CS 225 Data Structures April 16 Graph Traversal Wad ade Fag - - PowerPoint PPT Presentation

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CS 225 Data Structures April 16 Graph Traversal Wad ade Fag - - PowerPoint PPT Presentation

Jun 01, 2023 231 likes •449 views

CS 225 Data Structures April 16 Graph Traversal Wad ade Fag agen-Ulm lmschneid ider Graph ADT Functions: - insertVertex(K key); Data: - insertEdge(Vertex v1, Vertex v2, K key); - Vertices - Edges - removeVertex(Vertex v); - Some data

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SLIDE 1

CS 225

Data Structures

April 16 – Graph Traversal

Wad ade Fag agen-Ulm lmschneid ider

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SLIDE 2

Graph ADT

Functions:

  • insertVertex(K key);
  • insertEdge(Vertex v1, Vertex v2, K key);
  • removeVertex(Vertex v);
  • removeEdge(Vertex v1, Vertex v2);
  • incidentEdges(Vertex v);
  • areAdjacent(Vertex v1, Vertex v2);
  • origin(Edge e);
  • destination(Edge e);

Data:

  • Vertices
  • Edges
  • Some data structure

maintaining the structure between vertices and edges.

X V W Z Y b e d f g h

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SLIDE 3

Edge List

v u w a c b z d u v w z u v a v w b u w c w z d

Vertex List Edge List

Key Ideas:

  • Given a vertex, O(1) lookup in vertex list
  • Implement w/ a hash table, etc
  • All basic ADT operations runs in O(m) time
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SLIDE 4

Adja jacency Matrix

v u w a c b z d u v w z

u v w z u

Ø Ø

v

Ø Ø

w

Ø

z

Ø u v a v w b u w c w z d

Key Ideas:

  • Given a vertex, O(1) lookup in vertex

list

  • Given a pair of vertices (an edge),

O(1) lookup in the matrix

  • Undirected graphs can use an upper

triangular matrix

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SLIDE 5

Adja jacency List

v u w a c b z d u v w z u v a v w b u w c w z d a c a b b c d d

d=2 d=2 d=3 d=1

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SLIDE 6

Adja jacency List

v u w a c b z d u v w z u v a v w b u w c w z d a c a b b c d d

Key Ideas:

  • O(1) lookup in vertex list
  • Vertex list contains a doubly-linked

adjacency list

  • O(1) access to the adjacent vertex’s

node in adjacency list (via the edge list)

  • Vertex list maintains a

count of incident edges, or deg(v)

  • Many operations run

in O(deg(v)), and deg(v) ≤ n-1, O(n).

d=2 d=2 d=3 d=1

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SLIDE 7

Expressed as big-O

Edge List Adjacency Matrix Adjacency List Space

n+m N2 n+m

insertVertex(v)

1 n 1

removeVertex(v)

m n deg(v)

insertEdge(v, w, k)

1 1 1

removeEdge(v, w)

1 1 1

incidentEdges(v)

m n deg(v)

areAdjacent(v, w)

m 1 min( deg(v), deg(w) )

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SLIDE 8

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
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SLIDE 9

Traversal: BFS

A C D E B F G H

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SLIDE 10

Traversal: BFS

A C D E B F G H d p Adjacent Edges

0 A A C B D 1 A B A C E 1 A C B A D E F 1 A D A C F H 2 C E B C G 2 C F C D G 3 E G E F H 2 D H D G

G H F E D B C A

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SLIDE 11

BFS(G): Input: Graph, G Output: A labeling of the edges on G as discovery and cross edges foreach (Vertex v : G.vertices()): setLabel(v, UNEXPLORED) foreach (Edge e : G.edges()): setLabel(e, UNEXPLORED) foreach (Vertex v : G.vertices()): if getLabel(v) == UNEXPLORED: BFS(G, v) 1 2 3 4 5 6 7 8 9 10 11 12 BFS(G, v): Queue q setLabel(v, VISITED) q.enqueue(v) while !q.empty(): v = q.dequeue() foreach (Vertex w : G.adjacent(v)): if getLabel(w) == UNEXPLORED: setLabel(v, w, DISCOVERY) setLabel(w, VISITED) q.enqueue(w) elseif getLabel(v, w) == UNEXPLORED: setLabel(v, w, CROSS) 14 15 16 17 18 19 20 21 22 23 24 25 26 27

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SLIDE 12

BFS Analysis

Q: Does our implementation handle disjoint graphs? If so, what code handles this?

  • How do we use this to count components?

Q: Does our implementation detect a cycle?

  • How do we update our code to detect a cycle?

Q: What is the running time?

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SLIDE 13

Running time of f BFS

A C D E B F G H d p v Adjacent

0 A A C B D 1 A B A C E 1 A C B A D E F 1 A D A C F H 2 C E B C G 2 C F C D G 3 E G E F H 2 D H D G

G H F E D B C A

While-loop at :19? For-loop at :21?

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SLIDE 14

BFS(G): Input: Graph, G Output: A labeling of the edges on G as discovery and cross edges foreach (Vertex v : G.vertices()): setLabel(v, UNEXPLORED) foreach (Edge e : G.edges()): setLabel(e, UNEXPLORED) foreach (Vertex v : G.vertices()): if getLabel(v) == UNEXPLORED: BFS(G, v) 1 2 3 4 5 6 7 8 9 10 11 12 BFS(G, v): Queue q setLabel(v, VISITED) q.enqueue(v) while !q.empty(): v = q.dequeue() foreach (Vertex w : G.adjacent(v)): if getLabel(w) == UNEXPLORED: setLabel(v, w, DISCOVERY) setLabel(w, VISITED) q.enqueue(w) elseif getLabel(v, w) == UNEXPLORED: setLabel(v, w, CROSS) 14 15 16 17 18 19 20 21 22 23 24 25 26 27

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SLIDE 15

BFS Observations

A C D E B F G H d p v Adjacent

0 A A C B D 1 A B A C E 1 A C B A D E F 1 A D A C F H 2 C E B C G 2 C F C D G 3 E G E F H 2 D H D G

Q: 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?

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SLIDE 16

BFS Observations

  • Obs. 1: Traversals can be used to count components.
  • Obs. 2: Traversals 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

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SLIDE 17

Traversal: DFS

A C D E B F G H J K

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SLIDE 18

BFS(G): Input: Graph, G Output: A labeling of the edges on G as discovery and cross edges foreach (Vertex v : G.vertices()): setLabel(v, UNEXPLORED) foreach (Edge e : G.edges()): setLabel(e, UNEXPLORED) foreach (Vertex v : G.vertices()): if getLabel(v) == UNEXPLORED: BFS(G, v) 1 2 3 4 5 6 7 8 9 10 11 12 BFS(G, v): Queue q setLabel(v, VISITED) q.enqueue(v) while !q.empty(): v = q.dequeue() foreach (Vertex w : G.adjacent(v)): if getLabel(w) == UNEXPLORED: setLabel(v, w, DISCOVERY) setLabel(w, VISITED) q.enqueue(w) elseif getLabel(v, w) == UNEXPLORED: setLabel(v, w, CROSS) 14 15 16 17 18 19 20 21 22 23 24 25 26 27

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SLIDE 19

DFS(G): Input: Graph, G Output: A labeling of the edges on G as discovery and back edges foreach (Vertex v : G.vertices()): setLabel(v, UNEXPLORED) foreach (Edge e : G.edges()): setLabel(e, UNEXPLORED) foreach (Vertex v : G.vertices()): if getLabel(v) == UNEXPLORED: DFS(G, v) 1 2 3 4 5 6 7 8 9 10 11 12 DFS(G, v): Queue q setLabel(v, VISITED) q.enqueue(v) while !q.empty(): v = q.dequeue() foreach (Vertex w : G.adjacent(v)): if getLabel(w) == UNEXPLORED: setLabel(v, w, DISCOVERY) setLabel(w, VISITED) DFS(G, w) elseif getLabel(v, w) == UNEXPLORED: setLabel(v, w, BACK) 14 15 16 17 18 19 20 21 22 23 24 25 26 27

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SLIDE 20

Running time of f DFS

Labeling:

  • Vertex:
  • Edge:

Queries:

  • Vertex:
  • Edge:

A C D E B F G H J K