Automated Map Labeling with an introduction to computability, computational complexity, and evolutionary algorithms

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Automated Map Labelingwith an introduction to
computability, computational complexity, and
evolutionary algorithms

• Wan Bae (Computer Science, University of Denver)
• Sada Narayanappa (Jeppesen)
• Petr Vojtechovský (Mathematics, University of
  Denver)

are working on the problem
is merely talking about it
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Map Labeling
association of labels to geographic entities

• point features (restaurants)
• line features (streets)
• boundaries (state parks)

• We focus on point features.

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Why automate?

• manual resolution of conflicts is time
  consuming
• most conflicts can be resolved by simple
  heuristics
• labeling rules easily enforced

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Parameters of labels

• font size and type
• relative position of labels and sites
• proximity to other labels and sites
• esthetic preference/tradition

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How difficult is Map labeling?
NP-complete difficult!
We (mankind) think thats very difficult!
But we really dont know!
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What is computation?
We dont know.
Church-Turing thesis Computation is precisely
what can be performed with a universal Turing
machine.
tape with symbols (0,1,B)
current state

• Depending on the state and symbol
• rewrite symbol
• move head left or right
• change state
• stop when an end-state is reached

head
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How realistic are Turing machines?

• universal Turing machine is a TM that can
  emulate any other Turing machine
• universal Turing machines exist
• digital computers (with potentially unlimited
  memory and storage) are UTMs
• we dont know of any process in nature that
• resembles computation and is not done on a TM

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Algorithm TM
Algorithm is a finite list of well-defined
instructions for accomplishing some task that,
given an initial state, will terminate in a
defined end-state.
Read list of instructions description of
TM initial state input tape end-state whats
on the tape when TM stops
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Can all problems be solved with UTM?
No! Here is a problem that cannot be solved
Halting problem Given a Turing machine T and an
input tape I, will T eventually stop when started
with I?
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Complexity of computation
Let A be an algorithm (TM). A(n) max running
time of A on an input of length n
We are interested in the behavior of A(n) for
large n. More precisely We need to find a
function f(n), such as log(n), n2, exp(n), for
which the limit of A(n)/f(n) as n approaches
infinity is equal to 1.
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Hierarchy of complexities
Depending on f(n), the algorithm is

• polynomial, P, if f(n) is a polynomial in n
• exponential, if f(n) is an exponential function
  of n
• logarithmic, linear,

Example Addition of two n-digit numbers takes
about n steps, hence the usual addition
algorithm is linear.
Example Factorization of integers into primes
the complexity is not known. (RSA Internet
security?)
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The class NP
We dont know complexities for most
algorithms we might only have lower and upper
estimates.
A problem is NP (non-deterministic polynomial) if
once a solution is presented, it requires no more
than polynomial time to verify that the solution
is correct.
The biggest open problem in CS (1,000,000) Is
NPP?
A more lucrative problem (4,000,000) Be the
first one to solve Eternity II.
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The class NP-complete
A problem X is NP-complete if

• X is NP,
• for every problem Y in NP,
• it is possible to translate input y for Y
• into an input x for X in polynomial time,
• so that the solution to Y with y is the same
• as the solution to X with x.

Briefly, X is at least as hard as any NP problem.
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3-SAT is NP-complete
NP-complete problems exist!!!
3-SAT problem Given a logical condition of the
form (A or B or C) and (D or E or F) and
, where each of A, B, C, is a variable or a
negation of a variable, is it possible to assign
values TRUE or FALSE to the variables so that
the condition evaluates as TRUE?
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Traveling salesman is NP-complete
A graph is a bunch of points (vertices) connected
by lines (edges). A graph has a Hamiltonian
cycle if it is possible to visit all of its
vertices and return to the starting vertex
without using any edge more than once.
Traveling salesman problem Does a given graph
have a Hamiltonian cycle?
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Map labeling is NP-complete
even in these situations

• determine if a labeling exists with no
  overlaps
• as above, with all labels the same
• find an optimum labeling (allow overlaps)
• determine if at least 50 of labels can be
  positioned without overlaps

It is therefore reasonable to resort to

• approximate (non-optimal) solutions
• heuristics (algorithms driven by educated
  guesses)

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Cost function
It is possible to approximate solutions if the
problem admits a cost function, that is, a
measure of how far the (partial) solution is from
an optimal solution.
For map labelings, we can measure

• of nonoverlaping labels
• total area of overlap,

How to reduce the cost?
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Reducing cost Divide and conquer
1) split the problem into smaller ones 2) solve
the smaller problems 3) piece the large solution
together
Example QUICKSORT, has complexity n log(n)
For map labelings, 1) and 2) are easy. 3) is hard.
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Reducing cost Evolutionary algorithms

• From a current situation, evolve many similar
  situations by small modifications (mutations)
• 2) Pick the best offspring and kill other
  children.
• 3) Kill the parent if the surviving child is
  better.

Example stock market software
If the situation is based on a DNA,
evolutionary algorithms are called genetic.
Easy to adopt for map labeling.
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Reducing cost What is really going on?
There are many algorithms for cost reduction. All
suffer from the same difficulty local minimum
trap
? optimal
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Reducing cost for map labelingInitial
configuration
Can we initially place the labels in a clever
way?

• random position
• preferred position (top right)
• locally optimized position

B
label1
A
D
label2
C
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Convex Hull Onion Peeling
A
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Onion Peeling is worse than random
Onion Peeling was proposed in the literature for
map labeling. But it does not work very well
(without additional changes).
Denver
Parker
the labels tend to be shifted in similar
directions
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Implementation
Java Applet by Sada and Wan