On Mathematical Contents of Computer Science Contests

1
On Mathematical Contentsof Computer Science
Contests

• Nikolay V. Shilov
• (Computer Science at KAIST Mathematics and
  Mechanics at Novosibirsk State University,
  Russia)
• Svetlana O. Shilova
• (Novosibirsk Institute of Mathematics, Russia)

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Science Contests and Olympiads

• Science Olympiads and contests brings
  spirit of competitiveness to
  Science education. They
  benefit best students, engage
  them with research, recruit new
  talented researchers and engineers.
• Simultaneously Science Olympiads
  and contests challenge faculty
  to enhance teaching
  so that regular
  student can enjoy Olympiad problems.

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Situation with Computer Science Contests

• Computer Science contests become very popular
  with undergraduate students. Simultaneously they
  become more similar to a technical sport than to
  Science Olympiad.
• Three corner-stones of a success in these
  contests are
• art of problem formal modeling,
• cooking book of algorithms in heads,
• rapid typing skills in hands.

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Situation with Computer Science Contests

• The art of modeling is especially important since
  it is about research (not about technical
  proficiency), it puts Computer Science contests
  in line with science Olympiad like Mathematics
  and Physics Olympiads.
• Research component in Computer Science is not
  limited by art of modeling. It includes formal
  mathematical proofs of model properties and
  program correctness. Without proofs, a utility of
  some programs is very conventional (in spite of
  extensive testing).

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Situation with Computer Science Contests

• We discuss a particular problem that fits
  Mathematics Olympiad and Computer Science
  contests format simultaneously. But there is a
  number of other Computer Science problems that
  fit contest format, and that extend Computer
  Science by interesting Mathematics.
• Some of these problems can become student
  research topics and can help to overcome
  alienation when Computer Science students
  consider Mathematics being too pure, and
  Mathematics students consider Computer Science
  being too poor.

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Sample Problem I Balancing Coins

• Math All valid coins have equal standard weight,
  a fake
• coin has another weight. Is it possible to
  identify a unique
• fake in a set of 13 coinsbalancing them 3 times
  at most?

• Comp. Sci Write a program that inputs a number
  of coins
• M and balance limit N and outputs
• impossible, if it is impossible to identify a
  unique fake in set of M coins balancing them N
  times
• an executable program that implements a scenario
  for fake coin identification otherwise.

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Sample Problem II Black and White vs. Mars
Mission

• Math There are N black and N white points on the
  plane,
• non three are collinear. Prove that it is
  possible to couple
• black and white points in 1-1 manner by intervals
  without
• intersections.

• Comp. Sci There is a number of robots in a
  crater on Mars and
• the same number of shades. It is state of a sandy
  storm alarm.
• Every robot has no obstacle to move to every
  shade directly by a
• straight line. To exclude collisions,
  intersections of routes are
• prohibited. Write a program that inputs positions
  of robots and
• shades and then assigns individual shade for
  every robot.

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Mathematics solves Mars Mission Problem

• Geometric model for Mars Mission is a set of N
  black and N
• white points on the plane without collinear
  triples.
• Mathematical approach to Mars Mission problem
  consists
• in the following steps
• proving existence of any intersection-free
  coupling
• extracting algorithm from existential proof
• implementing the algorithm.
• Proving and algorithm engineering are research
• components, but testing of implementation can not
  assess
• neither proving nor originality of algorithm.

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Computer Science solves Mars Mission Problem

• But one can think about pure Computer Science
  approach
• to Mars Mission problem.
• There are two (at least) choices for algorithm
  design
• without preliminary proving of model properties
• exhaustive search,
• heuristic search.
• Let us examine both at some extend...

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Computer Science solves Mars Mission Problem

• The most stupid exhaustive search can be
  described at
• informal level as follows
• In this informal algorithm coupling is any set
  of N intervals
• that couples black and white points in 1-1
  manner, a good
• coupling is a coupling without intersections.
  The algorithm
• is too inefficient, since it obligatory generates
  and stores all
• possible N! couplings. Termination is guaranteed,
  but
• termination with good coupling is not
  guaranteed-(

• input initial positions of BW points
• generate and store all couplings for BW points
• search a good coupling among generated.

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Computer Science solves Mars Mission Problem

• Improved exhaustive search algorithm can be
  described at
• informal level as follows
• The improved algorithm avoids storage of all
  possible
• couplings. It exploits abstract data type
  (coupling) and
• standard structured control constructs (while
  loop).
• Termination guaranties correctness (i.e. that a
  good
• coupling is found), but termination is not
  guaranteed-(

VAR X coupling Xinitial WHILE not good(X)
DO X next(X)
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Computer Science solves Mars Mission Problem

• The idea of the heuristic search for Mars Mission
  is trivial
• start with initial coupling and try to resolve
  local conflicts
• (i.e. eliminate local intersections).
• This idea can be represented as follows
• where flip is operation that finds some
  intersection (if any)
• in the coupling and switches it off.

VAR X coupling Xinitial WHILE not good(X)
DO Xflip(X)
VAR X coupling Xinitial WHILE not good(X)
DO Xnext(X)
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Computer Science solves Mars Mission Problem
?
flip
(Flip eliminates an intersection, but may
introduce several new!)
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Computer Science solves Mars Mission Problem
?
flip
?
flip
For the initial coupling depicted above the
heuristic search algorithm terminates in 3
steps.
?
flip
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Computer Science solves Mars Mission Problem

• The design of heuristic search algorithm is based
  on a
• naïve believe that local improvements can solve
  the
• problem. Termination of the algorithm guaranties
  that a
• good coupling is found. But termination is not
  guaranteed

flip
2nd interval
2nd interval
1st interval
1st interval
flip
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How Computer Science Proves Algorithm Termination

• Method of R. Floyd
• A well-founded set (WFS) is a partial order (D,?)
  without infinite decreasing sequences d1 gt d2 gt
  ...
• Assume that F is a total mapping from
  configurations of a program into WFS (D,?).
• If algorithm precondition guarantees that each
  loop iteration decreases the value of the
  function F, then it guarantees program
  termination.

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Heuristic Search How to Prove Termination

• For given N black and given N white points on the
  plain
• there exist N! different couplings. In
  particular, all possible
• 6 coupling for given 3 black and given 3 white
  points are
• depicted below by different colors.

1
5
2
4
6
1
2
6
2
3
5
3
1
1
3
4
3
5
1
4
2
2
6
3
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Heuristic Search How to Prove Termination

• For every coupling X, let F(X) be the sum ?X of
  length of all
• intervals in the coupling. For given black and
  white points
• let D be ?X X is a coupling ?R. Let ? be
  standard
• linear order on R. Then (D, ?) is WES (since D is
  finite).
• (In particular, there are at most 6 different
  values ??X1,
• ?X2,??X3,??X4,??X5 and ??X6 in the example above.)

1
5
2
4
6
1
2
6
2
3
5
3
1
1
3
4
3
5
1
4
2
2
6
3
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Heuristic Search How to Prove Termination

• It remains to remark that every application of
  flip
• to two intersecting intervals that are not on a
• straight line decreases the value of F due to
  the
• triangle inequality
• gt
  
• Hence the loop WHILE not good(X) DO Xflip(X)
• always terminates, if there is no collinear
  triples!

d
3
1
a
b
2
4
1
2
3
4
b
a
d
c
d
a
b
c
c
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Computer Science solves Mathematical Problem

• It implies that heuristic search algorithm
• always terminates with good coupling, if there
  are no
• collinear triples. It also implies that we solve
  the following
• mathematical problem

VAR X coupling Xinitial WHILE not good(X)
DO Xflip(X)
There are N black and N white points on the
plane, non three are collinear. Prove that it is
possible to couple black and white points in 1-1
manner by intervals without intersections.
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Conclusion

• Observe that
• without proof the heuristic search algorithm has
  a very doubtful value
• the proof is mathematically strict but Computer
  Science in nature.
• Unfortunately, Computer Science contests do not
  take
• proofs in to account

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Conclusion

• But we would like to hope that Mathematics and
  Computer
• Science faculty can work together toward
• better Mathematics Contents of Computer
• Science Contests. We do believe that it
• enhance Mathematics and Computer
• Science university teaching.