Voronoi Graph

1
Voronoi Graph

• Voronoi region Vor(p) (p in set S)
• the set of points on the plane that are closer to
  p than to any other point in S
• Voronoi Graph VOR(S)
• dual to Voronoi region graph
• two points are adjacent if their Voronoi regions
  have common contiguous boundary (segment)

2
Voronoi Graph

• Voronoi Graph in the rectilinear plane
• Rectilinear distance p (x, y) p(x,y)

Voronoi region of b
ab
b
a
bc
c
ac
3
Voronoi Graph

• Fact In any cycle the longest edge does not
  belong to MST.
• Proof if it is the longest edge in a cycle, it
  would be the longest edge in MST.
• Claim 1 Voronoi graph contains minimum spanning
  tree.
• Claim 2 For any x,y ?S if (x,y) ?Vor (S), then
• x,y ?MST(S)
• Proof ? x,y-path P in Vor(S) s.t. ? e?P e lt
  xy

4
Voronoi Graph
Proof
(at least 2 edges in Voronoi groph will intersect
the red edge)
w
y
a
p
pw lt yppw (pwltpx, p closer to w than to x -in
Voronoi region on w) lt yppx xy
x
b
altb
5
Voronoi Graph
Proof in general case Let xy intersect
Voronoi diagram in points p1,p2,.... pipi1 will
be in a same Voronoi region (of some point) pi
pi1 ? Vor(xi) Construct x x1 x2 ... xi-1 y any
edge on this path is shorter than xy xi xi1 lt
pi y
x2
p3
y
p2
p1
x1
x
6
Voronoi Graph
Theorem For n points in the plane MST can be
found in O(n log n) Claim Finding Voronoi
graph for n points is at least as difficult as
sorting n numbers. Proof if the points are
on the same line
xn
x1
x2
x3
it is equivalent to sorting n numbers by Master
Theorem (for divide and conquer) T(n)2T(n/2)
O(n)
for merging
O(n log n)
7
Mehlhorn (88)
O( E V log V) 2-approximation of Steiner
trees G(V,E,w) S ? V terminals find T ? G, T ?
S, w(T) ? min. 2-approx. algorithm 1) Find all
shortest paths between any pair of points
Find GS(S,E,w) , GS is a complete graph
w(s,s)w(e)weight of the shortest path between
s and s in G 2) Find MST of this graph, replace
any edge with paths in G TSMST(GS) 3) T
? pathG(e) (this may not be a tree), last step
e?TS 4) TMST(T) Running time of this
algorithm S (V log VE) (all shortest path)
(bottle neck) S2S log SO(S2) (to find
T)
8
Definition of Voronoi graph in graphs
(not in the plane) - well be able to get O(V log
VE)
1
Voronoi regions
1.3
2
1.5
0.5
1
shortest path conn.these 2 points
1
terminals S
v
Voronoi region of v Vor(v)
weight shortest path weight in G between
2 terminals - all the weights we find well be in
the shortest path
9
Definition of Voronoi graph in graphs
For any v?Vor(s) (s ? S) we need to know ?
(s,v) ? (v) ( ? ) - shortest path between
s1,s2 for each e?E, e(a,b) v ? r(v)
(terminals closest to v) if r(a) ? r(b)
(they do not belong to same region)
w(a,b) ? (a) w(a,b) ? (v) if ? (a)
w(a,b) ? (v) lt ? (r(a),r(b)) then ?
(r(a),r(b)) w(a,b)as result well find all the
edges of Vor. Graph Problem find ( ? ) how to
find for any node distance to its root? Find ?
(v) for any v!
10
How to find ? (v) for any v
r(v)
(auxiliary node)
v
A
terminals
other nodes
- run Dijkstra (find shortest path to any
node) - any shortest path should go through 1
terminal (if it goes through some terminal, it
belongs to that region) ? (v) ? (A,v) ?
(r(v),v) O(EV log V)