Computing Close to Optimal Weighted Shortest Paths in Practice 30 th - PowerPoint PPT Presentation

Introduction Proposed method Experimental results Computing Close to Optimal Weighted Shortest Paths in Practice 30 th International Conference on Automated Planning and Scheduling (ICAPS 2020) Nguyet Tran 1 , Michael J. Dinneen, Simone Linz School of Computer Science, University of Auckland, New Zealand October 29-30, 2020 1. Email: ntra770@aucklanduni.ac.nz Nguyet Tran 2 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 1 / 20

Introduction Proposed method Experimental results Contents Introduction 1 Proposed method 2 Experimental results 3 Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 2 / 20

1 14 5 4 3 2 0 24 85 17 56 80 46 82 10 37 Introduction Proposed method Experimental results Introduction Problem Definition 5000 4000 WS = ( T , E , V ) : a continuous 3000 two-dimensional workspace Each region t i ∈ T is a triangle, and 2000 assigned a unit weight (or cost) w i > 0. Let p and q be two points on a region 1000 t i ∈ T , d ( p , q ) : the Euclidean distance between p 0 0 1000 2000 3000 4000 5000 and q D ( p , q ) = w · d ( p , q ) : the weighted length F IGURE – An example of WRP problem with 10 regions (or cost) between p and q , where w is the and three very-close optimum paths between unit weight of t i or the edge that the vertices (0 and 1), (2 and 3) and (4 and 5). segment ( p , q ) is on. T : the set of non-overlapping regions E : the set of edges V : the set of vertices Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 3 / 20

0 14 5 4 3 2 1 24 85 17 56 80 46 82 10 37 Introduction Proposed method Experimental results Introduction Problem Definition 5000 4000 For a pair of two vertices u , v ∈ V , the 3000 weighted region problem (WRP) asks for the minimum cost (or the weighted shortest) path 2000 P ∗ ( u , v ) = ( u = o 0 , o 1 , . . . , o k , o k + 1 = v ) 1000 such that the weighted length 0 D ( P ∗ ( u , v )) = � k 0 1000 2000 3000 4000 5000 i = 0 D ( o i , o i + 1 ) is minimum, F IGURE – An example of WRP problem with 10 regions where every o i , i ∈ { 1 , . . . , k } , called a and three very-close optimum paths between crossing point , can be a point on an edge in E vertices (0 and 1), (2 and 3) and (4 and 5). or a vertex in V . T : the set of non-overlapping regions E : the set of edges V : the set of vertices Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 4 / 20

85 14 5 4 3 2 1 0 24 17 56 80 46 82 10 37 Introduction Proposed method Experimental results Introduction Difficulties 5000 4000 3000 2000 The problem is NP-hard or not: Unknown Currently, there is no known polynomial or 1000 exponential time algorithm for finding the exact weighted shortest path. 0 0 1000 2000 3000 4000 5000 The exiting algorithms to solve WRP are all approximations. F IGURE – An example of WRP problem with 10 regions and three very-close optimum paths between vertices (0 and 1), (2 and 3) and (4 and 5). T : the set of regions E : the set of edges V : the set of vertices Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 5 / 20

0 14 5 4 3 2 1 24 85 17 56 80 46 82 10 37 Introduction Proposed method Experimental results Introduction Existing approaches 5000 4000 3000 An overview of the existing approaches: Exploiting Snell’s law 2000 (impractical solutions) Using heuristic methods 1000 (unpredictable results) Applying decomposition ideas, with a grid 0 0 1000 2000 3000 4000 5000 of cells or a graph of discrete points, called F IGURE – An example of WRP problem with 10 regions Steiner-points and three very-close optimum paths between (time-consuming for a close optimal result). vertices (0 and 1), (2 and 3) and (4 and 5). T : the set of regions E : the set of edges V : the set of vertices Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 6 / 20

0 14 5 4 3 2 1 24 85 17 56 80 46 82 10 37 Introduction Proposed method Experimental results Introduction Our approach 5000 4000 3000 An overview of the existing approaches: Exploiting Snell’s law 2000 (impractical solutions → practical solution) Using heuristic methods 1000 (unpredictable results) Applying decomposition ideas, with a grid 0 0 1000 2000 3000 4000 5000 of cells or a graph of discrete points, called F IGURE – An example of WRP problem with 10 regions Steiner-points and three very-close optimum paths between (time-consuming for a close optimal result). vertices (0 and 1), (2 and 3) and (4 and 5). T : the set of regions E : the set of edges V : the set of vertices Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 7 / 20

• • • 3 1 2 1 5 4 1 • • • • • Introduction Proposed method Experimental results Proposed method 1. Weighted shortest path crossing an edge sequence S v b i = b k c j = c k 14 P b P c e k q k Given an ordered sequence of k edges 12 b i c j S = ( e 1 , e 2 , . . . , e k ) , where three consecutive SP ( u, v ) p k edges in S cannot be in the same triangles a . 10 W = ( w 0 , . . . , w k ) is the weight list of S , where o 2 8 every w i , i ∈ { 0 , . . . , k } , is the unit weight of o r g the region between e i and e i + 1 , with o 1 P a q 3 6 e 0 = ( u , u ) and e k + 1 = ( v , v ) . e 3 a g P ( u , v ) = ( u = r 0 , r 1 , . . . , r k , r k + 1 = v ) is a r 3 4 path between two vertices u , v ∈ V , crossing p 2 = p 3 e 2 a 2 an edge sequence, where r i is on e i with every r 2 q 1 = q 2 2 i ∈ { 1 , . . . , k } . r 1 a 1 e 1 p 1 a . Otherwise, we present how to process it in the paper. 0 u 0 2 4 6 8 10 12 14 F IGURE – Illustration of Snell rays. Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 8 / 20

2 3 1 5 4 1 • • • • • • • • 1 Introduction Proposed method Experimental results Proposed method 1. Weighted shortest path crossing an edge sequence S v b i = b k c j = c k n ( r j ; e j ) w j 14 β j e j r j +1 r j w j − 1 P b P c e k q k α j 12 r j − 1 b i c j SP ( u, v ) n ( r i ; e i ) r i +1 p k β i w i 10 r i e i w i − 1 α i o 2 8 o r g o 1 P a q 3 6 r i − 1 e 3 a g F IGURE – Illustration of Snell’s law. r 3 4 Snell’s law : p 2 = p 3 e 2 a 2 r 2 P ( u , v ) has the minimum weighted length q 1 = q 2 2 r 1 crossing S if and only if at every crossing point a 1 e 1 r i on e i , for which r i is not an endpoint of e i , the p 1 0 u 0 2 4 6 8 10 12 14 following condition holds: w i − 1 sin α i = w i sin β i F IGURE – Illustration of Snell rays. Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 9 / 20

• 5 • • • • • • 3 1 2 1 • 4 1 Introduction Proposed method Experimental results Proposed method 1. Weighted shortest path crossing an edge sequence S Snell ray: v b i = b k c j = c k Let a 1 be a point on e 1 ∈ S . 14 Apply Snell’s law from u , crossing e 1 at a 1 , P b P c e k q k we can find the out-ray R a 12 1 . b i c j SP ( u, v ) p k Suppose that R a 1 intersects e 2 ∈ S at a 10 point a 2 . Then, we can continue calculating the path P a = ( u , a 1 , a 2 , . . . , a g , R a g ) , o 2 8 where 1 ≤ g ≤ k and R a g is the out-ray of o r g o 1 the path at e g ∈ S . P a q 3 6 e 3 We define P a to be a Snell ray of u , a g starting at the point a 1 , crossing S , from e 1 r 3 4 to e g . p 2 = p 3 e 2 a 2 r 2 Snell path: P ( u , v ) is a Snell path if q 1 = q 2 2 r 1 a 1 Every point r i , i ∈ { 1 , . . . , k } , is on the e 1 p 1 0 interior of e i , which cannot be one of the u 0 2 4 6 8 10 12 14 two endpoints of e i , and F IGURE – Illustration of Snell rays. Snell’s law is obeyed at every r i . Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 10 / 20

4 • • 3 1 2 1 5 1 • • • • • • Introduction Proposed method Experimental results Proposed method 1. Weighted shortest path crossing an edge sequence S v b i = b k c j = c k 14 P b P c e k q k 12 b i c j SP ( u, v ) p k 10 o 2 8 Two Snell rays P b and P c : o r g P b = ( u , b 1 , . . . , b i , R b i ) o 1 P a q 3 P c = ( u , c 1 , . . . , c j , R c 6 j ) , where b 1 � = c 1 e 3 a g P b and P c cannot intersect each other. r 3 4 p 2 = p 3 e 2 a 2 r 2 q 1 = q 2 2 r 1 a 1 e 1 p 1 0 u 0 2 4 6 8 10 12 14 F IGURE – Illustration of Snell rays. Nguyet Tran 1 , School of Computer Science, University of Auckland, New Zealand Michael J. Dinneen, Simone Linz October 29-30, 2020 11 / 20