Voronoi game

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Voronoi game

• Daniel Graff
• Seminar über Algorithmen

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Content

1. Introduction
2. The game
3. The circle game
4. The line segment game
5. Whites defense
6. Conclusion

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Voronoi game in 2D (1)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Game based on voronoi cells
• Played in 2D using euclidean distance
• Each player has n points
• Alternate playing
• Objective cover most of the arena
• VoronoiGameApplet.html
• Used for competitive facility location

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Voronoi game in 2D (2)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Difficult to give winning strategies for 2D
• Unless game is reduced to a sinlge round
• Focus on 1D games
• Circle game
• Line segment game

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Rules (1)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• 2 player (white and black)
• Playing n points (n gt 1)
• Alternatve playing
• arena is a curve C (open/ closed)
• White starts (like in chess)
• Points cannot lie upon each other
• W set of white points
• B set of black points

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Rules (2)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Points per round are not specified
• k number of rounds k ? n
• ?i number of points white plays in round I
• ?i number of points black plays in round I
• ? 1 ? i ? k, ?i, ?i gt 0
• ? 1 ? j ? k, ?ji1 ?i ? ?ji1 ?I
• ?ki1 ?i ?ki1 ?i n
• ?1 lt n (for the circle game)
• ?1 1 (for the line segment game)

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Rules (3)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• After all 2n points have been played
• Compute score
• W x ? C min d(x, w) lt min d(x, b)
• B x ? C min d(x, b) lt min d(x, w)

w?W
b?B
w?W
b?B
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Winning strategy
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Black always has a winning strategy
• Exclude the degenerate game, n 1
• Line segment white playes on the midpoint, black
  loose
• Circle ends in a tie (no matter where the points
  have been played)

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Definition
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• A set of W points and a set of B points has been
  placed on a closed curve
• Interval arc between two white or black points
• Monochromatic endpoints have same color
• Bichromatic endpoints have different colors
• White/ black interval monochromatic interval
• n(W) number of white intervals
• n(B) number of black intervals
• Keypoints special positions on the curve
• Key interval interval and its endpoints are
  keypoints

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Lemma 1
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• n(W) - n(B) w - b
• Proof
• Each point has two faces to each adjacent
  interval
• i number of bichromatic intervals
• 2w - i white faces facing white intervals
• 2b - i black faces facing black intervals? n(W)
  (2w i)/2 n(B) (2b i)/2? n(W) - n(B)
  w - b

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Lemma 2
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• n number of keypoints
• W set of w ? n white points
• B set of b lt w black points
• If W ? B covers all keypoints
• only one white interval (no key interval)? it
  exists a bichromatic key interval

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Lemma 2 - Proof
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• n arcs formed by n keypoints
• W ? B ? 2n - 1
• n - 1 points inside the n curve arcs? one
  interval free of points ? key interval
• Interval is not black (? Lemma 1)
• Interval in not white? only one white none key
  interval
• ? bichromatic key interval

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Keypoint strategy
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage I Black plays onto an empty keypoint
• Stage I ends after last keypoint is played
• Stage II Black plays into a white key
  interval(breaking white interval)
• Stage II ends when the last white key interval is
  broken
• Stage III Black breaks a white interval
• Stage III ends before plays his last point(not
  included in the basic keypoint strategy)

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Lemma 3
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• After Stage III there is no white key interval
• Proof
• k number of keypoints played by white
• k gt 1 (k ? 1 ? no white interval)
• ?1 lt n ? black has at least one keypoint (k lt
  n)? whits gets at most k - 1 white key intervals
• Black covers the remaining n - k keypoints (end
  of Stage I)
• Black plays k - 1 points in Stage II and III?
  after Stage II all white key intervals are
  broken(no new ones can be added)

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Lemma 4
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• After Stage III all black intervals are key
  intervals
• Proof
• End of Stage I black has only played onto
  keypoints
• Stage II and III black breaks white (key)
  intervals? creates bichromatic only
• White cannot create black intervals? all black
  intervals are key intervals

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Definition
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Circle C parameterized in 0,1
• Wm total length of all white intervals
• Bm total length of all black intervals
• Length of each bichromatic interval is divided
  equally among the players? B - W Bm - Wm?
  black wins iff Bm gt Wm
• n keypoints are the points ui i/n, i 0,1,,
  n - 1

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Circle strategy (1)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage I Black plays onto an empty keypoint
• Stage I ends after the last keypoint is played
• Stage II Black breaks a largest white interval
• Stage II ends before blacks last move

1
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Circle strategy (2)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage III (Blacks last move) Two
  possibilities(i) n(W) gt 1 ? black breaks a
  largest one(ii) n(W) 1 l length of white
  interval black plays in a bichromatic key
  interval (with distance lt (1/n) - l from white
  endpoint)

lt 0,05
l 1/5
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Proof of circle strategy (1)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Whites first move covers the first keypoint?
  Black plays onto at most n - 1 keypoints? Stage
  I ends before blacks last move
• Stage II
• At each play by black we have b lt w? Lemma 1 (
  n(W) ? 1 )
• White interval is key interval or length lt 1/n
• Stages I II implementation of keypoint strategy

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Proof of circle strategy (2)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage III (shows that black wins)
• White has played all points and n(W) ? 1
• n(W) gt 1
• Black breaks a largest white interval? n(B)
  n(W) ? 1 (Lemma 1)? all black intervals are key
  intervals (Lemma 4)? all white intervals lt 1/n
  (Lemma 3)? Bm gt Wm (black wins)
• n(W) 1
• White interval has length l lt 1/n (Lemma 3)
• It exists a bichromatic key interval (Lemma 2)
• Black places his last point there? Bm gt l Wm
  (black wins)

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Line segment game
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Game played on a line segment C, parameterized in
  0,1
• To reuse lemmas of section 2
• Extend C into a closed curve C
• Connecting 0 and 1 with a curve (border arc)
• Border interval
• Monochromatic only the parts on C is counted for
  Wm
• Bichromatic not shared equally
• Wb part on C for white
• Bb part on C for black
• Keypoints ui 1/(2n) i/n, i 0, 1, ..., n - 1

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Line strategy (1)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage I Black plays onto u0 or un-1
• Stage II Black plays onto an empty keypoint
• Stage II ends after the last keypoint is played

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1
3
4
Stage I II
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Line strategy (2)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage III
• (i) n(W) ? 1 (non-border interval) ? black
  breaks a largest non-border one
• (ii) n(W) 1 (border interval)
• (a) one white endpoint is a keypoint assume it
  is un-1 the other endpoint is 1 - l black
  places his new point in l, un-1
• (b) no white endpoints are keypoints l length
  of the white border interval black places in a
  bichromatic key interval with distance lt 1/n - l
  from white endpoint

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Line strategy (3)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stage IV (black's last move)
• (i) n(W) ? 1 ? black breaks a largest non-border
  one
• (ii) n(W) 1 l length of the white
  interval black places in a bichromatic key
  interval with distance lt 1/n - l from white
  endpoint

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Proof of line strategy (1)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Stages I II are well defined
• White cannot cover both u0 and un-1 (?1 1)
• Stage II cover all keypoints
• Stage III n(W) ? 1 (Lemma 1), maybe border
  interval
• (i) clearly well defined
• (ii)
• (a) clearly well defined
• (b) it exists a bichromatic key interval (Lemma 2)

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Proof of line strategy (2)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Scenario I
• White playes onto at least one keypoint
• Assume Stage III case (i) occurs (n(W) ? 1,
  non-border)? Stages I - III are an
  implementation of the keypoint strategy (apply
  Lemma 3 and 4)
• Assume Stage IV case (ii) occurs? white ends up
  with length l, black has gt l (black wins)
• Assume Stage IV case (i) occurs ( n(W) n(B) )
  ? ?x?n(W), x lt 1/n (Lemma 3)? ?y?n(B), y key
  interval (Lemma 4)? Wm lt Bm and border interval
  is... (i) monochromatic Wb Bb 0 ? black
  wins (ii) bichromatic Wb ? Bb 1/2n ? black
  wins

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Proof of line strategy (3)
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Scenario II
• Assume Stage III case (ii) occurs (n(W) 1,
  border)
• No key intervals for the rest of the game
• Assume Stage IV case (ii) occurs? white ends up
  with length l, black has gt l (black wins)

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Whites defense
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Margin by which black wins is very small
• White controls the margin
• Theorem White captures at least ½ - ? of the
  curve, ? ? gt 0
• Proof
• White plays n points within distance ?/2n of the
  n keypoints
• All white intervals have length at least 1/n -
  ?/n
• All black intervals have length at most 1/n ?/n
• If ( n(W) n(B) lt n ) Then Bm - Wm ? 2?(n-1)/n
• Wb ? 1/2n - ?/2n, Bb ? 1/2n ?/2n? B -W ? 2? ?
  W ? ½ - ?

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Conclusion
1. Introduction 2. The game 3. Circle game
4. Line segment game 5. Whites defense 6.
Conclusion

• Strategies for 1D competitive facility location
  problems
• Showing the 2nd player, black, to win
• White controls the margin by which black wins
• For practical purposes, 1D Voronoi game ends in a
  tie