Slide Cấu trúc dữ liệu và giả thuật - Lecture11 - Graphs - Phạm Bảo Sơn - UET - Tài liệu VNU

incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud.. It adds vertices by increasing distance..[r]

Data Structures and

Algorithms


"

Graphs!

•   A graph is a pair (V, E) , where!

–  V is a set of nodes, called vertices!

–  E is a collection of pairs of vertices, called edges!

–  Vertices and edges are positions and store elements!

•   Example:!

–  A vertex represents an airport and stores the three-letter airport code! –  An edge represents a flight route between two airports and stores the mileage of the route!

MIA DFW

SFO

LAX

LGA HNL

Phạm Bảo Sơn - DSA 5

Edge Types"

•   Directed edge!

–  ordered pair of vertices ( u , v )

–  first vertex u is the origin!

–  second vertex v is the destination!

–  e.g., a flight!

•   Undirected edge!

–  unordered pair of vertices ( u , v )

–  e.g., a flight route

•   Directed graph!

–  all the edges are directed!

–  e.g., route network!

•   Undirected graph!

–  all the edges are undirected!

–  e.g., flight network!

ORD flight AA 1206 PVD

John David

Phạm Bảo Sơn - DSA 7

Terminology"

•   End vertices (or endpoints) of

an edge!

–  U and V are the endpoints of a!

•   Edges incident on a vertex!

–  a, d, and b are incident on V!

P 1

Terminology (cont.)"

•   Path!

–  sequence of alternating

vertices and edges !

–  begins with a vertex!

–  ends with a vertex!

–  each edge is preceded and

followed by its endpoints!

•   Simple path!

–  path such that all its vertices

and edges are distinct!

Phạm Bảo Sơn - DSA 9

Terminology (cont.)"

•   Cycle!

–  circular sequence of alternating

vertices and edges !

–  each edge is preceded and

followed by its endpoints!

•   Simple cycle!

–  cycle such that all its vertices

and edges are distinct!

Phạm Bảo Sơn - DSA 11

Main Methods of the Graph ADT"

•   Vertices and edges!

–  are positions!

–  store elements!

•   Accessor methods!

–  endVertices(e): an array of

the two endvertices of e!

–  opposite(v, e): the vertex

opposite of v on e!

–  areAdjacent(v, w): true iff v

and w are adjacent!

an edge (v,w) storing element o!

–  removeVertex(v): remove vertex v (and its incident edges)!

–  removeEdge(e): remove edge e!

•   Iterator methods!

–  incidentEdges(v): edges incident to v!

–  vertices(): all vertices in the graph!

–  edges(): all edges in the graph!

Data Structures for Graphs"

•   Edge List!

•   Adjacency List!

•   Adjacency Matrix!

Phạm Bảo Sơn - DSA 13

–   origin vertex object!

–   destination vertex object!

–   reference to position in edge

Adjacency List Structure "

•   Edge list structure!

Phạm Bảo Sơn - DSA 15

•   Edge list structure!

•   Augmented vertex

objects!

–  Integer key (index)

associated with vertex!

•   The “ old fashioned ”

version just has 0 for no

edge and 1 for edge!

•   A subgraph S of a graph

G is a graph such that !

–   The vertices of S are a

subset of the vertices of G!

–   The edges of S are a

subset of the edges of G!

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Trees and Forests"

This definition of tree is

different from the one of a

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Spanning Trees and Forests"

•   A spanning tree of a

connected graph is a

spanning subgraph that is

a tree!

•   A spanning tree is not

unique unless the graph is

a tree!

•   Spanning trees have

applications to the design

–   Find and report a path between two given vertices!

–   Find a cycle in the graph!

•   Depth-first search is to graphs what Euler tour

is to binary trees!

Phạm Bảo Sơn - DSA 23

DFS Algorithm"

•   The algorithm uses a mechanism for

setting and getting “ labels ” of

Input graph G and a start vertex v of G

Output labeling of the edges of G

in the connected component of v

as discovery edges and back edges

else setLabel(e, BACK)

Algorithm DFS(G)

Input graph G

Output labeling of the edges of G

as discovery edges and

Phạm Bảo Sơn - DSA 25

DFS and Maze Traversal "

intersection, corner and

dead end (vertex) visited!

–  We mark each corridor

(edge ) traversed!

–  We keep track of the

path back to the entrance

(start vertex) by means

of a rope (recursion

stack)!

Phạm Bảo Sơn - DSA 27

Properties of DFS"

Property 1!

! DFS ( G, v ) visits all the

vertices and edges in

Analysis of DFS"

•   Setting/getting a vertex/edge label takes O (1) time!

•   Each vertex is labeled twice !

–   once as UNEXPLORED!

–   once as VISITED!

•   Each edge is labeled twice!

–   once as UNEXPLORED!

–   once as DISCOVERY or BACK!

•   Method incidentEdges is called once for each vertex!

•   DFS runs in O ( n + m ) time provided the graph is

represented by the adjacency list structure!

–   Recall that Σ v deg(v) = 2m

Phạm Bảo Sơn - DSA 29

as the start vertex!

•   We use a stack S to keep

track of the path between

the start vertex and the

current vertex!

•   As soon as destination

vertex z is encountered,

we return the path as the

contents of the stack !

S.pop(v)

keep track of the path

between the start vertex

and the current vertex!

•   As soon as a back edge

( v, w ) is encountered,

we return the cycle as

the portion of the stack

from the top to vertex w !

–   Find and report a path with the minimum

number of edges between two given vertices !

–   Find a simple cycle, if there is one!

Phạm Bảo Sơn - DSA 33

BFS Algorithm"

•  The algorithm uses a

mechanism for setting and

getting “ labels ” of vertices

for all v ∈ L i elements()

for all e ∈ G.incidentEdges(v)

Output labeling of the edges

and partition of the

Phạm Bảo Sơn - DSA 35

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!The discovery edges labeled by BFS( G,

s ) form a spanning tree T s of G s

Property 3!

!For each vertex v in L i

–   The path of T s from s to v has i edges !

–   Every path from s to v in G s has at least i

•   Setting/getting a vertex/edge label takes O (1) time!

•   Each vertex is labeled twice !

–   once as UNEXPLORED!

–   once as VISITED!

•   Each edge is labeled twice!

–   once as UNEXPLORED!

–   once as DISCOVERY or CROSS!

•   Each vertex is inserted once into a sequence L i !

•   Method incidentEdges is called once for each vertex!

•   BFS runs in O ( n + m ) time provided the graph is

represented by the adjacency list structure!

–   Recall that Σ v deg(v) = 2m

Phạm Bảo Sơn - DSA 39

Applications"

–   Compute a spanning forest of G !

–   Find a simple cycle in G , or report that G is a forest! –   Given two vertices of G , find a path in G between them with the minimum number of edges, or report that no such path exists!

Spanning forest, connected

Phạm Bảo Sơn - DSA 41

–   w is in the same level as

v or in the next level in

the tree of discovery edges!

Directed Graphs"

JFK BOS

MIA

ORD

SFO

Phạm Bảo Sơn - DSA 43

Digraph Properties"

–   Each edge goes in one direction:!

•   Edge (a,b ) goes from a to b , but not b to a.!

•   If G is simple, m < n*(n-1).!

adjacency lists, we can perform listing of

in-edges and out-in-edges in time proportional to

Phạm Bảo Sơn - DSA 45

Digraph Application"

completed before b can be started!

The good life

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vertex s determines the

vertices reachable from s

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•   Pick a vertex v in G.!

•   Perform a DFS from v in G.!

–   If there ’ s a w not visited, print “ no ” !

•   Let G ’ be G with edges

reversed.!

•   Perform a DFS from v in G ’ !

–   If there ’ s a w not visited, print “ no ” !

–   Else, print “ yes ” !

•   Running time: O(n+m).!

•   Maximal subgraphs such that each vertex can reach

all other vertices in the subgraph!

•   Can also be done in O(n+m) time using DFS, but is

more complicated (similar to biconnectivity).!

Phạm Bảo Sơn - DSA 51

Transitive Closure"

•   Given a digraph G , the

transitive closure of G is the

digraph G* such that!

–   G* has the same vertices

Computing the Transitive Closure"

DFS starting at

each vertex!

–   O(n(n+m))!

If there's a way to get

dynamic programming: The Floyd-Warshall

Algorithm

Phạm Bảo Sơn - DSA 53

Floyd-Warshall Transitive Closure"

Uses only vertices

numbered 1,…,k-1 Uses only vertices numbered 1,…,k (add this edge if it ’ s not already in)

–  G k has a directed edge ( v i , v j )

if G has a directed path from

•   Running time: O(n 3 ),

assuming areAdjacent is O(1)

(e.g., adjacency matrix)!

Algorithm FloydWarshall(G) Input digraph G

Output transitive closure G* of G

Phạm Bảo Sơn - DSA 55

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Phạm Bảo Sơn - DSA 59

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Phạm Bảo Sơn - DSA 61

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Phạm Bảo Sơn - DSA 63

DAGs and Topological Ordering"

•  A directed acyclic graph (DAG) is a

digraph that has no directed cycles!

•  A topological ordering of a digraph

is a numbering !

v 1 , …, v n !

!of the vertices such that for every

edge ( v i , v j ), we have i < j

•  Example: in a task scheduling

digraph, a topological ordering a

task sequence that satisfies the

precedence constraints!

Theorem!

!A digraph admits a topological

ordering if and only if it is a DAG!

write c.s program play

dream about graphs

A typical student day

10

11 make cookies

Phạm Bảo Sơn - DSA 65

!

!

!

!

Algorithm for Topological Sorting"

Method TopologicalSort(G)

H ← G // Temporary copy of G

n ← G.numVertices()

while H is not empty do

Let v be a vertex with no outgoing edges Label v ← n

n ← n - 1

Remove v from H

Topological Sorting Algorithm using DFS"

•  Simulate the algorithm by using

depth-first search!

•  O(n+m) time.!

Algorithm topologicalDFS(G, v)

Input graph G and a start vertex v of G

Output labeling of the vertices of G

in the connected component of v

else {e is a forward or cross edge}

Label v with topological number n

Phạm Bảo Sơn - DSA 67

Topological Sorting Example "

Topological Sorting Example "

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Phạm Bảo Sơn - DSA 69

Topological Sorting Example "

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Topological Sorting Example "

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Phạm Bảo Sơn - DSA 71

Topological Sorting Example "

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Topological Sorting Example "

Phạm Bảo Sơn - DSA 73

Topological Sorting Example "

Topological Sorting Example "

Phạm Bảo Sơn - DSA 75

Topological Sorting Example "

Topological Sorting Example "

–  In a flight route graph, the weight of an edge represents the

distance in miles between the endpoint airports!

MIA DFW

SFO

LAX

LGA HNL

Phạm Bảo Sơn - DSA 79

•   Given a weighted graph and two vertices u and v, we want to find

a path of minimum total weight between u and v

–  Length of a path is the sum of the weights of its edges.!

SFO

LAX

LGA HNL

Shortest Path Properties"

SFO

LAX

LGA HNL

Phạm Bảo Sơn - DSA 81

•   The distance of a vertex

v from a vertex s is the

length of a shortest path

between s and v!

•   Dijkstra ’ s algorithm

computes the distances

of all the vertices from a

given start vertex s!

•   Assumptions:!

–  the graph is connected!

–  the edges are

undirected!

–  the edge weights are

nonnegative !

•   We grow a “cloud” of vertices,

beginning with s and eventually

covering all the vertices!

•   We store with each vertex v a label d ( v ) representing the

distance of v from s in the

subgraph consisting of the cloud and its adjacent vertices!

•   At each step!

–  We add to the cloud the vertex

u outside the cloud with the smallest distance label, d( u ) –  We update the labels of the

vertices adjacent to u !

Edge Relaxation"

•   Consider an edge e = ( u,z )

such that!

–  u is the vertex most recently

added to the cloud!

–  z is not in the cloud!

•   The relaxation of edge e

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Phạm Bảo Sơn - DSA 85

•   A priority queue stores

the vertices outside the

for all v ∈ G.vertices()

if v = s

setDistance(v, 0) else

setDistance(v, ∞ )

l ← Q.insert(getDistance(v), v) setLocator(v,l)

while ¬ Q.isEmpty()

u ← Q.removeMin() for all e ∈ G.incidentEdges(u)

Analysis of Dijkstra ’ s Algorithm"

•   Priority queue operations!

–  Each vertex is inserted once into and removed once from the priority

queue, where each insertion or removal takes O(log n ) time!

–  The key of a vertex in the priority queue is modified at most deg( w )

times, where each key change takes O(log n ) time !

•   Dijkstra ’ s algorithm runs in O (( n + m ) log n ) time provided the

graph is represented by the adjacency list structure!

–  Recall that Σ v deg( v ) = 2 m

•   The running time can also be expressed as O ( m log n ) since the graph is connected!

Phạm Bảo Sơn - DSA 87

Shortest Paths Tree"

•   Using the template

method pattern, we

can extend Dijkstra ’ s

algorithm to return a

tree of shortest paths

from the start vertex to

all other vertices!

•   We store with each

vertex a third label:!

–  parent edge in the

shortest path tree!

•   In the edge relaxation

step, we update the

Why Dijkstra ’ s Algorithm Works"

•   Dijkstra ’ s algorithm is based on the greedy

method It adds vertices by increasing distance.!

n   Suppose it didn ’ t find all shortest

distances Let F be the first wrong

vertex the algorithm processed

n   When the previous node, D, on the

true shortest path was considered,

its distance was correct

n   But the edge (D,F) was relaxed at

that time!

n   Thus, so long as d(F)>d(D), F ’ s

distance cannot be wrong That is,

there is no wrong vertex

Phạm Bảo Sơn - DSA 89

Negative-Weight Edges"

–   If a node with a negative

incident edge were to be added

late to the cloud, it could mess

up distances for vertices already

! Dijkstra ’ s algorithm is based on the greedy

method It adds vertices by increasing distance

C ’ s true distance is 1, but it

is already in the cloud with d(C)=5!

Bellman-Ford Algorithm"

•   Works even with

negative-weight edges!

•   Must assume directed

edges (for otherwise we

would have negative-weight

cycles)!

•   Iteration i finds all shortest

paths that use i edges.!

•   Running time: O(nm).!

•   Can be extended to detect

setDistance(v, ∞ )

for i ← 1 to n-1 do for each e ∈ G.edges()

Phạm Bảo Sơn - DSA 91

DAG-based Algorithm"

•   Works even with

negative-weight edges!

•   Uses topological order!

•   Doesn ’ t use any fancy

setDistance(v, ∞ )

Perform a topological sort of the vertices

for u ← 1 to n do {in topological order}

for each e ∈ G.outEdges(u)