11142003 10:00AM

1
11/14/2003 1000AM

• Valentin Polishchuk
• Preliminary Examination
• Geometric Motion Planning Problems with
  Applications
• Adviser Professor Joseph Mitchell

2
Research Directions

• Motion Planning Problems
• Geometric Algorithms for Air Traffic Management

3
Motion Planning Problems
4
A Puzzle

• Target configuration
• Sink

• Geometric domain
• Items in it
• Agent (Robot), which travels and moves the items
• Initial position
• Rules/Constraints
• What happens after a move
• What is forbidden

5
Integral Orthohedral Version
Rectilinear domain, possibly with holes Vertices
have integral coordinates Domain is pixeled The
agent occupies 1 pixel An item occupies 1 pixel
The dual graph
6

• Feasibility
• Is it possible at all to move from the initial
  configuration to the target?
• Optimality
• Cost associated with agents move
• What is the min cost of rearranging the items?

7
The Snow Blower Problem

• Domain driveway
• Items snow of depth 1 on every pixel
• Agent snowblower
• Rules
• the snow from a pixel entered (if any) is put
  onto an adjacent pixel (Left, Forward or Right)
• if a boundary pixel is entered, the snow from it
  (if any) is thrown away
• its forbidden to put the snow on the pixel
  already containing snow
• Target configuration depth 0 snow everywhere,
  clear driveway
• Related problems
• Lawn-mowing Problem
• Milling Problem

8
Optimization

• Cost proportional to the number of pixels (with
  or without snow) visited
• NP-hard
• Reduction from Hamiltonian Cycle problem for
  cubic subgrid graphs (PV, Buro)
• PV Christos H. Papadimitriou, Umesh V.
  Vazirani On Two Geometric Problems Related to
  the Traveling Salesman Problem. J. Algorithms
  5(2) 231-246 (1984)
• Buro M. Buro, Simple Amazons Endgames and
  their Connection to Hamilton Circuits in Cubic
  Subgrid Graphs, The Second International
  Conference on Computers and Games (CG2000),
  Hamamatsu Japan

9
Reduction

• The region is clearly clearable (feasible
  instance)
• The graph is Hamiltonian iff the SBP can be
  solved in at most n moves. n is the number of
  nodes in the graph.
• Optimization problem is NP-hard even for regions
  with maximum pixel degree 3

10
An Approximation

• Assumption no deg 1 nodes
• Take any SB tour
• Detour (cost 2) at degree 3 nodes
• Get a Chinese Postman tour (visiting every edge)
  in the dual graph

11
Analysis

• Thus, CP SB 2d3
• CP/SB 1 2d3/SB 3 since SB d3
• 3-APX for max pixel degree 3 regions

SBP and TSP

• SBP is TSP in the dual graph (if its maxdeg 3)
• TSP APX SBP APX
• Christofides heuristic, 3/2-APX
• Can we do better (at least in some graphs)?

12
TSP in Grid Graphs

• For general grid graphs CP TSP 2d3 4d4
• CP/TSP 1 (2d3 4d4)/TSP
• better guarantee than 3/2 when d3 2d4 lt TSP/4
• sparse grid graphs d3 2d4 lt n/4
• sparse cubic subgrid graphs d4 0, d3 n/4
• Open Question
• Can CP ever actually do better than Christofides
  algorithm?

13
Comparison with Previous Work

• Dense (simple, solid) grid graphs rather then
  sparse were studied Ntafos, AFM
• 6/5-APX
• When is an APX necessary?
• When the problem is NP-hard
• TSP is hard if HC is hard
• HC for dense graphs is poly UL
• HC is hard for sparse grid graphs
• modification of the graph from PV

• AFM E.M. Arkin, S.P. Fekete, J. S. B. Mitchell
  Approximation Algorithms for Lawn Mowing and
  Milling, CGTA 17(1-2), October 2000, pp. 2550
• Ntafos S. Ntafos. Watchman routes under limited
  visibility. Comp. Geom. Theory and Appl,
  1(3)149--170, 1992
• UL C. Umans and W. Lenhart. Hamiltonian cycles
  in solid grid graphs. In Proc. 38th Annu. IEEE
  Sympos. Found. Comput. Sci., pages 496--507, 1997
  
• PV Christos H. Papadimitriou, Umesh V.
  Vazirani On Two Geometric Problems Related to
  the Traveling Salesman Problem. J. Algorithms
  5(2) 231-246 (1984)

14
Sidewalks

• Sidewalk contains no 2-by-2 square
• Dual graph contains no 4-cycle thin graph

Thin, but not maxdeg 3 Not clearable
maxdeg 3, but not thin Clearable
15
Sidewalks (cont.)

• HC for cubic subgrid graphs reduces to SBP, both
  hard
• HC for thin grid graphs hard?
• No.
• A by-product Polynomial algorithm for HC search
  problem in thin grid graphs

16
Feasibility

• maxdeg 3 regions
• always feasible
• Sidewalks
• DFS from a given entrance pixel O(n), n is the
  number of pixels in the domain SBARG
• General case
• DFS Partial clearing doesnt change feasibility.
  In O(n) moves every pixel is visited, in O(n) the
  snow from a pixel is thrown away so its O(n2)
• A by-product membership of the optimization
  problem in NP. So, SBP is NP-complete
• SBARG The Stony Brook Algorithms Reading
  Group, 2001

17
Variations on SBP

• D max snow depth through which the SB can move
• D gt 1
• if D gt 1 then any region is clearable starting
  from any pixel on the outer boundary of the
  region
• if D gt 2 then any region is clearable starting
  from any pixel on the boundary of the region
• if D gt 2 then any region is clearable starting
  from any pixel on the boundary of the region and
  for any initial direction of entrance of the SB
  into the region
• Snow Shovel Problem
• Scoop snow and walk through cleared region
• Non-square tiling
• Beehive Clearing Problem any region is
  clearable
• Non-convex tiling ask Mauritz Escher

18
Open Problems

• Snow Plower Problem SBP with fixed throw
  direction
• Analyze/improve the complexity of the algorithm
  for HC finding in thin grid graphs
• APX for TSP in thin grid graphs
• APX

19
The Box Mover Problem

• Domain warehouse
• Items boxes
• Agent warehouse-keeper (robot), pushing and/or
  pulling and/or lifting boxes
• Rules
• dont step on boxes
• not more than k boxes pushed at once
• not more than p boxes pulled at once
• not more than l boxes lifted at once
• BMP(k,p,l)
• Related problems
• SOKOBAN BMP(1,0,0)
• Push-k BMP(k,0,0), Push- BMP(8,0,0),
  PushPush, Push-X

20
Complexity of BMP(1,0,0)

• Previous work Feasibility
• P?
• No non-trivial problem known open question
• NP?
• NP-hard, but only 1 version (-X) is in NP
• PSPACE-complete
• Optimization only workload is counted (unlike
  SBP)
• Total travel or workload or mix is the same for
  feasibility
• NP-hard in the general setting

21
Making the Puzzle more tractable

• k ? , p ? , l ? ?
• introducing powerful robot makes it relatively
  easy to construct intractable puzzles (DDO)
• objection Mosaic Rearranging Problem (flying
  robot) is in P solved by assignment
• Limiting the robots capabilities
• the exact complexity of most problems (even the
  more tractable ones) is unknown, some are
  PSPACE-complete
• Our optimization problem is in NP for any k, p,
  l, including infinite values
• DDO E. Demaine, M. Demaine and J. O'Rourke,
  PushPush and Push-1 are NP-hard in 2D, in
  Proceedings of the12th Annual Canadian Conference
  on Computational Geometry (CCCG 2000),
  Fredericton, New Brunswick, Canada, August 16-18,
  2000, pages 211-219

22
Making the Puzzle more tractable (contd.)

• All blocks movable, no walls, infinite plane
• leakage is a problem
• l 0
• k, p gt1 a wall of thickness max(k, p)1 is
  rigid
• p 0 k1-by-k1 square is unmovable
• k 0 p1-by-p1 square is unmovable
• Palliative, what if k, p gt 0 or l gt 0 ?
• In our proof the wall thickness is constant

23
Another Problem - Crossovers

• DH, DDHO contrasted their work to all
  previous approaches of building circuits based on
  graphs, which seem to inherently require
  problematic crossings
• Our construction has no crossings the reduction
  is from the HC problem for planar graphs

DH E. Demaine and M. Hoffman, Pushing blocks
is NP-complete for non-crossing solution paths,
Proc. 13th Canad. Conf. Comput. Geom. (2001),
65-68DDHO E. Demaine, M. Demaine, M. Hoffmann,
and J. O'Rourke, Pushing Blocks is Hard,
Computational Geometry Theory and Applications,
Special issue of selected papers from the 13th
Canadian Conference on Computational Geometry,
2001.
24

• Cul1

Required third dimension to work
Cul2 Cul1 J. Culberson, Sokoban is
PSPACE-complete Proc. Internet Conf. Fun with
Algorithms (1998), N. S. E. Lodi, L. Pagli, Ed.,
Carelton Scientific, 65-76 Cul2 J. Culberson,
Private Communication, 2003
25
The Reduction

• HC for planar directed graphs with each node v
  satisfying out(v) in(v) 3 Pl.
• Pl J. Plesnik, The NP-completeness of the
  Hamilton cycle problem for planar digraphs of
  degree bound two, Inform. Process. Lett., 8, No.
  4(1979), 199-201
• JP D.S. Johnson and C. H. Papadimitriou,
  Computational complexity and the travelling
  salesman problem, in The Travelling Salesman
  Problem'' (E. W. Lawler, J. K. Lenstra and A.G.
  Rinnooy Kan, Eds.), Chap. 3, Wiley, New York, 1982

26
The Reduction (contd.)
Edges corridors of width 1Nodes
T-intersections
27
The Gadgets

• Node gadget

• Edge gadget

Checked box initial position. Shaded box
target position. Yes, in the edge gadget they
coincideThe robot is initially somewhere inside
28
Analysis

• If G is Hamiltonian the puzzle is solved in
  2(n-1) n 3n 2 pushes
• If not, then not less than 3n pushes is required
• So, our problem is NP-complete.
• BMP(1,0,0) with some boxes fixed to the floor
  (not all movable) is NP-complete

29
NP-Completeness Results

• BMP(0,1,0) with some boxes fixed
• same reduction, just the direction of the edge
  gadget is reversed
• BMP(0,0,1) with some boxes fixed
• although the directionality is lost and every
  edge requires work of 1 unit to pass, the same
  reduction holds with the bound replaced by 2n 1
• BMP(k,p,l) with some boxes fixed
• nothing changes the edge is just an energy
  waster
• BMP(k,p,l)-X with some boxes fixed
• since the proposed solution path is non-crossing
• BMP(k,p,l) with some boxes fixed
• could have assigned numbers to the boxes and
  target locations
• BMP(k,p,l)-X with some boxes fixed

30
All Blocks Movable

• Modified node and edge gadgets
• Breaking through a wall gives no benefit

31
Main Result

• All variations of our problem are
• NP-complete
• BMP(k,p,l)-X with or without fixed boxes is
  NP-complete for any (k,p,l) ? (0,0,0), including
  infinite values of k, p, l.

32
Open Problems

• An interesting problem in P (since DH, DDHO)
• maybe, an optimization problem?
• if the initial and target can be separated by a
  line greedy SBARG
• More optimization problems in NP
• PushPush version (feasibility is hard)

DH E. Demaine and M. Hoffman, Pushing blocks
is NP-complete for non-crossing solution paths,
Proc. 13th Canad. Conf. Comput. Geom. (2001),
65-68DDHO E. Demaine, M. Demaine, M. Hoffmann,
and J. O'Rourke, Pushing Blocks is Hard,
Computational Geometry Theory and Applications,
Special issue of selected papers from the 13th
Canadian Conference on Computational Geometry,
2001SBARG The Stony Brook Algorithms Reading
Group, 2003
33
Related Problems

• Lawn-mowing, Milling
• SBP
• BMP

34
Geometric Algorithms for Air Traffic Management

• Air space free space and no-fly zones
• K flights. (sk, tk)
• Route the flights
• Simplifying assumptions
• No time dependence
• 2D

35
Constrained Path

• Model for a flyable path
• Thick path
• Link-constrained path discrete model for
  curvature constrained path
• Rectilinear
• Monotone

36
Paths

• Non-crossing
• Short

37
Objective Function

• Length of individual path
• L1, L2
• link length
• Length of all paths
• sum of lengths of individual paths
• VLSI wire routing rectilinear version
• the length of the longest path
• hasnt been studied earlier

38
Minsum vs. Minmax

• Minmax tend to be harder
• in graphs minsum s-t paths min cost flow,
  minmax NP-hard
• May be different

• K-approximation to each other

• NP-hard if K is not constant BP
• BP O. Bastert, S.P. Fekete. "Geometrische
  Verdrahtungsprobleme." Technical Report ZPR
  96-247. 1996

39
K Short Non-Crossing Constrained Paths

• Connect the pairs of points inside a polygon by
  non-crossing constrained paths so that the length
  of the longest path is as small as possible
• Notation

40
Constraints/Restrictions

• Constraint condition on a path
• Restriction condition on P or on placement of
  (sk, tk) inside P
• Restrictions make the problem easier.
  Constraints generally make the problem harder
• Constraints considered earlier

41
Restrictions

• h 0 P is a simple polygon
• further restricted to be monotone
• further restricted to be convex

(sk, tk) placement inside P

• sk aligned
• si and sj coincide for some i and j
• sk are on the boundary of P
• all sk are on one edge of P combination of the
  above two
• Same restrictions on tk placement

42
Problem Formulation

• In a polygon P, such that P is restrictions on
  P, K pairs of points (sk, tk) restrictions on
  (sk, tk) placement are given. Also given is a
  bound B.
• Find K non-crossing constrained polygonal
  paths in P connecting sk and tk, such that the
  length definition length of the longest path is
  not more than B.

43
A Special Case

• sk, tk are on the boundary of P
• Unconstrained paths
• No path is allowed to go around a terminal of
  another path
• monotone

44
Solution

• Ordering around the boundary of P determines
  feasibility
• not s1, s2 sK, tK, tK-1,, t1 not feasible
• o.w. route in any order, all shortest paths
  will be found

45
All sk, tk are on the Boundary of P

• O(nK), Pap
• Pap E. Papadopoulou, K-Pairs non-crossing
  shortest Paths in a Simple Polygon, ISAAC 1996
  305-314

• Restriction on sk, tk placement is saved for
  future use
• polynomial algs work with no restriction
• hardness results work with all restrictions
  active

46
K 1

• Solved
• Unconstrained
• Rectilinear
• Monotone
• Thick
• new_obst old_obst 0 ,12
• Find unconstrained path avoiding new_obst

47
K 1, Link-Constrained Path

• Link-Constrained Path
• the length of each link is L
• the angle between consecutive links is ?
• Robotics
• given links
• the angle between consecutive links is ?
• What is the region reachable by such robot arm?
• in a polygon?

48
Reachable Region

• Different link lengths (1,2,4,8,)
• at least
• exponential complexity

• All links are 1-links
• simulations

49
Wobbly Link Lengths

• Link length is 1 d
• fuzzy contours

50
Wobbly Path

• Link length is 1 d
• Angle between links is ? ?

51
Wobbly Graph

• Why only paths? General wobbly graphs
• GIVEN Graph (not necessarily planar) in very
  general form (adjacency matrix)
• FIND Possible planar layouts of the graph
  subject to
• Each edge length is 1 d
• Angle between edges is ? ?
• Precisely the Map Extraction Problem
• Robots sense with angle and length tolerance
• GIVEN Robots adjacency matrix who sees whom
• FIND Possible shape of the region where the
  robots are dispersed

52
Map Extraction

• In general the goal is not very clear
• distribution over possible shapes?
• one particular shape?
• decision can the robots be in a given region?
• An easier problem
• fix the position of one of the robots
• for every robot plot a (convex) superset of its
  feasible positions

53
Two Ideas

• Andy Wildenberg
• fix a direction
• project the distances onto the direction
• choose another direction
• Tien-Ruey Hsiang
• an LP formulation

54
Combining the Ideas
55
Another Problem

• All links are 1-links
• Angle is free
• 1-TSP (Traveling Salesperson Problem)
• GIVEN Points in the plane
• FIND An optimal polygonal tour visiting every
  point exactly once and such that every link is
  1-link

56
1-TSP

• Crossings allowed
• NP-complete (from TSP)
• 3/2 approximation
• ceiling preserves triangle inequality
• Points inside unit square, non-crossing path
• Feasibility established SBARG
• Hardness is open
• O(n) approximation
• SBARG The Stony Brook Algorithms Reading Group

57
Awkward Path Problems
58
K 1, a Hard Problem

• In a polygon P find shortest path from s to t
• Restrictions
• h 1
• smooth (circular) obstacles
• Constraints
• (self-)crossings allowed
• link length is L
• angle is free

59
An Instance

• r and R are circles radii
• s is on the larger circle
• L2 4(R2 r2) so happened
• every link hits a pre-defined point
• s1 hit after a full turn

s1
s
s
60
Too Many Links

• (ss1) can be arbitrarily small
• The number of links in s-t path is proportional
  to (st)/(ss1)
• can be astronomically large (rational case)
• t may be reachable from s in only infinitely many
  links (irrational case)

t
s1
s
s
61
K gt 2

• No polynomial algorithm is known
• All NP-hardness proofs hold even for the case K
  2 (and all restrictions active)
• So, we concentrate on K 2

62
Polynomially Solvable Cases

• Known Results
• Not known
• New Results
• h0

63
K gt 2

• Optimum none of the paths is the shortest

64
NP-Complete Problems

• Known results
• Not known
• New results
• Translate graph problems into geometric setting
• Idea vertex disjoint paths in planar graphs
  correspond to non-crossing constrained paths in
  polygonal domains

65
The Basic Graph Problem

• Two Length-Bounded (Internally) Vertex-Disjoint
  s-t Paths in a Weighted Planar Graph HP
• GIVEN A planar graph G(V,E), vertices s and t,
  a bound B, a length assigned to every edge
• FIND Two (internally) vertex-disjoint paths
  connecting s and t and such that the length of
  each path is not more than B
• Reduction from the Partition problem
• GIVEN A set of integers S c1,, cm
• FIND A subset of S such that the sum of the
  elements in the subset is half the sum of the
  elements in S
• HP H. van der Holst and J.C. de Pina,
  Length-bounded disjoint paths in planar graphs.
  Sixth Twente Workshop on Graphs and Combinatorial
  Optimization (Enschede,1999). Discrete Applied
  Mathematics 120 (1-3) (2002), 251-261

66
The Reduction

• Consider the graph G0. The edges with no label
  close to them are unit-length edges
• G0 contains 2 internally vertex-disjoint paths
  from
• s to t each of length bounded by B02(m1) ½
  ?ci
• iff the corresponding instance of Partition is
  feasible

67
Rectilinearize

• G contains two vertex disjoint s-t paths each of
  length bounded by B10m-62?ci iff G0 contains
  two vertex disjoint s-t paths each of length
  bounded by B0

68
A By-Product

• Two Length-Bounded (Internally) Vertex-Disjoint
  s-t Paths in an Unweighted (Euclidean edge
  length) Planar Graph of Maxdeg 3
• is NP-complete
• Doesnt prove for grid graph!

69
From Graph to Geometric Domain

• P the Minkowski sum of the planar layout of G
  and the unit square. G is also drawn for clarity

70
The Reduction

• P contains two non-crossing thick s-t paths each
  of Euclidean length bounded by B iff G contains
  two vertex disjoint s-t paths each of length
  bounded by B

71
NP-Hardness Results

• Finding two short non-crossing thick paths in a
  polygon
• Finding two short non-crossing thick rectilinear
  paths
• since the two s-t paths in P are rectilinear
• Finding two short non-crossing thick monotone
  paths
• since the two s-t paths in P are monotone

72
The Idea of the Reduction

• 2 paths do not fit into the region corresponding
  to the node of G

73
More Hardness Results

• Finding 2 link-constrained paths
• Finding 2 link-constrained monotone paths
• Finding 2 link-constrained paths with no angle
  constraint

74
More Hardness Results (contd.)

• Finding 2 thick paths through weighted region
• interesting for air traffic routing
• Finding 2 separated paths (generic case)
• paths are separated if they never come close to
  one another

75
Moving Obstacles

• No bound on the velocity of the moving object
• polynomial if the obstacles move algebraically in
  space-time RSh (very theoretical not
  suitable for implementation and not interesting
  from practical point of view)
• Upper bound on the velocity of the moving object
• NP-hard CaRe, even if the obstacles move with
  constant velocity (Asteroid Avoidance Problem)
• Lower bound on the velocity of the moving object
• hasnt been considered earlier
• natural for air traffic routing
• NP-hard

RSh J. Reif and M. Sharir. Motion Planning in
the presence of Moving Obstacles. Proc. 25th IEEE
symp. FOCS, (1985), pp. 144-154CaRe J. Canny
and J. Reif. New lower bound techniques for robot
motion planning problems. In Proceedings of the
27th Annual IEEE Symposium on the Foundations of
Computer Science, pages 49-60, Los Angeles, USA,
1987
76
NP-Hardness Proof

• Following the lines of 3D case from RSh
• point object
• discrete time
• but can be modified for continuous time model
• Reduction from 3-SATISFIABILITY (3SAT)

RSh J. Reif and M. Sharir. Motion Planning in
the presence of Moving Obstacles. Proc. 25th IEEE
symp. FOCS, (1985), pp. 144-154
77
3-SATISFIABILITY (3SAT)

• U x1, , xn set of (Boolean) variables
• Definition For x from U x and x are literals
  over U
• Definition A clause over U is a set of literals
  over U
• Clause represents the disjunction of the
  literals in it
• Satisfied by a truth assignment for U iff at
  least one of its elements is true
• GIVEN Set U x1, , xn of variables,
  collection C of clauses over U such that each
  clause c in C has c3.
• FIND A truth assignment for x1, , xn such
  that each clause in C is satisfied

78
The Idea

• Reduction from 3-SATISFIABILITY (3SAT)
• Path encoding
• exponentially many paths, each represents a
  truth assignment
• Time encoding
• time t an integer in 0 2n -1
• binary representation for t, t xn xn-1 x2x1
• each bit is a truth assignment for a variable

79
The Variable Gadget

• xi
• oscillating obstacle
• switches back and forth every 2i-1 time units

80
The Clause Gadget
(x1 V x2 V x3 )
81
An Example
target
target
(x1 V x2 V x3 ) (x1 V x2 V x4 )
t 0 0000
t 1 0001
start
start
82
Discussion

• In the problem formulation time t is present
• Can the object move in t time units from start to
  target positions?
• Projects far (exponentially far) into the future
• Make the algorithm run through all 2n time
  units to report all feasible times is unfair,
  compare given n, list all integers from 1 to n
• Careful analysis of the solution must be done

83
Feasibility

• General case, Kgt2 (not constant)
• Non-crossing constrained paths feasibility
  problem
• dont care about the length
• NP-hard
• Reduction from Constrained Planar Disjoint
  Connecting Paths Problem Rich
• GIVEN A planar (grid) graph G, maxdeg 3 2K
  vertices are labeled to form K pairs
• FIND K vertex-disjoint paths between the K
  pairs over the edges of G
• The old idea vertex-disjoint paths in a graph
  correspond to separated paths in polygonal
  domain
• Rich D. Richards, Complexity of Single-Layer
  Routing, IEEE Transactions on Computers, Vol.
  C-33, No. 3, March 1984, pp. 286-288

84
Feasibility (A Special Case)

• K rectilinear paths with at most 2 bends each in
  grid

Balance and route 1-by-1 from the bottom Light
right from the origin and left from the
destination Take the rightmost vertical line
were the rays see each other
85
Conclusion

• P is simple (or h const), number of paths fixed
  
• polynomial,O(1) homotopy classes
• P is not simple and paths are constrained (the
  interesting case) NP-hard
• What to do?
• Grid
• Route unconstrained
• Heuristic
• 1-by-1 routing

86
Unconstrained Paths in Grid

• GIVEN A rectangular grid (20X50) with some
  nodes marked as obstacles
• FIND As many rectilinear paths as possible
  from one side of the grid to the opposite
• Maxflow in the grid graph s, t (super)

87
Mincost Maxflow

• Routes maximum number of paths with minimum total
  length

88
45 degree Turns

• A node in the center of every cell of the grid
  linear number of additional nodes
• More paths fit in the same region
• Smoother paths (look better subjective)

89
45 degree Turns (more pictures)
90
Min-Turns Curvature-Constrained Paths in Grid

• In grid
• monotone (the only found way to avoid
  self-intersection)
• fixed lanes (m in total)
• switching lanes along a circular (90-degree)
  arc of pre-defined radius (r possible values)
• avoid obstacles
• Idea of Wilfong Wi
• O(m2r) turns
• consider only feasible turns (no collision with
  obstacles)
• Directed graph G (N, A)
• N the set of feasible turns
• an edge from turn 1 to turn 2 iff the exit of
  turn 1 is in the direction of the entrance of the
  turn 2 (the same lane)
• Shortest path in G is min-turns
  curvature-constrained path in the environment

Wi Wilfong, G. T., Motion planning for an
autonomous vehicle, Proc. IEEE Int. Conf. on
Robotics and Automation, 1988, pp. 529-533
91
Implementation

• 1 by 1 routing
• previously routed paths treated as obstacles
• forbid straight line routes

92
Greedy Min-Turns Curvature-Constrained Routing
93
Different Radii
94
1-by-1 Unconstrained Paths Routing (The General
Case)

• Not always OPT
• how bad can it be with worst possible ordering?

95
Notation and Assumptions

• G(s, t) shortest s-t path
• G(s1, t1) the longest of G(sk, tk), k 1K
• G(s1, t1) 1
• High interaction between sk, tk , k 1K all
  G(sk, tk) intersect G(s1, t1)
• G complete graph on sk, tk

96
Homotopy

• Definition Two paths are homotopic
  (homotopically equivalent) (wrt sk, tk, k 1K)
  if they can be continuously transformed to one
  another never intersecting any of sk, tk, k 1K
  
• Definition A homotopy class for the family of
• sk-tk, k 1K paths is a sketch of K
  non-intersecting sk-tk paths, showing how each
  path winds around sk, tk and the other paths
• Examples
• Any routing defines a homotopy class
• 1-by-1 routing defines a homotopy class
• OPT routing defines a homotopy class (our Holy
  Grail)

97
Why G?

• Shortest paths given homotopy class (paths pull
  taut) use edges of G

98
1-by-1 routing is an O(K)-approximation

• 1-by-1 routing specifies a homotopy class
• Find shortest paths given this homotopy class
• Find the longest of the K shortest paths given
  the homotopy class it is a subgraph of G,
  planar!
• 2K nodes in G a planar subgraph of G has O(K)
  edges
• All G(sk, tk) intersect G(s1, t1) (the longest!)
• all sk, tk live not far from s1, t1
• not far from one another
• the edges of G are O(1) length
• the longest path, GREEDY O(K)
• OPT G(s1, t1) 1
• GREEDY/OPT O(K)

99
1-by-1 routing is an O(K)-approximation

• So, its a ?(K) approximation

100
GK2K vs. VG((sk, tk))

• VG((sk, tk)) the segment endpoints visibility
  graph is not always enough

101
Points Inside a Polygon

• G(s, t) geodesic path
• G the complete graph of geodesic paths
• All the discussion above valid
• 1-by-1routing ?(K)-approximation for points
  inside a polygon as well

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Towards O(1) Approximation

• Definition Pairs (sk, tk) are in convex ordered
  position if they can be ordered so that each pair
  is outside the convex hull of all previous pairs
• Examples
• points in convex position

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Convex Ordered Position

• If the pairs are in convex ordered position
• route the paths according to the ordering
• O(1) approximation
• Given K pairs of points how hard is to establish
  if they are in convex ordered position? Open
  problem

104
Research Directions

• Link distance
• 2 vertex-disjoint paths in grid graphs
• Establish strong NP-completeness or find a pseudo
  polynomial time algorithm
• Finding 2 curvature-constrained paths
• a path is curvature-constrained if its average
  curvature is bounded everywhere along the path
• Analyze/improve the complexity of heuristics
• 1-by-1 routing drop the restriction that all
  paths intersect the longest path
• O(1) approximation for unconstrained paths
  routing
• An approximation for the general problem of
  constrained disjoint paths routing
• APX for 2 thick non-crossing rectilinear paths
  routing problem Done (2-APX)