Voronoi game Daniel Graff Seminar ber Algorithmen Voronoi

Voronoi game Daniel Graff Seminar über Algorithmen Voronoi game

Content 1. 2. 3. 4. 5. 6. 1/5/2022 Introduction The game The circle game The line segment game White‘s defense Conclusion Voronoi game 2

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Voronoi game in 2 D (1) Game based on voronoi cells Played in 2 D using euclidean distance Each player has n points Alternate playing Objective: cover most of the “arena” Voronoi. Game. Applet. html Used for competitive facility location 1/5/2022 Voronoi game 3

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Voronoi game in 2 D (2) Difficult to give winning strategies for 2 D Unless game is reduced to a sinlge round Focus on 1 D games Circle game Line segment game 1/5/2022 Voronoi game 4

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Rules (1) 2 player (white and black) Playing n points (n > 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 1/5/2022 Voronoi game 5

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Rules (2) 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 > 0 1 j k, ji=1 i ji=1 I ki=1 i = n 1 < n (for the circle game) 1 = 1 (for the line segment game) 1/5/2022 Voronoi game 6

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Rules (3) After all 2 n points have been played Compute score W = |{ x C: min d(x, w) < min d(x, b) }| w W b B B = |{ x C: min d(x, b) < min d(x, w) }| b B w W 1/5/2022 Voronoi game 7

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Winning strategy 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) 1/5/2022 Voronoi game 8

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Definition 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 1/5/2022 Voronoi game 9

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Lemma 1 n(W) - n(B) = w - b Proof Each point has two faces to each adjacent interval i = number of bichromatic intervals 2 w - i white faces facing white intervals w 2 b - i black faces facing black intervals b w n(W) = (2 w – i)/2 n(B) = (2 b – i)/2 b n(W) - n(B) = w - b w 1/5/2022 Voronoi game 10

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Lemma 2 n = number of keypoints W = set of w n white points Keypoint B = set of b < w black points If W B covers all keypoints only one white interval (no key interval) w b it exists a bichromatic key interval w b w 1/5/2022 Voronoi game 11

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Lemma 2 - Proof n arcs formed by n keypoints |W B| 2 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 w b w bichromatic key interval 1/5/2022 Voronoi game 12

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Keypoint strategy w_1 b_6 w_3 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 b_2 b_4 (breaking white interval) w_5 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) 1/5/2022 Voronoi game 13

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Lemma 3 w_1 b_6 After Stage III there is no white key interval Proof k = number of keypoints played by white b_2 k > 1 (k 1 no white interval) w_5 1 < n black has at least one keypoint (k < 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) 1/5/2022 Voronoi game w_3 b_4 14

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Lemma 4 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 1/5/2022 Voronoi game 15

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Definition 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 > Wm n keypoints are the points: ui = i/n, i = 0, 1, …, n - 1 1/5/2022 Voronoi game 16

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Circle strategy (1) 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 black´s last move 1 keypoints 6 0 4 3/4 1/4 5 1/2 7 2 3 Stage I 1/5/2022 Stage II Voronoi game 17

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Circle strategy (2) Stage III: (Black´s last move) Two possibilities: (i) n(W) > 1 black breaks a largest one (ii) n(W) = 1 l = length of white interval black plays in a bichromatic key interval (with distance < (1/n) - l from white endpoint) 8 < 0, 05 l = 1/5 Stage III 1/5/2022 Voronoi game 18

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Proof of circle strategy (1) White‘s first move covers the first keypoint Black plays onto at most n - 1 keypoints Stage I ends before black‘s last move Stage II At each play by black we have b < w Lemma 1 ( n(W) 1 ) White interval is key interval or length < 1/n Stages I + II: implementation of keypoint strategy 1/5/2022 Voronoi game 19

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Proof of circle strategy (2) Stage III (shows that black wins) White has played all points and n(W) 1 n(W) > 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 < 1/n (Lemma 3) Bm > Wm (black wins) n(W) = 1 White interval has length l < 1/n (Lemma 3) It exists a bichromatic key interval (Lemma 2) Black places his last point there Bm > l = Wm (black wins) 1/5/2022 Voronoi game 20

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Line segment game 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 border interval Wb = part on C for white Bb = part on C for black 0 1 Keypoints: ui = 1/(2 n) + i/n, i = 0, 1, . . . , n - 1 1/5/2022 Voronoi game 21

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Line strategy (1) Stage I: Black plays onto u 0 or un-1 Stage II: Black plays onto an empty keypoint Stage II ends after the last keypoint is played 2 3 4 1 Stage I + II 1/5/2022 Voronoi game 22

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Line strategy (2) 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 < 1/n - l from white endpoint 5 1/5/2022 7 Voronoi game 6 23

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Line strategy (3) 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 < 1/n - l from white endpoint 8 1/5/2022 l = 1/8 Voronoi game 24

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Proof of line strategy (1) Stages I + II are well defined White cannot cover both u 0 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) 1/5/2022 Voronoi game 25

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Proof of line strategy (2) 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 > l (black wins) Assume Stage IV case (i) occurs ( n(W) = n(B) ) x n(W), x < 1/n (Lemma 3) y n(B), y = key interval (Lemma 4) Wm < Bm and border interval is. . . (i) monochromatic: Wb = Bb = 0 black wins (ii) bichromatic: Wb Bb = 1/2 n black wins 1/5/2022 Voronoi game 26

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Proof of line strategy (3) 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 > l (black wins) 1/5/2022 Voronoi game 27

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion White‘s defense Margin by which black wins is very small White controls the margin Theorem: White captures at least ½ - of the curve, > 0 Proof White plays n points within distance /2 n 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) < n ) Then Bm - Wm 2 (n-1)/n Wb 1/2 n - /2 n, Bb 1/2 n + /2 n B -W 2 W ½ - 1/5/2022 Voronoi game 28

1. Introduction – 2. The game – 3. Circle game – 4. Line segment game – 5. White‘s defense – 6. Conclusion Strategies for 1 D competitive facility location problems Showing the 2 nd player, black, to win White controls the margin by which black wins For practical purposes, 1 D Voronoi game ends in a tie 1/5/2022 Voronoi game 29