Introduction to Algorithms
Elementary Graph Algorithms
My T. Thai @ UF
Outline
 Graph representation
 Graph-searching algorithms
 Breadth-first search
 Depth-first search
 Topological sort
 Strongly connected components
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Graphs
 Given a graph G=(V, E)
 V: set of vertices
 E: set of edges
 Types of graphs:





Undirected:
Directed:
Weighted: each edge e has a weight w(e)
Dense:
Sparse: |E| = o(V2)
 |E| = O(|V|2)
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Representations of graphs
 Two common ways to represent for algorithms:
 Adjacency lists
 Adjacency matrix
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Adjacency lists
 Array Adj of |V| lists, one per vertex
 List of vertex u, Adj[u], include all vertices v
such that
 If edges have weights, can put the weights in
the lists
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Adjacency matrix
 Store adjacent matrix |V| × |V| matrix A = (aij )
 If edges have weights, aij is the weight of edge
(i, j)
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Adjacency lists vs Adjacency matrix
Adjacency lists
Adjacency matrix
 Space:
 Time: to list all vertices adjacent to u:
 Time: to determine if (u, v) ∈ E:
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Graph searching algorithms
 Visiting all the nodes in a graph in a particular
manner following the edges of the graph
 Used to discover the structure of the graph
 Two common searching algorithms:
 Breadth-first search (BFS): visit the sibling nodes
before visiting the child nodes (visit node layer by
layer based on the distance to the root)
 Depth-first search (DFS): visit the child nodes
before visiting the sibling nodes (follow the “depth”
of the graph)
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Breadth-first search
 Input: Graph G = (V, E), either directed or
undirected, and source vertex s ∈ V
 Output:
 d[v] = distance (smallest # of edges, or shortest path)
from s to v, for all v  V

such that (u, v) is last edge on shortest path s
to v

u is v’s predecessor
 Builds breadth-first tree with root s that contains all
reachable vertices
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Breadth-first search
 Discover vertices in a wave manner from the
source s
 First hit all vertices 1 edge from s, then vertices 2 edges
from s …
 A vertex is discovered at the first time that it is encountered during
the search.
 A vertex is finished if all vertices adjacent to it have
been discovered.
 Colors the vertices to keep track of progress.
 White: Undiscovered.
 Gray: Discovered but not finished.
 Black: Finished.
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Queue Q contains all
discovered but not finished
vertices – gray vertices
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Example
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Analysis of BFS
 Initialization takes O(V)
 In while loop
 Each vertex is enqueued and dequeued at most once =>
total time for queuing is O(V)
 The adjacency list of each vertex is scanned at most
once. The sum of lengths of all adjacency lists is (E)
=>Total running time: O(V+E) linear in the size of
the adjacency list representation of the graph
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Correctness of BFS
Where
is the shortest distance from s to v
Proof (sketched):


:
 If
t is the sequence of vertices in
the queue, then
 If v is reached from uMywhere
d.u =
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Depth-first search
 Input: G = (V, E), directed or undirected. No
source vertex given.
 Output: 2 timestamps on each vertex:
 d[v] = discovery time
 f [v] = finishing time
 [v] : predecessor of v = u, such that v was
discovered during the scan of u’s adjacency list
 Build depth-first forest comprising several
depth-first trees
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Depth-first search
 From a vertex v
 Follow an outgoing edge to explore
 Keep searching as deep as possible
 When all vertices reachable through the edge is
explored, follow another outgoing edge of v
 When all vertices reachable from the
original source are discovered, choose an
undiscovered vertex as a new source then
repeat the process
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Depth-first search
white: undiscovered
gray: discovered
black: finished
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Example
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Analysis of DFS
 In DFS: loops on lines 1-2 & 5-7 take (V)
time, excluding time to execute DFS-Visit
 DFS-Visit is called once for each white
vertex vV when it’s colored gray the first
time. Lines 3-6 of DFS-Visit is executed
|Adj[v]| times. The total cost of executing all
DFS-Visit is vV|Adj[v]| = (E)
 Total running time of DFS is (V+E)
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Parenthesis Theorem
Theorem 22.7 For all u, v, exactly one of the following holds:
1. d[u] < f [u] < d[v] < f [v] or d[v] < f [v] < d[u] < f [u] and
neither u nor v is a descendant of the other.
2. d[u] < d[v] < f [v] < f [u] and v is a descendant of u.
3. d[v] < d[u] < f [u] < f [v] and u is a descendant of v
 d[u] < d[v] < f [u] < f [v] cannot happen
 Like parentheses:
 OK: ( ) [ ] ( [ ] ) [ ( ) ]
 Not OK: ( [ ) ] [ ( ] )
Corollary 22.8
v is a proper descendant of u if and only if d[u]<d[v]<f [v]<f [u]
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Example of parenthesis theorem
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Proof of Parenthesis Theorem
 W.l.g suppose d[u] < d[v], there are two cases:
 d[v] < f[u]
 v is discovered while u is gray  v is descendant of u
 v is discovered more recently than u  v is finished
before the algorithm comes back and finishes u
 Interval [d[v], f[v]] is included in [d[u], f[u]]
 d[v] > f[u]
 d[v] < f[v]
 Two intervals [d[u], f[u]] and [d[v], f[v]] are disjoint
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White-path theorem
Theorem 22.9 In a depth-first forest of a (directed or undirected)
graph G =(V,E), vertex v is a descendant of vertex u if and
only if at the time d[u] that the search discovers u, there is a
path from u to v consisting entirely of white vertices.
Proof:
 => at time d[u], all vertices on the path from u to v, including
v are descendant of u

Apply corollary 22.8, d[u] < d[v], these vertices are white at time d[u]
 <= Suppose there is a white path from u to v but v is not a
descendant of u
 Let v’ the nearest vertex to u on the path s.t. v’ is not descendant of u
 w is the prior vertex of v’ on the path  w is descendant of u  d[u]
< d[w] < d[v’] < f[v’] < f[w]< f[u]  v’ is the descendant of u
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Classification of edges
 Tree edge: in the depth-first forest. Found by exploring
(u, v)
 Back edge: (u, v), where u is a descendant of v
 Forward edge: (u, v), where v is a descendant of u, but
not a tree edge
 Cross edge: any other edge. Can go between vertices in
same depth-first tree or in different depth-first trees
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
 Directed acyclic graph (DAG)
 A directed graph with no cycles
 Good for modeling processes and structures that
have a partial order, edge (u, v) represent order u>v
 Topology sort: make a total order of a given
DAG such that vertex u appears before v if
there is edge (u, v)
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Example
Order for
getting dressed
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Characterizing DAGs
Lemma 2.11. A directed graph G is acyclic if and only if a
depth-first search of G yields no back edges.
Proof:
 ⇒: Show that back edge ⇒cycle
 Suppose there is a back edge (u, v) => v is ancestor of u in depthfirst forest => there is a path from v to u => there is a cycle
 ⇐: Show that cycle ⇒back edge
 Suppose G contains cycle c. Let v be the first vertex discovered in
c, and let (u, v) be the preceding edge in c.
 At time d[v], vertices of c form a white path v to u
 By white-path theorem, u is descendant of v in depth-first forest
=>(u, v) is a back edge.
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TOPOLOGICAL-SORT
 Running time: (V + E)
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Strongly connected components
 Given directed graph G = (V, E), a strongly
connected component (SCC) of G is a maximal
set of vertices C ⊆ V such that for all u, v ∈ C,
there are paths both from u to v and from v to u
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Component Graph
 GSCC = (VSCC, ESCC).
 VSCC has one vertex for each SCC in G
 ESCC has an edge if there’s an edge between the
corresponding SCC’s in G
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GSCC is a DAG
Proof
 Suppose there is a path v v in G
 there are paths u u v and v v u in G
 u and v are reachable from each other, so they
are not in separate SCC’s.
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Transpose of a Directed Graph
 GT = transpose of directed graph G
 GT = (V, ET), ET = {(u, v) : (v, u)  E}.
 GT is G with all edges reversed.
 Can create GT in Θ(V + E) time if using
adjacency lists.
 Observation: G and GT have the same SCC’s.
(u and v are reachable from each other in G if
and only if reachable from each other in GT.)
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Algorithm to compute SCCs
 Time: (V + E)
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Example
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How does it work?
 Idea:
 By considering vertices in second DFS in decreasing order of
finishing times from first DFS, we are visiting vertices of the
component graph in topologically sorted order.
 Because we are running DFS on GT, we will not be visiting any v
from u, where v and u are in different components.
 Notation:




d[u] and f [u] always refer to first DFS.
Extend notation for d and f to sets of vertices U  V:
d(U) = minuU{d[u]} (earliest discovery time)
f (U) = maxuU{ f [u]} (latest finishing time)
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SCCs and DFS finishing times
Proof:
 Case 1: d(C) < d(C)




Let x be the first vertex discovered in C.
At time d[x], all vertices in C and C are white.
=> there exist paths of white vertices from x to
all vertices in C and C.
By the white-path theorem, all vertices in C and
C are descendants of x in depth-first tree.
By the parenthesis theorem, f [x] = f (C) > f(C).
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SCCs and DFS finishing times
Proof:
 Case 2: d(C) > d(C)





Let y be the first vertex discovered in C
At time d[y], all vertices in C are white and
there is a white path from y to each vertex in C
 all vertices in C become descendants of y
Again, f [y] = f (C).
At time d[y], all vertices in C are also white.
By earlier lemma, since there is an edge (u, v),
we cannot have a path from C to C => no
vertex in C is reachable from y => at time f [y],
all vertices in C are still white
=> for all w  C, f [w] > f [y] => f (C) > f (C)
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SCCs and DFS finishing times
Proof:
 (u, v)  ET  (v, u)  E
 Since SCC’s of G and GT are the same, f(C) >
f (C), by Lemma 22.14.
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Correctness
 When we do the second DFS, on GT, start with
SCC C such that f(C) is maximum.
 The second DFS starts from some x  C, and it
visits all vertices in C.
 Corollary 22.15 says that since f(C) > f (C) for all
C  C, there are no edges from C to C in GT.
=> DFS will visit only vertices in C.
=> the depth-first tree rooted at x contains exactly the
vertices of C.
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Correctness
 The next root chosen in the second DFS is in SCC C such
that f (C) is maximum over all SCC’s other than C.
 DFS visits all vertices in C, but the only edges out of C go to C,
which we’ve already visited.
 Therefore, the only tree edges will be to vertices in C.
 Each time we choose a root for the second DFS, it can
reach only
 vertices in its SCC—get tree edges to these,
 vertices in SCC’s already visited in second DFS—get no tree edges
to these.
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Summary
 Two common ways to represent a graph: adjacency
lists and adjacency matrix. Choosing representation
depends on:
 Type of graph: dense or sparse
 Type of operation that is used frequently in the problem
 adjacency lists saves more space than adjacency matrix
 BFS starts from a given source and computes the
shortest paths from the source to other vertices. Some
vertices may be not discovered
 BFS starts from an arbitrary vertex and discovers the
whole graph to capture the graph’s structure
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 Both BFS and DFS can provide a set of reachable
vertices from a given vertex
 Topology sort is used to form the total order from
partial orders
 Strongly connected components list all strongly
connected components in their topology order
 BFS, DFS, Topology Sort, Strongly Connected
Component run in linear time in term of the graph’s
size
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