Advice in the Context of Some Geometric Problems (Matching and Packing) Shahin Kamali (U. Manitoba) August 28, 2020 Joined work with Prosenjit Bose 1 , Paz Carmi 2 , Stephane Durocher 3 and Arezoo Sajadpour 3 1Carleton University 2Ben-Gurion University 3University of Manitoba (Matching and Packing) Advice in the Context of Some Geometric Problems 1 / 20 �
Geometric Matching https://www.houseandgarden.co.uk/gallery/ animals-cities-coronavirus-lockdown (Matching and Packing) Advice in the Context of Some Geometric Problems 2 / 20 �
Non-Crossing Matching Monochromatic Non-crossing Matching The input is a set of n points in general position. (Matching and Packing) Advice in the Context of Some Geometric Problems 3 / 20 �
Non-Crossing Matching Monochromatic Non-crossing Matching The input is a set of n points in general position. The goal is to form a maximum matching s.t. the line segments between the matched points do not intersect. (Matching and Packing) Advice in the Context of Some Geometric Problems 3 / 20 �
Non-Crossing Matching Monochromatic Non-crossing Matching The input is a set of n points in general position. The goal is to form a maximum matching s.t. the line segments between the matched points do not intersect. In the offline setting, one can sort items (say by their x -coordinate) and match consecutive points. (Matching and Packing) Advice in the Context of Some Geometric Problems 3 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. 1 (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 match p with an existing unmatched point or leave it unmatched. 2 (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 3 match p with an existing unmatched point or leave it unmatched. 2 Greedy algorithms do not leave a point unmatched if they can match it with some existing point. (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 3 match p with an existing unmatched point or leave it unmatched. 2 Greedy algorithms do not leave a point 4 unmatched if they can match it with some existing point. Not all points can be matched in the online setting. Given an adversarial sequence of n points, how many points can be matched? (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 3 match p with an existing unmatched point or leave it unmatched. 2 5 Greedy algorithms do not leave a point 4 unmatched if they can match it with some existing point. Not all points can be matched in the online setting. Given an adversarial sequence of n points, how many points can be matched? (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 3 match p with an existing unmatched point or leave it unmatched. 2 5 Greedy algorithms do not leave a point 4 unmatched if they can match it with some 6 existing point. Not all points can be matched in the online setting. Given an adversarial sequence of n points, how many points can be matched? (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 3 match p with an existing unmatched point or leave it unmatched. 2 5 Greedy algorithms do not leave a point 4 unmatched if they can match it with some 6 existing point. 7 Not all points can be matched in the online setting. Given an adversarial sequence of n points, how many points can be matched? (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can 1 3 match p with an existing unmatched point or leave it unmatched. 2 8 5 Greedy algorithms do not leave a point 4 unmatched if they can match it with some 6 existing point. 7 Not all points can be matched in the online setting. Given an adversarial sequence of n points, how many points can be matched? (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the online setting, points arrive one by one. Upon arrival of a point p an algorithm can match p with an existing unmatched point or 1 3 leave it unmatched. 2 Greedy algorithms do not leave a point 8 5 unmatched if they can match it with some 4 6 existing point. 9 Not all points can be matched in the online 7 setting. Given an adversarial sequence of n points, how many points can be matched? (Matching and Packing) Advice in the Context of Some Geometric Problems 4 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the worst case, a greedy algorithm has one unmatched point per each pair of matched points (it matches roughly 2 n / 3 points). (Matching and Packing) Advice in the Context of Some Geometric Problems 5 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the worst case, a greedy algorithm has one unmatched point per each pair of matched points (it matches roughly 2 n / 3 points). 1 3 The proof is based on partitioning the plane based on an extension of the greedy line 2 8 5 segments. 4 There is at most one unmatched point per 6 each convex partition. 9 7 (Matching and Packing) Advice in the Context of Some Geometric Problems 5 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the worst case, a greedy algorithm has one unmatched point per each pair of matched points (it matches roughly 2 n / 3 points). 1 3 The proof is based on partitioning the plane based on an extension of the greedy line 2 8 5 segments. 4 There is at most one unmatched point per 6 each convex partition. 9 7 An adversarial argument shows that no deterministic algorithm can match more than 2 n / 3 points in the worst case. (Matching and Packing) Advice in the Context of Some Geometric Problems 5 / 20 �
Non-Crossing Matching Online Monochromatic Matching In the worst case, a greedy algorithm has one unmatched point per each pair of matched points (it matches roughly 2 n / 3 points). 1 3 The proof is based on partitioning the plane based on an extension of the greedy line 2 8 5 segments. 4 There is at most one unmatched point per 6 each convex partition. 9 7 An adversarial argument shows that no deterministic algorithm can match more than 2 n / 3 points in the worst case. Greedy algorithms are the optimal deterministic online algorithms. (Matching and Packing) Advice in the Context of Some Geometric Problems 5 / 20 �
Non-Crossing Matching Monochromatic Matching with Advice Question 1: How many advice bits are necessary/sufficient to match all points? (Matching and Packing) Advice in the Context of Some Geometric Problems 6 / 20 �
Non-Crossing Matching Monochromatic Matching with Advice Question 1: How many advice bits are necessary/sufficient to match all points? Upper bounds: 1 . 5 n bits are sufficient. (Matching and Packing) Advice in the Context of Some Geometric Problems 6 / 20 �
Non-Crossing Matching Monochromatic Matching with Advice Question 1: How many advice bits are necessary/sufficient to match all points? Upper bounds: 1 . 5 n bits are sufficient. Mimic an optimal matching based on x -coordinates. For each point encode whether its partner I) appears later II) appears earlier on its left III) appears earlier on its right. The online algorithm matches p with the leftmost point on its right or rightmost point on its left if its partner appears earlier. 8 (10) 1 (0) 7 (11) 6 (0) 8 (10) 2 (0) 5 (0) 8 (10) 4 (11) 3 (0) (Matching and Packing) Advice in the Context of Some Geometric Problems 6 / 20 �
Non-Crossing Matching Monochromatic Matching with Advice Question 1: How many advice bits are necessary/sufficient to match all points? (Matching and Packing) Advice in the Context of Some Geometric Problems 7 / 20 �
Non-Crossing Matching Monochromatic Matching with Advice Question 1: How many advice bits are necessary/sufficient to match all points? Can we use a reduction from the binary-guessing problem [ B¨ ockenhauer et al., 2014] to show a lower bound of Ω( n )? Maybe; we tried and failed! (Matching and Packing) Advice in the Context of Some Geometric Problems 7 / 20 �
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