Some Graph Algorithms

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Some Graph Algorithms
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1. Topological sort

• 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
• This is a partial orderingof the tasks

• Some possible total orderings
• A B E D H I F C G
• B A E D H G I F C
• A B E D H I G F C

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Informal algorithm

• Extracting a total ordering from a partial
  ordering is called a topological sort
• Heres the basic idea
• Repeatedly,
• Choose a node all of whosepredecessors have
  alreadybeen chosen

C
G

• Example
• 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

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Implementing topological sort

• 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) in-edges, 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

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Using buckets

• We can start with an array of linked lists
  arrayn points to the linked list of nodes with
  n in-edges
• At each step,
• Remove a node N from array0
• For each node M that N points to,
• Get the in-degree d of node M
• Remove node M from bucket arrayd
• Add node M to bucket arrayd-1
• Quit when bucket array0 is empty
• As always, it doesnt make sense to use a high
  efficiency (but more complex) algorithm if the
  problem size is small

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2. Connectivity

• 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 connectivity table tells us whether it is
  possible to get from one node to another by
  following one or more edges

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Transitive closure

• 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

• Warshalls algorithm is a systematic method of
  finding the transitive closure of a graph

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Warshalls algorithm

• 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
• Warshalls 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 B1
• Warshalls observation makes it possible to avoid
  most of the searching that would otherwise be
  required

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Warshalls algorithm Implementation

• for (i 1 i lt N i)
• for (j 1 j lt N j)
• if (aji)
• for (k 1 k lt N k)
• if (aik) ajk true
• Its easy to see that the running time of this
  algorithm is O(N3)

Algorithm adapted from Algorithms in C by Robert
Sedgewick
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3. All-pairs shortest path

• Closely related to Warshalls algorithm is
  Floyds 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 c1c2
• 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)

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State graphs

• 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

• The edges should be labelled with the causes of
  the state transitions, but in this example they
  are too verbose

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4. Automata

• 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

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Operation of an automaton

• An automation represents a program to accept or
  reject a sequence of inputs
• It operates as follows
• 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
• 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.

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Example automaton

• 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

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Example automaton

• A hard-wired automaton is easy to implement in
  a programming language

• state q0loop 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 caseend loop

A non-hard-wired automaton can be implemented
as a directed graph
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5. Nondeterministic automata

• A nondeterministic automaton is one in which
  there may be more than one out-edge with the same
  label

• 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
• 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

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6. String searching

• Automata can be used to implement efficient
  string searching
• Example Search ABAACAAABCAB for ABABC

• The stands for everything else

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The End