Problem Solving with Algorithms and Data Structure — Graph

Bruce Tsai

http://interactivepython.org/runestone/static/pythonds/Graphs/Objectives.html

Graph

A set of objects where some pairs of objects are connected by links

road map

airline flights

internet connection

skill tree

Tree is special kind of graph

Vertex(node)

Edge(arc)

3

Goal

Represent problem in form of graph

Use graph algorithms to solve problem

4

(Undirected) Graph

G = (V, E)

V = {v_1, v_2, v_3, v_4, v_5, v_6}

E = {(v_1, v_2), (v_1, v_5), (v_2, v_3), (v_2, v_5), (v_3, v_4), (v_4, v_5), (v_4, v_6)}

(v_1, v_5) == (v_5, v_1)

5

Directed Graph

D = (V, A)

V = {v_0, v_1, v_2, v_3, v_4, v_5}

A = {(v_0, v_1), (v_0, v_5), (v_1, v_2), (v_2, v_3), (v_3, v_5), (v_3, v_4), (v_4, v_0), (v_5, v_2), (v_5, v_4)}

6

Attributed Graph

Attribute

Characters of graph vertices or edges

weight, color, anything you want

7

Path and Cycle

Path

Sequence of vertices that are connected by edges

[v_1, v_5, v_4, v_6]

Cycle

A path that starts and ends at same vertex

[v_2, v_3, v_4, v_5, v_2]8

Adjacency Matrix

9

from

to

Discussion Question

AB

C

DE

F

75

1

2

7

3

2

8

1

2

4

5

6

1

8

10

Adjacency List

11

Discussion Question

12

3

45

6

1015

57

7

10

713

5

12

Word Ladder Problem

Make change occur gradually by changing one letter at a time

FOOL -> POOL -> POLL -> POLE -> PALE -> SALE -> SAGE

13

Word Ladder Graph

14

O(V*L)

O(2E)

O(V+E)

Breadth First Search (BFS)

18

BFS Analysis

O(V+E)

19

Discussion Question - BFS

[1]

[1, 2, 3, 6]

[2, 3, 6]

[3, 6, 4]

[6, 4, 5]

[4, 5]

[5]

12

3

45

6

20

Knight’s Tour Problem

21

Legal Move Graph

22

Knight Graph

24

Knight’s Tour

27

Choose Better Next Vertex

Visit hard-to-reach corners early

Use the middle square to hop across board only when necessary

28

General Depth First Search (DFS)

Knight’s Tour is special case of depth first search

create the depth first tree

General depth first search is to search as deeply as possible

30

O(V+E)

O(V)

queue

stack

Computational Complexity

P versus NP problem

P: questions for which some algorithms can provide an answer in polynomial time

NP: questions for which an answer can be verified in polynomial time

Subset sum problem

{-7, -3, -2, 5, 8} -> {-3, -2, 5}

34

decision problem set

optimization problemssearch problems

NP-complete

Reduction: algorithm for transforming one problem into another problem

NP-complete: a problem p in NP is NP-Complete if every other problem in NP can be transformed into p in polynomial time

hardest problems in NP

36

NP-hard

Decision problem: any arbitrary yes-or-no question on an infinite set of inputs

NP-hard: a decision problem H in NP-hard when for any problem L in NP, there is polynomial-time reduction from L to H

at least as hard as the hardest problems in NP

37

Topological Sorting

Linear ordering of all vertices of a directed graph

(u, v) is an edge of directed graph and u is before v in ordering

38

Start and Finish Time

Implementation

1. Call dfs(g)

2. Store vertices based on decreasing order of finish time

3. Return the ordered list

41

Lemma

A directed graph G is acyclic if and only if a depth-first search of G yields no back edges.

Back edge: edge (u, v) connecting vertex u to an ancestor v in depth-first tree (v ↝ u → v)

=>: back edge exists then cycle exists

<=: cycle exists then back edge exists

42

Proof of Implementation

For any pair of distinct vertices u, v ∈ V, if there is an edge in G from u to v, then f[v] < f[u].

Consider edge (u, v) explored by DFS, v cannot be gray since then that edge would be back edge. Therefore v must be white or black.

If v is white, v is descendant of u, and so f[v] < f[u]

If v is black, it has already been finished, so that f[v] < f[u]

43

Strongly Connected Components

G = (V, E) and C ⊂ V

such that ∀ (v_i, v_j) ∈ C, path v_i to v_j and path v_j to v_i both exist

C is strongly connected components (SCC)

44

Implementation

1. Call dfs(g)

2. Compute transpose of g as g_t

3. Call dfs(g_t) but explore each vertex in decreasing order of finish time

4. Each tree of step 3 is a SCC

original transpose

46

dfs(g)

dfs(g_t)

Lemma

Let C and C’ be distinct SCCs in directed graph G = (V, E), let u, v ∈ C, let u’, v’ ∈ C’, and suppose that there is a path u ↝ u’ in G. Then there cannot also be a path v’ ↝ v in G.

if v’ ↝ v exists then both u ↝ u’ ↝ v’ and v’ ↝ v ↝ u exist

C and C’ are not distinct SCCs49

Lemma

Let C and C’ be distinct SCCs in directed graph G = (V, E). Suppose that there is an edge (u, v) ∈ E, where u ∈ C and v ∈ C’. Then f(C) > f(C’)

from x ∈ C to w ∈ C’

from y ∈ C’ cannot reach any vertex in C

50

Corollary

Let C and C’ be distinct SCCs in directed graph G = (V, E). Suppose that there is an edge (u, v) ∈ ET, where u ∈ C and v ∈ C’. Then f(C) < f(C’).

(v, u) ∈ E then f(C’) < f(C)

51

Proof of Implementation

Shortest Path Problem

Find the path with the smallest total weight along which to route any given message

53

Dijkstra’s Algorithm

Iterative algorithm providing the shortest path from one particular starting node to all other nodes in graph

Path distance, priority queue

54

O(V)

O(logV)

O(logV)O(E*logV)

O(V*logV)

O((V+E)*logV)

Broadcast Problem

Minimum Spanning Tree

T is acyclic subset of E that connects all vertices in V

The sum of weight of edges in T is minized

61

Prim’s Spanning Tree Algorithm

Special case of generic minimum spanning tree

Find shortest paths in graph

62

Kruskal’s Algorithm

Sort edges in E into ascending order by weight

For (u, v) ∈ E, if set of u not equal to set of v

A ← A ∪ {(u, v)}

65

Referencehttp://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/AproxAlgor/TSP/tsp.htm

http://en.wikipedia.org/wiki/Graph_%28mathematics%29

http://en.wikipedia.org/wiki/P_versus_NP_problem

http://en.wikipedia.org/wiki/NP-complete

http://en.wikipedia.org/wiki/NP-hard

http://en.wikipedia.org/wiki/Reduction_(complexity)

http://en.wikipedia.org/wiki/Knight%27s_tour

http://staff.ustc.edu.cn/~csli/graduate/algorithms/book6/chap24.htm

Introduction to Algorithms, 2nd edition