Department of Computer Science

Algorithms and Data Structures for Data Science

CS 277 Brad Solomon

November 8, 2021

Graph Traversals

mp_huffman review

Average:

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Learning Objectives

Introduce graph coloring and graph traversals

Identify utility of traversal labelings

Define key graph functions and discuss implementation details

Graph Implementation: Edge List

v

u

w z

u

v

w

z

Vertex Storage:

Edge Storage:

|V|= n,|E|= m

Graph Implementation: Adjacency Matrix

v

u

w z

u

v

w

z

Vertex Storage:

Edge Storage:

u v w z

u 0 1 1 0

v 1 0 1 0

w 1 1 0 1

z 0 0 1 0

Adjacency List

v

u

w z

Vertex Storage:

Edge Storage:u

v

w

z

v w

u w

v u z

z

d=2

d=2

d=3

d=1

Graph Coloring

5

36

4

2

1

4

Graph Coloring

A

G F

HB

E

D

I

C

Example modified from Princeton (Math Alive) — Author unknown

Graph Coloring

F

C I

BG

H

E

A

D

Example modified from Princeton (Math Alive) — Author unknown

Graph Coloring

An optimal solution can be found — but is NP-complete

Heuristic algorithms have no guarantees (but are usually fast)

Our heuristic has an upper bound but not a clear lower bound

Traversal: Objective: Visit every vertex and every edge in the graph.

Purpose: Search for interesting sub-structures in the graph.

We’ve seen traversal before ….but it’s different:

• Ordered • Obvious Start •

• • •

Traversal: BFS

A

C D

E

B

G H

F

Traversal: BFS v d P Adjacent Edges

A

B

C

D

E

F

G

H

A

C D

E

B

G H

F

Traversal: BFS

G H F E D C B A

v d P Adjacent Edges

A 0 - B C D

B 1 A A C E

C 1 A A B D E F

D 1 A A C F H

E 2 C B C G

F 2 C C D G

G 3 E E F H

H 2 D D G

A

C D

E

B

G H

F

Traversal: BFS

G H F E D B C A

v d P Adjacent Edges

A 0 - C B D

B 1 A A C E

C 1 A A B D E F

D 1 A A C F H

E 2 C B C G

F 2 C C D G

G 3 E E F H

H 2 D D G

A

C D

E

B

G H

F

class vertObj(): def __init__(self, v): self.val = v self.eList = []

# Added for BFS self.visited = 0 self.pred = None self.dist = -1

1 2 3 4 5 6 7 8 9

10 11 12

class edgeObj(): def __init__(self, e): self.v2 = e self.parallel = None

# Added for BFS self.label = 0

1 2 3 4 5 6 7 8 9

10 11 12

BFS AnalysisQ: What do we do if we have a disjoint graph?

Q: Does our implementation detect a cycle?

Q: What is the running time?

Running time of BFS

G H F E D B C A

v d P Adjacent Edges

A 0 - C B D

B 1 A A C E

C 1 A B A D E F

D 1 A A C F H

E 2 C B C G

F 2 C C D G

G 3 E E F H

H 2 D D G

A

C D

E

B

G H

F

BFS ObservationsQ: What is a shortest path from A to H?

Q: What is a shortest path from E to H?

Q: How does a cross edge relate to d?

Q: What structure is made from discovery edges?

v d P Adjacent Edges

A 0 - C B D

B 1 A A C E

C 1 A B A D E F

D 1 A A C F H

E 2 C B C G

F 2 C C D G

G 3 E E F H

H 2 D D G

A

C D

E

B

G H

F

BFS ObservationsObs. 1: BFS can be used to count components.

Obs. 2: BFS can be used to detect cycles.

Obs. 3: In BFS, d provides the shortest distance to every vertex.

Obs. 4: In BFS, the endpoints of a cross edge never differ in distance, d, by more than 1: |d(u) − d(v) | ≤ 1

Traversal: DFS

A

C

D

E

BF

G

H

JI

Traversal: DFS

Discovery Edge

Back Edge

A

C

D

E

BF

G

H

JI

def DFS_recur(self, s): self.V[s].visited = 1

el = self.getEdges(s) for e in el: l = e.label v = self.V[l] if v.visited == 0: self.DFS_recur(l)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

Traversal: DFS

DFS ObservationsObs. 1: DFS can be used to count components.

Obs. 2: DFS can be used to detect cycles.

Obs. 3: In DFS, distance provides no clear meaning

DFS vs BFSDFS: BFS:

Pros: Pros:

Cons: Cons:

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