Design of Spatial Information Systems

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Design of Spatial Information Systems
• Lecture 11b
• Path Finding
• Alexander Kolesnikov

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Navigation in network model
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Navigation problem (1)
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Navigation problem (2)
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Problem formulation

• Input is a graph G (V, E) and the source
  vertex v_src and
• destination vertex v_dst.
• Here V is the set of vertices and E the set of
  edges in graph G.
• Further we have an edge weight function C(u,v)
  for edge e(u,v)?E.
• Find such sequence of edges from v_src to v_dst
  that
• summa of weights is minimal.

v
u
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Dijkstras algorithm Notation (1)

• The algorithm divides the vertices, V, into two
  disjoint sets, V S ? Q.
• The set, S, contains those vertices whose final
  shortest path weights
• have been determined and the set, Q V-S.
• When a vertex is moved from Q to S, we say that
  it is retired.

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Dijkstras algorithm Notation (2)

• For every vertex, v, we store a currently
  minimum shortest-path estimate value, Cv (or
  cost function).
• Since we are interested in finding the shortest
  path, we will also store, for every vertex a
  predecessor edge, prevv, that is the edge from
  a neighbor to the vertex in the currently best
  path to the vertex in question. This can then be
  used to reconstruct the path by backtracking from
  the destination v_dst to v_src.

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Dijkstras algorithm Main idea

• Maintain a set S of vertices whose final
  shortest-path
• from the source v_src have already be
  determined.
• Repeatedly select the vertex u ?V-S with the
  minimum
• shortest path Cu estimate, add u to S and
  relax all
• edges leaving u.

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umin
umin
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40
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Relaxation
// Relax (u,v) IF Cv gt Cu weight(u,v) THEN
Cv Cu weight(u,v) prevv
u ENDIF
c(u)8 c(v)14 weight(u,v)3 83 11lt
14 ?c(v)11 prev(v)u
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u
v
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Dijkstras algorithm
function Dijkstra(Ggraph, v_src, v_dst) 2
FOR each vertex v IN VG DO
// Initialization 3 Cv ? 4
previousv undefined 5 Cv_src 0 6
S ? // Set of processed vertices
(empty so far) 7 Q VG // Set of all
vertices of G 8 WHILE (u ! v_dst) 9
DO u Extract_Min_C(Q) // Get vertex u with
min Cu 10 S S ? u // Add
vertex u to the set S 11 FOR each edge
(u,v) outgoing from u DO 12 IF
Cv gt Cu weight(u,v) // Relax
(u,v) 13 THEN Cv Cu
weight(u,v) 14 prevv
u
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Dijkstras algorithm Demo
http//www-b2.is.tokushima-u.ac.jp/ikeda/suuri/di
jkstra/DijkstraApp.shtml?demo6
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Result Tokushima ? Ikeda
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Result Suvikatu 19 ? BePop
BePop
Suvikatu
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Navigation in 2-Dimensional space
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Navigation problem
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1) Construct graph on regular grid
Edge weight is defined as distance between
vertices of the edge.
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2) Find the shortest path from start to dest.
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Surface properties
Weight of edge is defined by properties of the
surface.
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Result Example 2
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Navigation problem 2
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Navigation problem 2
The problem of navigation among obstacles can be
solved in two ways 1) Shortest path in graph 2)
Shortest path in road map
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Configuration space
Work space Configuration space
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Trapezoidal map
Configuration space
Trapezoidal map
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Road map construction
1) One node at the center of each trapezoid 2)
One node at the midlle of eah vertical
extension Connect two nodes iff one node is in
the center of a trapezoid and other node is on
the border of the same trapezoid.
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Road map
Find path in the road map
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Result Example
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Navigation in 3-Dimensional space
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Problem Path finding on the terrain
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Slope obstacles
Slope obstacles ? gt 22?
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Slope calculation
Slope
y(j-1)
x(i-1)
x(i1,j)
z(xi ,yj)
y(j)
y(j)
z(xi-1 ,yj)
y(j1)
x(i-1)
x(i)
Weight depends on gradient of elevation
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Result