Transcript Graph Algorithms

Some Graph Algorithms
1. Topological sort
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Suppose a project involves doing a number of tasks, but
some of the tasks cannot be done until others are done
Your job is to find a legal order in which to do the tasks
For example, assume:
A must be done before D
 B must be done before C, D, or E
 D must be done before H
 E must be done before D or F
 F must be done before C
 H must be done before G or I
 I must be done before F
 Some possible total orderings:
This is a partial ordering
 A B E D H I F C G
of the tasks
 B A E D H G I F C
 A B E D H I G F C
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Informal algorithm
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Extracting a total ordering from a partial ordering is called
a topological sort
A
B
C
A
Here’s the basic idea:
B
C
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Example:
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Repeatedly,
 Choose a node all of whose
“predecessors” have already
been chosen
D
D
E
F
G
G
H
H
I
I
Only A or B can be chosen. Choose A.
Only B can be chosen. Choose B.
Only E can be chosen. Choose E.
Continue in this manner until all nodes have been chosen.
If all your remaining nodes have predecessors, then there is
a cycle in the data, and no solution is possible
Implementing topological sort
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The graph structure can be
implemented in any convenient
way
We need to keep track of the
number of in-edges at each node
Whenever we choose a node, we
need to decrement the number of
in-edges at each of its successors
Since we always want a node with the fewest (zero) inedges, a priority queue seems like a good idea
To remove an element from a priority queue and reheap it
takes O(log n) time
There is a better way
Using buckets
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We can start with an array of linked lists; array[n]
points to the linked list of nodes with n in-edges
At each step,
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Remove a node N from array[0]
For each node M that N points to,
 Get the in-degree d of node M
 Remove node M from bucket array[d]
 Add node M to bucket array[d-1]
Quit when bucket array[0] is empty
As always, it doesn’t make sense to use a high
efficiency (but more complex) algorithm if the
problem size is small
Bucket example
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Buckets:
0 → A B
1 → E G H I
2 → C F
3 → D
Buckets after choosing B:
0 → A E
1 → G H I C
2 → F D
3 →
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2. Connectivity
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Suppose you want to find out quickly (O(1) time) whether it
is possible to get from one node to another in a directed graph
You can use an adjacency matrix to represent the graph
A
E
D
G
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B
F
C
A
B
C
D
E
F
G
AB CD EF G
A
B
C
D
E
F
G
AB CD EF G
A connectivity table tells us whether it is possible to get
from one node to another by following one or more edges
Transitive closure
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Reachability is transitive: If you can get from A to E, and
you can get from E to G, then you can get from A to G
A
B
C
D
E
F
G
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AB CD EF G
A
B
C
D
E
F
G
AB CD EF G
new
Warshall’s algorithm is a systematic method of finding the
transitive closure of a graph
Warshall’s algorithm
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Transitivity: If you can get from A to B, and you can
get from B to C, then you can get from A to C
Warshall’s observation: If you can get from A to B
using only nodes with indices less than B, and you can
get from B to C, then you can get from A to C using only
nodes with indices less than B+1
Warshall’s observation makes it possible to avoid most
of the searching that would otherwise be required
Warshall’s algorithm: Implementation
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for (i = 1; i <= N; i++) {
for (j = 1; j <= N; j++) {
if (a[j][i]) {
for (k = 1; k <= N; k++) {
if (a[i][k]) a[j][k] = true;
}
}
}
}
It’s easy to see that the running time of this algorithm* is O(N3)
*Algorithm adapted from Algorithms in C by Robert Sedgewick
3. All-pairs shortest path
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Closely related to Warshall’s algorithm is Floyd’s
algorithm
Idea: If you can get from A to B at cost c1, and you can
get from B to C with cost c2, then you can get from A to
C with cost c1+c2
Of course, as the algorithm proceeds, if you find a lower
cost you use that instead
The running time of this algorithm is also O(N3)
State graphs
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The next couple of algorithms are for state graphs, in
which each node represents a state of the computation, and
the edges between nodes represent state transitions
Example: Thread states in Java
waiting
start
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ready
running
dead
The edges should be labelled with the causes of the state
transitions, but in this example they are too verbose
4. Automata
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Automata are a formalization of the notion of
state graphs
Each automaton has a start state, one or more
final states, and transitions between states
a
The start
state
A final
state
A state
transition
Operation of an automaton
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An automation represents a “program” to accept or reject a
sequence of inputs
It operates as follows:
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Start with the “current state” set to the start state and a “read head” at the
beginning of the input string;
While there are still characters in the string:
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Read the next character and advance the read head;
From the current state, follow the arc that is labeled with the character just
read; the state that the arc points to becomes the next current state;
When all characters have been read, accept the string if the current state is
a final state, otherwise reject the string.
Example automaton
1
q1
q0 1
0
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0
q2 1
1
0
0
q3
Example input string: 1 0 0 1 1 1 0 0
Sample trace: q0 1 q1 0 q3 0 q1 1 q0 1 q1 1 q0 0 q2 0 q0
Since q0 is a final state, the string is accepted
Example automaton
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A “hard-wired” automaton
is easy to implement in a
programming language
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q1
q0 1
0
0
q2 1
1
0
0
q3
• A non-hard-wired automaton
can be implemented as a
directed graph
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state := q0;
loop
case state of
q0 : read char;
if eof then accept string;
if char = 0 then state := q2;
if char = 1 then state := q1;
q1 : read char;
if eof then reject string;
if char = 0 then state := q3;
if char = 1 then state := q0;
q2 : read char;
if eof then reject string;
if char = 0 then state := q0;
if char = 1 then state := q3;
q3 : read char;
if eof then reject string;
if char = 0 then state := q1;
if char = 1 then state := q2;
end case;
end loop;
5. Nondeterministic automata
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A nondeterministic automaton is one in which there may
be more than one out-edge with the same label
a
A
a
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B
etc.
C
A nondeterministic automaton accepts a sequence of inputs
if there is any way to use that string to reach a final state
There are two basic ways to implement a nondeterministic
automaton:
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Do a depth-first search, using the inputs to choose the next state
Keep a set of all the states you could be in; for example, starting
from {A} with input a, you would go to {B, C}
6. String searching
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Automata can be used to implement efficient string
searching
Example: Search ABAACAAABCAB for ABABC
A
*
0 A 1
C*
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B
A
A
*
2
A 3
A
B 4
*
*
The “*” stands for “everything else”
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The End
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