Elementary Graph Algorithm

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Elementary Graph Algorithm

• 10144017 ???
• 10144019 ???
• 10144044 ???

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Contents

• Representations of graphs
• Breadth-first search
• Depth-first search
• Topological sort
• Strongly connected components

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Elementary Graph Algorithms

• Searching a graph means systematically following
  the edges of the graph so as to visit the
  vertices of the graph
• A graph-searching algorithm can discover much
  about the structure of a graph
• Other graph algorithms are organized as simple
  elaborations of basic graph-searching algorithms

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Representations of graphs

• Two standard ways to represent a graph G (V, E)
• As a collection of adjacency list preferred when
  graph is sparse
• As a collection of adjacency matrix preferred
  when graph is dense
• Either way is applicable to both directed and
  undirected graph

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Adjacency-list representation
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Adjacency-list representation

• Adjacency-list representation of a graph G (V,
  E) consists of an array Adj of V lists, one for
  each vertex in V.
• Sum of the lengths of all the adjacency lists
• Directed graph E
• Undirected graph 2E
• Memory requirement
• both directed and undirected T(V E)
• Adjacency lists can readily be adopted to
  represent weighted graph simply stores weight
  with vertex

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Adjacency-matrix representation
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Adjacency-matrix representation

• Adjacency-matrix representation of a graph G
  consists of a V x V matrix such A aij that
• Aij 1 if(I, j) ? E, 0 otherwise
• Transpose of a matrix A (aij) to be AT aTij
  (aji)
• Since in an undirected graph A AT, memory
  needed to store the graph almost in half
• Simplicity of an adjacency matrix may make it
  preferable when when graph are resonably small
• When a graph is undirected, adjacncy matrix uses
  only one bit per entry

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Breadth-first search

• Simplest algorithms for searching a graph
• Breadth-first search systematically explores the
  edges of graph to discover every vertex that
  reachable from source vertex
• Discovers all vertices at distance k from s
  before discovering any vertices at distance k1
• u is the predecessor or parent of v Vertex v
  discovered in the course of scanning
  adjacency-list of already discovered vertex u

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• BFS(G, s)
• 1 for each vertex u ? VG s
• 2 do coloru lt- WHITE
• 3 du lt- 8
• 4 pu lt- NIL
• 5 Colors lt- GRAY
• 6 ds lt- 0
• 7 ps lt- NIL
• 8 Q lt- Æ
• 9 ENQUEUE(Q, s)
• 10 while Q ¹ Æ
• 11 do u lt- DEQUEUE(Q)
• 12 for each v Î Adju
• 13 do if colorv WHITE
• 14 then colorv lt- GRAY
• 15 dv lt- du 1
• 16 pv lt- u
• 17 ENQUEUE(Q, v)
• 18 Coloru lt- BLACK

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Figure 22.3
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Figure 22.3
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Figure 22.3
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Figure 22.3
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Figure 22.3
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BFS Analysis

• Operations of enqueueing and dequeueing take O(1)
  time, so the total time devoted to queue
  operation is O(V)
• Total time spent in scanning adjacency lists is
  O(E)
• Total running time of breadth-first search is O(V
  E)

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Shortest Paths

• Shortest path distance d(s, v) from s to v as
  the minimum number of edges in any path from
  vertex s to vertex v
• If there is no path from s to v, then d(s, v)
  8
• Shortest path a path length d(s, v) from s to v
  is said to be shortest path
• Breadth first search actually computes shortest
  path distances

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Lemma 22.1

• Let G (V, E) be a directed or undirected graph,
  and let s?V be an arbitary vertex
• Then, for any edge (u,v) ? E, d(s,v) ?
  d(s,u)1
• Proof If u is reachable from s, then so is v. In
  this case, the shortest path from s to v cannot
  be longer than the shortest path from s to u
  followed by the edge(u,v), and thus the
  inequality holds. If u is not reachable from s,
  then d(s,u) ?, and the inequality holds

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Lemma 22.2

• Let G(V,E) be a directed or undirected graph,
  and suppose that BFS is run on G from a given
  source vertex s ? V
• Then upon termination, for each vertex v ? V, the
  value dv computed by BFS satisfies dv ?
  d(s,v)

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Lemma 22.3

• Suppose that during the execution of BFS on a
  graph G(V,E), the queue Q contains the vertices
  ltv1, v2, , vrgt, where v1 is the head of Q and
  vr is the tail.
• Then, dvr ? dv11 and dvi ? dvi12 for i
  1, 2, . , r-1

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Corollary 22.4

• Suppose that vertices vi and vj are enqueued
  during the execution of BFS, and that vi is
  enqueued before vj
• Then dvi ? dvj at the time that vj is
  enqueued
• Proof immediate from Lemma 22.3 and the property
  that each vertex receives a finite d value at
  most once during the course of BFS

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Theorem 22.5(Correctness of breadth-first search)

• Let G(V,E) be a directed or undirected graph,
  and suppose that BFS is run on G from a given
  source vertex s ? V
• Then, during its execution, BFS discovers every
  vertex v ? V that is reachable from the source,
  and upon termination, dv d(s,v) for all v ? V

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Breadth-first trees

• Predecessor subgraph of G as Gp(Vp,Ep), where
• Vp v ? V pv? NIL ? s
• Ep (pv,v) v ? Vp- s
• The predecessor subgraph Gp is a beadth-first
  tree if Vp consists of the vertices reachable
  from s and, for all v ? Vp, there is a unique
  simple path from s to v in Gp that is also a
  shortest path from s to v in G

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Lemma 22.6

• When applied to a directed or undirected graph G
  (V, E), procedure BFS constructs p so that the
  predecessor subgraph Gp (Vp,Ep) is a
  breadth-first tree

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Depth-first search

• In depth-first search, edges are explored out of
  the most recently discovered vertex v that still
  has unexplored edges leaving it
• Predecessor subgraph of a depth-first search
  forms a depth-first forest composed of several
  depth-first tree
• Gp(V, Ep) where Ep (pv,v) v?V and
  pv?NIL

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Depth-first search(cont)

• Besides creating a depth-first forest,
  depth-first search also timestamps each vertex
• Each vertex has two timestaps
• dv timestamp records when v is first
  discovered
• fv timestamp records when the search finishes
  examining vs adjacency list
• Timestamps are integers between 1 and 2V
• For every vertex v, dv lt fv
• The running time of DFS_VISIT is Q(E)
• the running time of DFS Q(VE)

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DFS(G) for each vertex u ? VG do coloru lt-
WHITE pu lt- NIL time lt- 0 for each vertex u ?
VG do if coloru WHITE then
DFS-VISIT(u) DFS-VISIT(u) coloru lt-
GRAY ?White vertex u has just been
discovered time lt- time 1 du lt- time for each
v ? Adju ? Explore edge (u, v) do if colorv
WHITE then pv lt- u DFS-VISIT(v) coloru
lt- BLACK ? Blacken u it is finished fu lt-
time lt- time 1
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Figure 22.4
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Figure 22.4
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Figure 22.4
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Figure 22.4
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Figure 22.4
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Figure 22.4
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Figure 22.4
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Figure 22.4
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Properties of depth-first search

• The predecessor subgraph Gp does indeed from a
  forest of trees, since the structure of the depth
  first-tree exactly mirrors the structure of
  recursive calls of DFS-VISIT
• Discovery and finishing times have parenthesis
  structure (figrure 22.5)

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Threorem 22.7 (Parenthesis theorem)

• In any depth-first search of a (directed of
  undirected) graph G (V, E), for any two
  vertices u and v, exactly one of the following
  three conditions hold
• The intervals du,fu and dv,fv are
  entirely disjoint, and neither u nor v is a
  descendant of the other in the depth-first forest
• ZThe intervals du,fu is contained entirely
  within the interval dv,fv, and u is a
  descendant of v in a depth-first tree,or
• The intervals dv,fv is contained entirely
  within the interval du,fu, and u is a
  descendant of u in a depth-first tree

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Figure 22.5
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Figure 22.5
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1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 (s (z (y (x x) y) (w w) z)
s) (t (v v) (u u) t)
(b)
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Figure 22.5
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t
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Corollary 22.8 (Nesting of descendants
intervals)

• Vertex is a proper descendant of vertex u in the
  depth-first forest for a (directed or undirected)
  graph G if and only if du lt dv lt fv lt fu
• Immediate from Theorem 22.7

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Threorem 22.9 (White-path theorem)

• In a depth-first forest of a (directed or
  undirected) graph G (V, E), vertex v is a
  descendant of vertex u if any only if at the time
  du that the search discovers u, vertex v can be
  reached from u along a path consisting entirely
  of white vertices

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Classification of edges

• Tree edges edges in the depth-first forest Gp.
  Edge(u,v) is a tree edge if v was first
  discovered by exploring edge(u, v)
• Back edges edges (u, v) connecting a vertex u to
  an ancestor v in a depth-first tree(include
  self-loop)
• Forward edges nontree edges (u, v) connecting a
  vertex u to a descendant v in a depth-first tree
• Cross edges all other edges

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Theorem 22.10

• In a depth-first search of an undirected graph G,
  every edge of G is either a tree edge or a back
  edge.

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Topological sort

• Topological sort of a dag G (V, E)
• Linear ordering of all its vertices such that if
  G contains an edge(u, v), then u appears before v
  in the ordering
• Can be viewed as an ordering of its vertices
  along a horizontal line so thatall directed edges
  go from left to right
• Different usual kind of sorting(ex. Quicksort)

• TOPOLOGICAL-SORT(G)
• call DFS(G) to compute finishing times fv for
  each vectex v
• as each vertex is finished, insert it onto the
  front of a linked list
• return the linked list of vertices

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Figure 22.7
undershorts
socks
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watch
9/10
pants
shoes
12/15
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shirt
1/8
belt
6/7
tie
2/5
jacket
3/4
(a)
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Figure 22.7
undershorts
pants
belt
shirt
tie
jacket
shoes
socks
watch
11/16
12/15
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1/8
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3/4
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(b)
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Lemma 22.11

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

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Figure 22.8
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Theorem 22.12

• TOPOLOGICAL-SORT(G) produces a topological sort
  of a directed acyclic graph G.

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Strongly connected components

• Many algorithm that work with directed graphs
  begin with decomposing a directed graph into its
  strongly connected components

• STRONGLY-CONNECTED-COMPONENTS(G)
• call DFS(G) to compute finishing times fu for
  each vertex u
• compute Gt
• call DFS(Gt), but in the main loop of DFS,
  consider the vertices in order of decreasing
  fu(as computed in line 1)
• Output the vertices of each tree in the
  depth-first forest formed in line 3 as a separate
  strongly connected component

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Figure 22.9
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Figure 22.9
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Figure 22.9
cd
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Lemma 22.13

• Let C and C be distinct strongly connected
  components in directed graph G (V,E), let
  u,v?C, let u, v ?C, and suppose that there is
  a path ugtu in G. Then there cannot also be a
  path v gtv in G.

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Lemma 22.14

• Let C and C be distinct strongly connected
  components in directed graph G(V,E). Suppose
  that there is an edge (u,v)?ET. where u?C
  andv?C. Then f(C)ltf(C).
• Proof 2 cases depending on which strongly
  connected component, C or C, had the first
  discovered vertex during the DFS.

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Corollary 22.15

• Let C and C be distinct strongly connected
  components in directed graph G(V,E). Suppose
  that there is an edge (u,v)?ET. where u?C
  andv?C. Then f(C)ltf(C).
• Proof Since(u,v)?ET, we have (v,u)?E. Since the
  strongly connected components of G and GT are the
  same, Lemma 22.14 implies that f(C)ltf(C).

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Theorem 22.16

• STRONGLY-CONNECTED-COMPONENTS(G) correctly
  computes the strongly connected components of a
  directed graph G.