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Elementary Graph Algorithms
CONTENTS
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

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Complexity Analysis (Section 3.2)
Search Algorithms (Section 3.4)
Topological Sorting (Section 3.4)
Flow Decomposition (Section 3.5)
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Complexity Analysis
 How to judge the goodness of an algorithm?
 How economically it uses the computing resources:
 Time efficiency
 Space efficiency
 Time is more crucial in practice.
 How to judge the time requirement of an algorithm?
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Two Type of Analysis
 Empirical Analysis
 Implementation dependent
 Often inconclusive
 Too expensive and time-consuming
 Worst-Case Analysis
 Obtain upper bounds on the number of steps
 Often easy to perform
 Mostly conclusive
 Bad examples determine the performance
 Most popular
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Worst-Case Analysis
 This type of analysis determines an upper bound on the
number of steps as a function of the problem size.
 The size of a problem is the number of bits required to
state the problem.
 The size of a network problem is a function of n, m, log
C, and log U, where C denotes largest cost and U
denotes the largest capacity.
 If each piece of data (cost and capacity) fits into a
single word, then the size of a network problem is a
function of n and m.
 The worst-case analysis for a network optimization
algorithm determines an upper bound on the number of
steps as a function of n, m, log C, and log U.
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Big O Notation
 Estimate the number of steps required to execute the
following program:
for i : = 1 to n do
for j : = 1 to n do
cij : = aij + bij;
 The precise number of steps is implementation dependent
and is very difficult to count.
 MORALE:
 Ignore the constants
 Count each operation as one step
 Consider the most dominant term when problem size
becomes sufficient large
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Order Notation
 An algorithm is said to run in O(f(n)) time if for some
numbers c and n0, the time taken by the algorithm is at
most cf(n) for all n  n0.
 Polynomial Time Algorithms
 Examples: O(n2), O(nm), O(m + n log n), O(m log C).
 Exponential Time Algorithms
 Examples: O(2n), O(n!)
 Pseudo-Polynomial Time Algorithms
 Examples: O(m+nC).
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Growth Rates of Various Functions
n
log n
n.5
n2
n3
2n
n!
10
3.32
3.16
102
103
103
3.6x106
100
6.64
10.00
104
106
1.27x1030
9.33x10157
1000
9.97
31.62
106
109
1.07x10301
4.02x102567
10,000
13.29
100.00 108
1012
0.92x103010
2.85x1035689
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Big  and Big  Notation
 An algorithm is said to run in (f(n)) time if for some
numbers c’ and n0 and for all n  n0, the algorithm takes at
least cf(n) time for some problem instances.
 Whereas Big O notation gives an upper bound on the
running time of an algorithm, Big  notation gives a
lower bound on the running time of an algorithm.
 The bubble sort algorithm is a (n2) time algorithm.
 An algorithm is said to be a (f(n)) algorithm if the
algorithm is both O(f(n)) and (f(n)).
 The bubble sort algorithm is a (n2) time algorithm.
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Search Algorithms
 Search algorithms are used to find all nodes in a network
satisfying a particular property.
 Example: Find all nodes in a network that are reachable
by directed paths from a specific node s, called the
source node.
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4
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s 1
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3
9
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Basic Idea
 Start at the source node and try to reach more and more
nodes.
 Marked Node: A node which is known to be reachable from
the source node.
 Unmarked Node: A node which is not marked.
 Admissible Arc: An arc (i, j) is called an admissible arc if
node i is marked and node j is unmarked.
 PROPERTY. If an arc (i, j) is admissible, then node j can be
marked.
s
i
Marked
Marked
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j
Unmarked
Numerical Example
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s 1
7
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2
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s 1
6
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11
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The Algorithm
algorithm search;
begin
unmark all nodes in N; mark node s;
pred(s) : = 0; LIST : = {s};
while LIST is non-empty do
begin
select a node i in LIST;
if node i has an admissible arc (i, j) then
begin
mark node j; pred(j) : = i;
add node j to LIST;
end
else delete node i from LIST;
end;
end;
 The predecessor indices define a directed-out-tree, called
the search tree.
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Running Time Analysis
 How to identify admissible arcs emanating from a node
quickly?
 current[i] : the next arc emanating from node i yet to be
examined
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5
4
1
7
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 This data structure examines an arc at most once; thus the
running time of the algorithm is O(m).
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Breadth-First and Depth-First Searches
Breadth-First Search: Maintain LIST as a queue. Nodes are
examined in the FIFO order.
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s 1
4
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5
1
6
7
6
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Depth-First Search: Maintain LIST as a stack. Nodes are
examined in the LIFO order.
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1
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Additional Applications
 An O(m) algorithm is possible for each following problem:
 Find all nodes that can reach a specified sink node to
using directed paths
 Identifying whether an undirected network is connected
 Identifying whether a directed network is connected
 Identifying all components of a directed network
 Identifying whether a network is strongly connected
 Identifying whether a network is bipartite
 Determining whether a directed network is acyclic
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Knight’s Tour Problem
 Determine a tour of a knight, if it exists, that starts at the
square denoted by s and ends at the square denoted by t and
does not visit any shaded square.
 Determine a Knight’s tour, if it exists, that visits the minimum
number of squares.
s
t
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Determining Acyclicity of a Network
 Topological Ordering: A graph G = (N, A) is said to have a
topological ordering if its nodes can be numbered so that for
each arc (i, j)  A, i < j.
 A graph may or may not have a topological ordering.
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Some Properties
 A graph containing a directed cycle cannot be topologically
ordered.
i
i<j
k<i
k
j
j<k
 An acyclic graph can always be topologically ordered.
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Numerical Examples
 Determine a topological order of the following graphs:
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Algorithm Description
Step 0: Compute indegree[i] for each node i. Let S be the set
of all nodes with zero indegree.
Step 1: Select a node i from S and label it.
Step 2: Delete node i and all arcs emanating from it. Update S
and go to Step 1.
THEOREM. The algorithm topologically sorts the network in
O(m) time.
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Flow Decomposition
While formulating network flow problems, we can adopt either
of two equivalent approaches: (a) flows on arcs; and (b) flows
on paths and cycles.
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1
0
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Arc flow x = {xij : (i, j)  A}
A path and cycle flow
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Notation
 P
P
 W
W
: a directed path
: collection of all directed paths
: a directed cycle
: collection of all directed cycles
 Path and Cycle Flow:
 f(P) for every P  P, and
 f(W) for every W 
W
 1 if (i, j)  W
ij (W)  
 0 if (i, j)  W
 1 if (i, j)  P
ij (P)  
 0 if (i, j)  P
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Decomposition Theorem
THEOREM 1: Every path and cycle flow has a unique
representation as nonnegative arc flows.
PROOF:
xij 
  (P) f(P)    ( W ) f( W )
ij
PP
ij
W W
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2
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2
1
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3
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3
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Decomposition Theorem (contd.)
THEOREM 2: Every nonnegative arc flow x can be expressed
as a path and cycle flow with the following properties:
(a) Each directed path with positive flow connects a supply
node to a demand node.
(b) At most (n+m) paths and cycles have nonzero flow; out of
these, at most m cycles have nonzero flows.
PROOF:
 First identify directed paths from supply nodes to demand
nodes using some search algorithm and eliminate flow on
them.
 When no such paths are left, identify directed cycles and
eliminate flows. We can eliminate at most m cycles, and at
most (n+m) cycles plus paths.
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Numerical Example
 Determine a flow decomposition for the following arc flow:
xij
i
2
13
j
4
11
2
1
13
5
4
3
1
25
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