A linear time algorithm for the
weighted lexicographic rectilinear
1-center problem in the plane
Nir Halman, Technion, Israel
This work is part of my Ph.D. thesis, held in Tel Aviv
University, under the supervision of Professor Arie Tamir
Planar 1-center problems
Input: a set P={p1,…,pn} of n points pi=(xi,yi)
a set W={w1,…,wn} of positive weights
Output: a center p=(x,y) which minimizes
f(p)=Maxi {wi d(p,pi )}
where d is a distance function
Motivation: customers, service center, 1/wi =speed
d=l2: solvable in linear time [M83],[D86],[SW96]
Rectilinear 1-center (d=l1 )
c = Minp Maxi {wi (|xi-x|+|yi-y|)}
Min
c
s.t. wi (|xi-x|+|yi-y|)  c, i=1,…,n
Min
c
s.t. wi ((xi-x)+(yi-y))  c, i=1,…,n,
wi (-(xi-x)+(yi-y))  c, i=1,…,n,
wi ((xi-x)-(yi-y))  c,
i=1,…,n,
wi (-(xi-x)-(yi-y))  c, i=1,…,n.
Transform the problem into one where d=l
1-center problem with d=l
c = Minp Maxi {wi Max{|xi-x|,|yi-y|}}
Min
s.t.
c
wi|xi-x|  c,
i=1,…,n,
wi|yi-y|  c, i=1,…,n.
Decomposition into 2 one-dimensional problems
Min
cx
s.t. wi|xi-x|  cx i=1,…,n.
c=Max{cx , cy}
Min
cy
s.t. wi|yi-y|  cy i=1,…,n.
Lex 1-center problem [O97]
Unique solutions to 1-dimensional problems and
Euclidean planer 1-center
This is not the case for planar 1-center with l1 , l
Input: a set P={p1,…,pn} of n points pi=(xi,yi)
a set W={w1,…,wn} of positive weights
Output: a center p=(x,y) which lex-minimizes
lf(p)=q(w1d(p,p1 ),…,wn d(p,pn ))
where q is the non-decreasing ordering of a multi-set
Example
Y
p3=(3,2)
p1=(-4,1)
p5=(5,0)
X
p2=(6,-1)
p4=(2,-3)
The distances vector is lf(p)=(5,5,4,2½, 2½)
No O(n log n) time algorithm exists
Talk outline
 • Introduction
• An (n log n) lower bound for calculating
the optimal value of the optimal solution
• A restricted problem, the min rays problem
and the relations between them
• Solving the min rays problem in linear time
• Putting it all together
Lower bound
Obs 1: it takes (n log n) time to compute the
distances vector of a given set of n real points
Proof: reduction from sorting
Given is a set R={r1,…,rn} of n real numbers
Suppose the minimal element is 0
Set P={r1,…,rn ,2rmax}
Center is rmax Distances vector yields sorting of R
Talk outline
 • Introduction
 • An (n log n) lower bound for calculating
the optimal value of the optimal solution
• A restricted problem, the min rays problem
and the relations between them
• Solving the min rays problem in linear time
• Putting it all together
Restricted problem
Input: P={p1,…,pn}, W={w1,…,wn} as before and
a real number x@
Output: a center p@ =(x@,y) which lex-minimizes
lf(p@)=q(w1d(p@,p1 ),…,wn d(p@,pn ))
Set ai = wi|x@- xi|, gi(y)=max{ai, wi|y-yi|}, then
lf(p@)=q(g1(y),…,gn(y ))
Obs 2: Each gi(y) has a
shape
Min rays problem
 gi(y) define r-i(y) as
and r+i(y) as
Let u(y) be upper envelop of {r-i(y),r+i(y) | i=1,…,n}
Goal: calculate c = min u(y)
+
r-1
Y
r
r-2
p3=(3,2)
p1=(-4,1)
r+5
r+2
C
p5=(5,0)
r-4
X

r-3
1
r+4
r+3
p2=(6,-1)
p4=(2,-3)
Y
Restricted vs. min rays
Thm 1: y$, c$ an optimal solution of the min
rays problem  (x@, y$) is an optimal center
in the corresponding restricted problem
Proof:
C
C
Y
Y
Talk outline
 • Introduction
 • An (n log n) lower bound for calculating
the optimal value of the optimal solution
 • A restricted problem, the min rays problem
and the relations between them
• Solving the min rays problem in linear time
• Putting it all together
Solving min rays problem
Let y$, c$ be an optimal solution of the min rays
problem, p@=(x@, y$) be an optimal center for the
corresponding restricted problem.
Oracle:
Input: same as for min rays problem with an
additional point p =(x@, y)
Output: y$=y or y$>y, or y$<y
Lem 1: the test oracle can be implemented in
linear time
Solving min rays problem (cont’)
Thm 2: the min rays problem is solvable in
linear time
Proof: partition the given 2n rays into n pairs.
Group the pairs of rays into the 12 distinct sets:
index
1
2
3
4
type
r
r’
r
mix
r’
inc
r
r
r’
r
r’
r’
r’
r
r’
r
r’
r
Solving min rays problem (cont’)
Proof (cont’):
r’
½ of the rays are dropped
r
r
r’
¼ of rays are dropped
r’
r
1/
8
of rays are dropped
Talk outline
 • Introduction
 • An (n log n) lower bound for calculating
the optimal value of the optimal solution
 • A restricted problem, the min rays problem
and the relations between them
 • Solving the min rays problem in linear time
• Putting it all together
Putting it all together
Algorithm:
• transform the input to l
• solve non-lex problem
• solve min rays problem
Generalizes to (in the same time bound):
• lex 1-center in Rd with l norm (quadratic
dependency in d )
• different weights on each of the axes
• all centers
Thank you !