A generic data structure for representing discrete paths on regular grids e and ´ Alexandre Blondin Mass´ E. Marcotte LaCIM Universit´ e du Qu´ ebec ` a Montr´ eal June 4th, 2016 GASCom 2016 Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 1 / 26
Discrete paths (a) (b) (c) Three discrete paths : ◮ (a) simple ; ◮ (b) nonsimple ; ◮ (c) closed and simple . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 2 / 26
Simple paths (self-avoiding walks) Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 3 / 26
Some questions Given some discrete path ... ◮ ...is it simple ? ◮ ...how many intersection points does it have? Given two discrete regions (closed paths)... ◮ ...what is their intersection ? ◮ ...what is their union ? ◮ ...what is their difference ? More generally, how efficiently can we answer these questions? Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 4 / 26
Solution 1: balanced tree/sorting ◮ We walk along the path, step by step ; ◮ We insert the points in a balanced tree ; ◮ If there is an intersection, it is immediately detected since the point already belongs to the tree; ◮ Complexity? ◮ O ( n log n ) time; ◮ O ( n ) space; ◮ Similarly, it is possible to store all points in a list and then sort it, yielding the same complexities . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 5 / 26
Solution 2: boolean matrix ◮ Using a boolean matrix , we can also mark the visited points; ◮ Let w and h be the width and height of the bounding box of the path; ◮ Then we have the following complexities : ◮ O ( n ) time; ◮ O ( wh ) space. ◮ It is of course easy to build examples where w, h ∈ Ω( n ), so that the space complexity is then Θ( n 2 ). Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 6 / 26
Solution 3: hash table ◮ We mark the visited points with a hash table ; ◮ It is well-known that hash tables have O (1) average complexity per operation; ◮ However, there is no guarantee of having O (1) in the worst case . ◮ Complexity? ◮ Time : depends on the quality of the hash function; ◮ Space : depends on the capacity of the hash table. ◮ Since the distribution of discrete paths is quite complexe, it is hard to image a function with guaranteed perfect hashing . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 7 / 26
Solution 4: linear sorting ◮ In some cases, one can sort in linear time : ◮ Bucket sort; ◮ Counting sort; ◮ Radix sort. ◮ In our case, it suffices to sort the points according to their x -coordinate and then their y -coordinate (using the radix sort ); ◮ Complexity? ◮ Time : Θ( n ); ◮ Space : Θ( n ). ◮ However, intersection points cannot be detected online . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 8 / 26
Brlek, Koskas and Proven¸ cal’s data structure ◮ S. Brlek, M. Koskas, X. Proven¸ cal, A linear time and space algorithm for detecting path intersection in Z d , Theoretical Computer Science (TCS) 412 , 2011, p. 4841-4850; ◮ The authors introduce a data structure called enriched radix tree , which allows the detection of self-intersection efficiently. ◮ As a first step, we present it for the N 2 grid, then we discuss how it can be extended to Z 2 , as well as other grids . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 9 / 26
The radix relation ◮ The radix relation on N 2 is defined by ( x, y ) → ( x ′ , y ′ ) ⌊ x ′ / 2 ⌋ , ⌊ y ′ / 2 ⌋ � � if and only if ( x, y ) = ◮ Clearly, → is a strict order relation; ◮ ( x, y ) is called the parent of ( x ′ , y ′ ); ◮ ( x ′ , y ′ ) is called the child of ( x, y ); ◮ It may be seen as a 2D generalization of the heap order for arrays: 1 2 3 4 5 6 7 8 9 10 16 14 10 8 7 9 3 2 4 1 Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 10 / 26
Combining two relations on N 2 (a) (b) ◮ (a) The radix (strict) order; ◮ (b) The 4 -adjacency relation; ◮ We need to understand how those two relations interact . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 11 / 26
Observation 1 Lemma (Reformulation of BKP, 2011) If p, p ′ ∈ N 2 are neighbors , then exactly one of those two conditions is verified: (i) p and p ′ are siblings , i.e. parent ( p ) = parent ( p ′ ); (ii) parent ( p ) and parent ( p ′ ) are neighbors . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 12 / 26
Observation 2 Lemma (Reformulation of BKP, 2011) Let p, p ′ ∈ Z 2 be two nonsibling neighbor points. Then parent ( p ′ ) = parent ( p ) + # » d , d = # » pp ′ ∈ { (1 , 0) , (0 , 1) , ( − 1 , 0) , (0 , − 1) } . where # » Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 13 / 26
Basic idea As we travel along the discrete path, we add ◮ a vertex for each new visited point; ◮ undirected edges between some of the neighbor points; ◮ directed edges between some of the points related by the parent/child relation ; Then 1. We start at the origin ; 2. We read the next move, which points toward of the four possible neighbors of the current point; 3. We find this neighbor in the data structure, adding whatever vertex or edge that is needed; 4. We repeat steps 2-3 as long as the discrete path has not been completely traveled. Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 14 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Example Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 15 / 26
Pseudocode (1/3) 1: function DiscretePathGraph (): discrete path graph u ← Node () 2: u .point ← (0 , 0) 3: u .parent ← u 4: u .visited ← true 5: G .current node ← u 6: return G 7: 8: end function 9: 10: procedure Move ( G : radix tree, d : direction) next node ← Neighbor ( G. current node , d ) 11: ⊲ Do something with next node and current node 12: G. current node ← next node 13: 14: end procedure Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 16 / 26
Pseudocode (2/3) 1: function Neighbor ( u : node, d : direction): node if u .neighbor[ d ] is not defined then 2: p ← u. point 3: p ′ ← p + d 4: if parent ( p ) = parent ( p ′ ) then 5: v ← Child ( u. parent , d ) 6: else 7: d ′ ← parents direction( p, p ′ , d ) 8: v ← Child ( Neighbor ( u. parent , d ′ ) , d ) 9: end if 10: u. neighbor[ d ] ← v 11: v. neighbor[ − d ] ← u 12: end if 13: return u .neighbor[ d ] 14: 15: end function Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 17 / 26
Pseudocode (3/3) 1: function Child ( u : node, d : direction): node if u .child[ i ] is not defined then 2: v ← Node () 3: i ← child index( d ) 4: v .point ← child coordinates( u. point , d ) 5: u .child[ i ] ← v 6: v .parent ← u 7: v .visited ← false 8: end if 9: return u .child[ i ] 10: 11: end function Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 18 / 26
The 4-step lemma Lemma (BKP, 2011) Let γ be a discrete path of length 4 and p i the i -th point of N 2 visited by γ , for 0 ≤ i ≤ 4. Then there exists a least two distinct indices j and j ′ such that p j and p j ′ are siblings . Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 19 / 26
Complexity ◮ It is proportional to the number of vertices in the graph, since the number of neighbor edges and parent/child edges is bounded by a constant ; ◮ But, � 0 ≤ i ≤ h, p ∈ points ( γ ) � points (parent i ( p )) � � points ( G ) = . where h is the height of the tree. ◮ Therefore, � 4 � i | points (parent i ( p )) | ≤ | points ( p ) | 5 and � 4 h ∞ � i � � | points (parent i ( p )) | ≤ � | points ( G ) | = n = 5 n = O ( n ) . 5 p ∈ points ( γ ) i =0 i =0 ◮ The amortized complexity of Move () is O ( n ) /n = O (1). Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 20 / 26
Other grids (1/3) (a) (b) (c) ◮ (a) 4-regular; ◮ (b) 6-regular; ◮ (c) 8-regular (nonplanar). Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 21 / 26
Other grids (2/3) ◮ The 4-step lemma does not hold ! Blondin Mass´ e and Marcotte (LaCIM, UQAM) June 4th, 2016 22 / 26
Recommend
More recommend
