nested: (x (y y) x) Not properly nested: (x (y x) y)

Proof in textbook p. 608



White Path Theorem

 In a DFS forest of a directed or undirected graph 𝐺 = 𝑉, 𝐸 ,

 vertex 𝑣 is a descendant of vertex 𝑢 in the forest  at the time 𝑢. 𝑑 that the search discovers 𝑢, there is a path from 𝑢 to 𝑣 in 𝐺 consisting entirely of WHITE vertices



Proof.

 

 Since 𝑣 is a descendant of 𝑢, 𝑢. 𝑑 < 𝑣. 𝑑

 Hence, 𝑣 is WHITE at time 𝑢. 𝑑

 In fact, since 𝑣 can be any descendant of 𝑢, any vertex on the path from 𝑢



Classification of Edges in 𝐺

 Tree Edge (GRAY to WHITE)

 Edges in the DFS forest

 Found when encountering a new vertex 𝑣 by exploring 𝑢, 𝑣

 Back Edge (GRAY to GRAY)

 𝑢, 𝑣 , from descendant 𝑢 to ancestor 𝑣 in a DFS tree

 Forward Edge (GRAY to BLACK)

 𝑢, 𝑣 , from ancestor 𝑢 to descendant 𝑣. Not a tree edge.

 Cross Edge (GRAY to BLACK)

 Any other edge between trees or subtrees. Can go between vertices in same DFS tree or in different DFS trees

51

In an undirected graph, back edge = forward edge.

To avoid ambiguity, classify edge as the first type in the list that applies.



Edge classification by the color of 𝑣 when visiting 𝑢, 𝑣

 WHITE: tree edge

 GRAY: back edge

 BLACK: forward edge or cross edge

 𝑢. 𝑑 < 𝑣. 𝑑  forward edge

 𝑢. 𝑑 > 𝑣. 𝑑  cross edge

Theorem 22.10

In DFS of an undirected graph, there are only tree edges and back edges



Connected Components



Strongly Connected Components



Topological Sort

53

54



Input: a graph 𝐺 = 𝑉, 𝐸



Output: a connected component of 𝐺



a maximal subset 𝑈 of 𝑉 s.t. any two nodes in 𝑈 are connected in 𝐺

Why must the connected components of a graph be disjoint? 55

1

2

5

3 4

6

7

8 9

57

58

Course Website: http://ada.miulab.tw Email: ada-ta@csie.ntu.edu.tw

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