Given a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C [0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangle R contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollary of this result is an 0( n log n) approximation algorithm for the worst-case Euclidean TSP. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n)

Snyder, Timothy. L

Steele, John M

English

Kosmopolis

University of Pennsylvania
ScholarlyCommons
Operations, Information and Decisions Papers Wharton Faculty Research
1993
Equidistribution of Point Sets for the Traveling
Salesman and Related Problems
Timothy. L. Snyder
John M. Steele
University of Pennsylvania
Follow this and additional works at: http://repository.upenn.edu/oid_papers
Part of the Other Mathematics Commons
This paper is posted at ScholarlyCommons. http://repository.upenn.edu/oid_papers/260
For more information, please contact repository@pobox.upenn.edu.
Recommended Citation
Snyder, T. L., & Steele, J. M. (1993). Equidistribution of Point Sets for the Traveling Salesman and Related Problems. SODA '93
Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms, 462-466. Retrieved from http://repository.upenn.edu/
oid_papers/260
Equidistribution of Point Sets for the Traveling Salesman and Related
Problems
Abstract
Given a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that is
of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set
S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C
[0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangle
R contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollary
of this result is an 0( n log n) approximation algorithm for the worst-case Euclidean TSP. Analogous results are
proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree.
These equidistribution theorems are the first results concerning the structure of worst-case point sets like
S(n).
Disciplines
Other Mathematics
This journal article is available at ScholarlyCommons: http://repository.upenn.edu/oid_papers/260
Chapter 50 
Equidistribution of Point Sets for the 
Traveling Salesman and Related Problems 
Timothy Law Snyder* 
Abstract 
Given a set S of n points in the unit square [0, 1)2, 
an optimal traveling salesman tour of S is a tour of 
S that is of minimum length. A worst-case point set 
for the Traveling Salesman Problem in the unit square 
is a point set S(n) whose optimal traveling salesman 
tour achieves the maximum possible length among all 
point sets S C [0, 1)2, where JSI = n. An open problem 
is to determine the structure of S(n). We show that 
for any rectangle R contained in [0, 1 F, the number of 
points in S(n) n R is asymptotic to n times the area of 
R. One corollary of this result is an 0( n log n) approx-
imation algorithm for the worst-case Euclidean TSP. 
Analogous results are proved for the minimum spanning 
tree, minimum-weight matching, and rectilinear Steiner 
minimum tree. These equidistribution theorems are the 
first results concerning the structure of worst-case point 
sets like S(n). 
1. Introduction 
This paper deals with worst-case arrangements of 
points for problems in combinatorial optimization; it is 
shown that these point sets are equidistributed. For 
specificity, we concentrate on the Traveling Salesman 
Problem. 
Given a set of points Sin the unit square [0, 1)2, an 
optimal traveling salesman tour of S is a tour consist-
ing of edges from the complete graph on S and having 
total length minr{ I:eET Jel :Tis a tour of S }. Here, 
Jel denotes the Euclidean length of the edge e. We use 
J. Michael Steele** 
TSP(S) to denote the set of edges of an optimal travel-
ing salesman tour of Sand JTSP(S)J to denote the sum 
of the Euclidean lengths of the edges in TSP(S). 
A worst-case optimal traveling salesman tour is a 
tour of total length 
PTsP(n) = max JTSP(S)J. (1.1) 
5C(0,1] 2 
ISJ=n 
In words, PTsP( n) is the maximum length, over all point 
sets in [0, 1)2 of size n, that an optimal traveling sales-
man tour can attain. Such a tour attaining this length 
is called a worst-case TSP tour and its associated point 
set a worst-case TSP point set. 
The first works on the sequence PTsP( n) were 
the lower bounds of Fejes-T6th (1940) and the up-
per bounds of Verblunsky (1951). Successive improve-
ments to these bounds and their higher-dimensional 
analogues appeared in Few (1955), Supowit, Reingold, 
and Plaisted (1983), Moran (1984), Goldstein and Rein-
gold (1988), Karloff (1989), and Goddyn (1990). One 
result we use later is that 
(1.2) 
where f3TsP > 0 is a constant (Steele and Snyder ( 1989)). 
All these results deal with the worst-case length 
PTsP(n). There are no results, however, concerning the 
locations of the points that give rise to a worst-case 
tour. Let S(n) be a worst-case TSP point set. An en-
gaging open problem is to determine the structure of 
* Department of Computer Science, Georgetown University, Washington, DC 20057. Research supported in part by 
Georgetown University 1991 Summer Research Award and Georgetown University 1992 Junior Faculty Research 
Fellowship. 
** Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104. Research 
supported in part by the following grants: NSF DMS92-11634, NSA MDA904-89-H-2034, AFOSR-89-0301, and 
ARO DAAL03-91-G-0110. 
462 
EQUIDISTRIBUTION OF POINT SETS FOR THE TRAVELING SALESMAN 463 
S( n). It is generally believed that S(n) is asymptot- and label the cells Q;, where 1 :::; i :::; m2, then, for all 
ically a lattice; for example, Supowit, Reingold, and 1 :::; i :::; m2, 
Plaisted (1983) conjectured a hexagonal grid for S(n). 
Our main result is that any sequence of worst-case 
TSP point sets is asymptotically equidistributed: 
Theorem 1. If { S(n) : 1 :::; n < oo} is a sequence 
of worst-case TSP point sets, then, for any rectangle 
Rc [0,1)2, 
lim .!.lsCn) n R I = Area(R). (1.3) 
n-+oo n 
A corollary to Theorem 1 is that any worst-case Eu-
clidean traveling salesman problem can be solved within 
a factor of 1 + E, for any E > 0, in 0( n log n) time us-
ing the algorithm of Karp (1976). Even though Karp's 
algorithm is for points drawn uniformly from the unit 
square, we discuss in the concluding section how The-
orem 1 and the limit (1.2) guarantee this performance 
for worst-case point sets. 
The proof of Theorem 1 rests on the limit ( 1.2) and 
a characterization of IS(n) n R I in probabilistic terms. 
Even though S(n) is deterministic, a simple probabilis-
tic identity plays a useful role. We show in a later sec-
tion that the method provides analogous theorems for 
other problems, including the minimum spanning tree, 
minimum weight matching, and rectilinear Steiner tree. 
In Section 2, we begin with two observations that 
simplify the proof of Theorem 1 in Section 3. Section 4 
contains results and proofs for problems other than the 
Traveling Salesman Problem, and Section 5 concludes 
with speculations on extensions and applications of our 
results, including algorithmic ones. 
2. Deviations from the Mean Number 
of Points per Cell 
To prove Theorem 1, it suffices to show that if we 
divide the unit square [0, 1)2 into m2 equally-sized par-
allel subsquares or cells, each having side length 1/m, 
. IS(n) n Q;J 1 
lim = -. 
n-+oo n m2 (2.1) 
Let s(n, i) = IS(n) n Q;J; in words, s(n, i) is the 
number of points in the ith cell of a worst-case TSP 
point set. Points that lie on the boundaries of the Q; 
can be arbitrarily assigned. Furthermore, let 
2 2 
1 m 1 m 2 
Vn = - 2 L({s(n,i)}l/2- - 2 L{s(n,j)}l/2) . 
m i=l m i=l 
(2.2) 
One should note that if X is a random variable taking 
on the value {s(n,i)}1/2 with probability 1/m2, then 
Vn is the variance of X. Our first lemma estimates the 
expected value of X. 
Lemma 1. As n -+ oo, 
2 
1 m nl/2 
-2 L {s(n,j)}l/2 =-+ o(nl/2). (2.3) 
m i=l m 
Proof. 
First write (1.2) as 
PTsP(n) = /3n1/ 2 + r(n), where r(n) = o(n 1/2); (2.4) 
for convenience, we have dropped the "TSP" from f3TsP· 
Let W denote a closed walk on S(n) = {x1 , x2, ... , 
xn}, i.e, W is a sequence of edges (x;" X; 2 ), (x; 2 , X;3 ), 
... , (x;k-1> x;k ), (x;k, x; 1 ) that visits each point of S(n) 
at least once and begins and ends at the same point. 
Since W is feasible for the traveling salesman problem 
on S(n) even if W visits some points more than once 
and traverses some edges more than once, LeEW lei :::; 
ITSP(SCn))l. 
We now construct a closed walk W on S(n) as fol-
lows. Label the cells Q; with the numbers 1 through 
m2 by traversing the cells row by row, proceeding left 
464 SNYDER AND STEELE 
to right in the first row, right to left in the second, and where h(n) = o(nl/2). This proves half of the 
so on, alternating the direction of traversal in each row 
until all cells have been labeled. Next, construct in each 
cell Q; an optimal traveling salesman tour of the points 
of S(n) that lie in Q;. These disjoint, within-cell tours 
are then connected by placing an edge e; between cell 
number i and cell number i + 1, for 1 ::; i ::; m 2 - 1. 
A closed walk W that traverses each of the within-cell 
tours once and each of thee; twice can then be obtained. 
To assess the length of W, let Q; and Qj be adja-
cent cells. Since any pair of points in Q; U Qj can be 
connected by a line segment of length at most 51/2 Jm, 
PTsP(n) = jTSP(S(n))j 
:S 2::: Jej 
eEW 
lemma. To obtain the other half, we note from the 
m2 / Schwarz inequality that Li=l { s( n, i) p 2 ::; mnl/2 
2 
since I':~ 1 s(n,i) = n. 
0 
We now use Lemma 1 and the characterization of 
Vn as a variance to show that Vn/n goes to zero. 
Lemma 2. For Vn as defined in Equation (2.2), we 
bave 
Vn = o(n) (2.8) 
as n _,. oo. 
m 2 m 2 -1 
= 2::: jTSP(S(n) n Q;)j + 2 2::: je;j (2.5) Proof. 
m2 
:S 2::: jTSP(S(n) n Q;)l + 2 ·51/ 2 m. 
i=1 
We now use (2.4) in (2.5) along with the fact that 
ITSP(S(n) n Q;)l is at most PTsP(s(n, i)) scaled by the 
cell size 1 J m to get 
PTsP(n) = j3n 112 + r(n) 
m2 
:S ~ 2::: PTsP( s( n, i)) + 2 ·51 12m 
i=l 
m2 
::; ~ l:f3{s(n, i)} 1/ 2 
(2.6) 
i=l 
m2 
+ ...!_ 2::: r(s(n, i)) + 2. 51/2m, 
m i=1 
where, for all 1 ::; i ::; m2, the value r( s( n, i)) = 
o( { s( n, i)}l/2) = o( n 112). Since m is fixed, we cancel 
j3 to find 
m2 
2:{s(n,i)}112 2: mn1/2 + h(n), (2.7) 
i=l 
First note that Vn can be written as 
using the remark following definition (2.2) and the iden-
2 
tity Var( X) = EX2 (EX)2. Since I':~ 1 s( n, i) = n, 
applying Lemma 1 to the second term of (2.9) gives 
n (nl/2 )2 
Vn =-- -- +o(n112) . 
m 2 m 
(2.10) 
On expanding, we see that Vn = o(n). 
3. Equidistribution in the Worst-
Case TSP 
0 
To prove Theorem 1, recall that it suffices to show 
that limn-.oo IS(n) n Q;lfn = 1/m2 , for all1::; i::; m2. 
EQUIDISTRIBUTION OF POINT SETS FOR THE TRAVELING SALESMAN 465 
,:,From the definition (2.2) of Vn, we have that for all i 
satisfying 1 :::; i :::; m2, 
(3.1) 
Applying first Lemma 2 then Lemma 1 to (3.1) gives 
m2 
{s(n, i)} 1/2 =-; I.:{s(n,j)}l/2 + o(nl/2) 
m i=l (3.2) 
nl/2 
=-+ o(nl/2). 
m 
Squaring (3.2) yields Theorem 1. 
4. Equidistribution in the MST, 
Matching, and Steiner Problems 
0 
The method just used for the TSP can be applied 
to the minimum spanning tree, the minimum-length 
matching, and the rectilinear minimum Steiner tree. 
If L = L( S) denotes the length associated with any 
of these, then we can define Pr.(n) = supS:ISI=n L(S) 
and let s~n) be such that L(S~n)) = PTsP(n). To show 
that s~n) is asymptotically equidistributed boils down 
to checking that L satisfies two conditions: 
1. Pr.(n) = f3Ln112+o(nlf2), where j3L > 0 is constant; 
2. pr.,(n) :::; m- 1 I:Z::1 PL(sL(n, i)) + o(n112), where 
Sr.( n, i) is the number of points of S~n) contained 
in the cell Q;. 
Condition 1 has been proved for the minimum span-
ning tree, minimum matching, and rectilinear Steiner 
tree problems ( cf., Steele and Snyder (1989) and Sny-
der (1992)), and Condition 2 can be verified for these 
problems by the method used in the proof of Lemma 1. 
For example, if L( S) = MST( S) denotes the to-
tal length of a minimum spanning tree of S, we first 
form minimum spanning trees MST( s~jT n Q;) on the 
points of sSnjT in the cells Q;, where 1 :::; i :::; m 2 . The 
trees within cells can then be interconnected at total 
cost O(m) = o(n112 ) by adding m2 - 1 edges, each 
costing no more than 51/2/m. This forms a heuris-
tic tree on sSnjT. Since the lengths JMST( s8'jT n Q;) I 
are no greater than the worst-case (within-cell) lengths 
m-1PMsT(sMsT(n, i)), Condition 2 follows. 
Checking these conditions for each of the problems 
yields the following. 
Theorem 2. If { S~n) : 1 :::; n < oo} is a sequence 
of worst-case point sets for the function L, where L is 
the minimum spanning tree, the minimum matching, 
or the rectilinear minimum Steiner tree, then, for any 
rectangle R C [0, 1]2, 
lim .!_ IS~n) n R I= Area(R). (4.1) 
n-+-oo n 
5. Concluding Remarks 
The asymptotic equidistribution of worst-case point 
sets for the problems we have considered offers some 
support to the conjectures of Supowit, Reingold, and 
Plaisted (1983) that worst-case point sets are approxi-
mated by lattices as n - oo. It is still a major open 
problem to resolve these conjectures. 
The results also prove that the worst-case travel-
ing salesman problem on S(n) is amenable to Karp's 
probabilistic algorithm for the TSP, even though S(n) is 
entirely deterministic. For n points selected uniformly 
from [0, 1)2, Karp's algorithm runs in O(nlogn) time 
with probability one, and, for all c > 0, the tour formed 
by the algorithm is within a factor of 1 + E of optimum 
with probability one, as n - oo. 
Using Karp's algorithm, given c > 0, we can guar-
antee the construction of a tour T of S(n) in O(nlogn) 
time such that the total length of T is at most 1 + E 
466 SNYDER AND STEELE 
times the optimal length PTsP(n), as n-+ oo. These re- Hence, this problem also remains open. 
suits are entirely deterministic, and they can be proved 
by observing that Karp's analysis uses a probabilistic 
counterpart to the limit theorem expressed by (1.2) 
along with asymptotic equidistribution of point sets 
uniformly selected from the unit square. Though the 
limit (1.2) for the worst-case length ofthe TSP has been 
known for several years, an equidistribution result for 
S(n) was required in order to guarantee both the time 
and performance bounds for Karp's algorithm applied 
to S(n). Until there are definitive tests for worst-case 
point sets, however, this result is only of theoretical in-
terest; whether it can be used for point sets other than 
S(n) is an open problem. 
Several other open problems are motivated by our 
results. For any finite set of points S, the Steiner prob-
lem is to find a minimum-length tree T = (V, E) such 
that V contains S. The added points Q = V - S are 
the Steiner points ofT. Though any metric can be used 
to assign costs to edges, two metrics of interest are the 
Euclidean and rectilinear (Ll). 
Let S( n) be a worst-case point set for the rectilin-
ear Steiner problem, and let T be a rectilinear minimum 
Steiner tree of S(n). The asymptotic equidistribution in 
Theorem 2 applies only to S(n), and not to the set Q of 
Steiner points ofT; we conjecture that Q is asymptoti-
cally equidistributed, as well. 
For the Euclidean Steiner problem, the limit result 
for Condition 1 in Section 4 has yet to be established. 
We believe such a result holds, and it would imply that a 
worst-case point set for the Euclidean Steiner problem is 
asymptotically equidistributed. It is also likely that the 
Steiner points in the Euclidean case are asymptotically 
equidistributed. 
A final open problem concerns the greedy matching. 
Though Condition 1 in Section 4 holds for this problem, 
the methods used here to verify Condition 2 fail since a 
greedy matching is not a matching of minimum length. 
References 
Fejes-T6th, L.T. (1940). "Uber ein geometrische Satz," 
Math. Zeit. 46, 299-327. 
Few, L. (1955). "The shortest path and the shortest 
road through n points in a region," Mathe-
matika 2, 141-144. 
Goddyn, L. (1990). "Quantizers and the worst-case Eu-
clidean traveling salesman problem," J. Comb. 
Theory, Series B 50, 65-81. 
Karloff, H.J. (1989). "How long can a Euclidean trav-
eling salesman tour be?" SIAM J. Disc. Math. 
2, 91-99. 
Karp, R.M. (1976). "The probabilistic analysis of 
some combinatorial search algorithms," in Al-
gorithms and Complexity: New Directions and 
Recent Results, J.F. Traub, ed., Academic 
Press, New York, 1-19. 
Moran, S. (1984). "On the length of optimal TSP cir-
cuits in sets of bounded diameter," J. Comb. 
Theory, Series B 37, 113-141. 
Snyder, T.L. (1992). "On minimal rectilinear Steiner 
trees in all dimensions," to appear in Discrete & 
Comp. Geom.; also in Proceedings of the Sixth 
Annual Symposium on Computational Geome-
try (1990), Association for Computing Machin-
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Steele, J.M. and Snyder, T.L. (1989). "Worst-case 
growth rates of some classical problems of com-


Equidistribution of Point Sets for the Traveling Salesman and Related Problems

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsOperations, Information and Decisions Papers Wharton Faculty Research1993Equidistribution of Point Sets for the TravelingSalesman and Related ProblemsTimothy. L. SnyderJohn M. SteeleUniversity of PennsylvaniaFollow this and additional works at: http://repository.upenn.edu/oid_papersPart of the Other Mathematics CommonsThis paper is posted at ScholarlyCommons. http://repository.upenn.edu/oid_papers/260For more information, please contact repository@pobox.upenn.edu.Recommended CitationSnyder, T. L., & Steele, J. M. (1993). Equidistribution of Point Sets for the Traveling Salesman and Related Problems. SODA '93Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms, 462-466. Retrieved from http://repository.upenn.edu/oid_papers/260Equidistribution of Point Sets for the Traveling Salesman and RelatedProblemsAbstractGiven a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that isof minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point setS(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C[0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangleR contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollaryof this result is an 0( n log n) approximation algorithm for the worst-case Euclidean TSP. Analogous results areproved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree.These equidistribution theorems are the first results concerning the structure of worst-case point sets likeS(n).DisciplinesOther MathematicsThis journal article is available at ScholarlyCommons: http://repository.upenn.edu/oid_papers/260Chapter 50 Equidistribution of Point Sets for the Traveling Salesman and Related Problems Timothy Law Snyder* Abstract Given a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C [0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangle R contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollary of this result is an 0( n log n) approx-imation algorithm for the worst-case Euclidean TSP. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n). 1. Introduction This paper deals with worst-case arrangements of points for problems in combinatorial optimization; it is shown that these point sets are equidistributed. For specificity, we concentrate on the Traveling Salesman Problem. Given a set of points Sin the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour consist-ing of edges from the complete graph on S and having total length minr{ I:eET Jel :Tis a tour of S }. Here, Jel denotes the Euclidean length of the edge e. We use J. Michael Steele** TSP(S) to denote the set of edges of an optimal travel-ing salesman tour of Sand JTSP(S)J to denote the sum of the Euclidean lengths of the edges in TSP(S). A worst-case optimal traveling salesman tour is a tour of total length PTsP(n) = max JTSP(S)J. (1.1) 5C(0,1] 2 ISJ=n In words, PTsP( n) is the maximum length, over all point sets in [0, 1)2 of size n, that an optimal traveling sales-man tour can attain. Such a tour attaining this length is called a worst-case TSP tour and its associated point set a worst-case TSP point set. The first works on the sequence PTsP( n) were the lower bounds of Fejes-T6th (1940) and the up-per bounds of Verblunsky (1951). Successive improve-ments to these bounds and their higher-dimensional analogues appeared in Few (1955), Supowit, Reingold, and Plaisted (1983), Moran (1984), Goldstein and Rein-gold (1988), Karloff (1989), and Goddyn (1990). One result we use later is that (1.2) where f3TsP > 0 is a constant (Steele and Snyder ( 1989)). All these results deal with the worst-case length PTsP(n). There are no results, however, concerning the locations of the points that give rise to a worst-case tour. Let S(n) be a worst-case TSP point set. An en-gaging open problem is to determine the structure of * Department of Computer Science, Georgetown University, Washington, DC 20057. Research supported in part by Georgetown University 1991 Summer Research Award and Georgetown University 1992 Junior Faculty Research Fellowship. ** Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104. Research supported in part by the following grants: NSF DMS92-11634, NSA MDA904-89-H-2034, AFOSR-89-0301, and ARO DAAL03-91-G-0110. 462 EQUIDISTRIBUTION OF POINT SETS FOR THE TRAVELING SALESMAN 463 S( n). It is generally believed that S(n) is asymptot- and label the cells Q;, where 1 :::; i :::; m2, then, for all ically a lattice; for example, Supowit, Reingold, and 1 :::; i :::; m2, Plaisted (1983) conjectured a hexagonal grid for S(n). Our main result is that any sequence of worst-case TSP point sets is asymptotically equidistributed: Theorem 1. If { S(n) : 1 :::; n < oo} is a sequence of worst-case TSP point sets, then, for any rectangle Rc [0,1)2, lim .!.lsCn) n R I = Area(R). (1.3) n-+oo n A corollary to Theorem 1 is that any worst-case Eu-clidean traveling salesman problem can be solved within a factor of 1 + E, for any E > 0, in 0( n log n) time us-ing the algorithm of Karp (1976). Even though Karp's algorithm is for points drawn uniformly from the unit square, we discuss in the concluding section how The-orem 1 and the limit (1.2) guarantee this performance for worst-case point sets. The proof of Theorem 1 rests on the limit ( 1.2) and a characterization of IS(n) n R I in probabilistic terms. Even though S(n) is deterministic, a simple probabilis-tic identity plays a useful role. We show in a later sec-tion that the method provides analogous theorems for other problems, including the minimum spanning tree, minimum weight matching, and rectilinear Steiner tree. In Section 2, we begin with two observations that simplify the proof of Theorem 1 in Section 3. Section 4 contains results and proofs for problems other than the Traveling Salesman Problem, and Section 5 concludes with speculations on extensions and applications of our results, including algorithmic ones. 2. Deviations from the Mean Number of Points per Cell To prove Theorem 1, it suffices to show that if we divide the unit square [0, 1)2 into m2 equally-sized par-allel subsquares or cells, each having side length 1/m, . IS(n) n Q;J 1 lim = -. n-+oo n m2 (2.1) Let s(n, i) = IS(n) n Q;J; in words, s(n, i) is the number of points in the ith cell of a worst-case TSP point set. Points that lie on the boundaries of the Q; can be arbitrarily assigned. Furthermore, let 2 2 1 m 1 m 2 Vn = - 2 L({s(n,i)}l/2- - 2 L{s(n,j)}l/2) . m i=l m i=l (2.2) One should note that if X is a random variable taking on the value {s(n,i)}1/2 with probability 1/m2, then Vn is the variance of X. Our first lemma estimates the expected value of X. Lemma 1. As n -+ oo, 2 1 m nl/2 -2 L {s(n,j)}l/2 =-+ o(nl/2). (2.3) m i=l m Proof. First write (1.2) as PTsP(n) = /3n1/ 2 + r(n), where r(n) = o(n 1/2); (2.4) for convenience, we have dropped the "TSP" from f3TsP· Let W denote a closed walk on S(n) = {x1 , x2, ... , xn}, i.e, W is a sequence of edges (x;" X; 2 ), (x; 2 , X;3 ), ... , (x;k-1> x;k ), (x;k, x; 1 ) that visits each point of S(n) at least once and begins and ends at the same point. Since W is feasible for the traveling salesman problem on S(n) even if W visits some points more than once and traverses some edges more than once, LeEW lei :::; ITSP(SCn))l. We now construct a closed walk W on S(n) as fol-lows. Label the cells Q; with the numbers 1 through m2 by traversing the cells row by row, proceeding left 464 SNYDER AND STEELE to right in the first row, right to left in the second, and where h(n) = o(nl/2). This proves half of the so on, alternating the direction of traversal in each row until all cells have been labeled. Next, construct in each cell Q; an optimal traveling salesman tour of the points of S(n) that lie in Q;. These disjoint, within-cell tours are then connected by placing an edge e; between cell number i and cell number i + 1, for 1 ::; i ::; m 2 - 1. A closed walk W that traverses each of the within-cell tours once and each of thee; twice can then be obtained. To assess the length of W, let Q; and Qj be adja-cent cells. Since any pair of points in Q; U Qj can be connected by a line segment of length at most 51/2 Jm, PTsP(n) = jTSP(S(n))j :S 2::: Jej eEW lemma. To obtain the other half, we note from the m2 / Schwarz inequality that Li=l { s( n, i) p 2 ::; mnl/2 2 since I':~ 1 s(n,i) = n. 0 We now use Lemma 1 and the characterization of Vn as a variance to show that Vn/n goes to zero. Lemma 2. For Vn as defined in Equation (2.2), we bave Vn = o(n) (2.8) as n _,. oo. m 2 m 2 -1 = 2::: jTSP(S(n) n Q;)j + 2 2::: je;j (2.5) Proof. m2 :S 2::: jTSP(S(n) n Q;)l + 2 ·51/ 2 m. i=1 We now use (2.4) in (2.5) along with the fact that ITSP(S(n) n Q;)l is at most PTsP(s(n, i)) scaled by the cell size 1 J m to get PTsP(n) = j3n 112 + r(n) m2 :S ~ 2::: PTsP( s( n, i)) + 2 ·51 12m i=l m2 ::; ~ l:f3{s(n, i)} 1/ 2 (2.6) i=l m2 + ...!_ 2::: r(s(n, i)) + 2. 51/2m, m i=1 where, for all 1 ::; i ::; m2, the value r( s( n, i)) = o( { s( n, i)}l/2) = o( n 112). Since m is fixed, we cancel j3 to find m2 2:{s(n,i)}112 2: mn1/2 + h(n), (2.7) i=l First note that Vn can be written as using the remark following definition (2.2) and the iden-2 tity Var( X) = EX2 (EX)2. Since I':~ 1 s( n, i) = n, applying Lemma 1 to the second term of (2.9) gives n (nl/2 )2 Vn =-- -- +o(n112) . m 2 m (2.10) On expanding, we see that Vn = o(n). 3. Equidistribution in the Worst-Case TSP 0 To prove Theorem 1, recall that it suffices to show that limn-.oo IS(n) n Q;lfn = 1/m2 , for all1::; i::; m2. EQUIDISTRIBUTION OF POINT SETS FOR THE TRAVELING SALESMAN 465 ,:,From the definition (2.2) of Vn, we have that for all i satisfying 1 :::; i :::; m2, (3.1) Applying first Lemma 2 then Lemma 1 to (3.1) gives m2 {s(n, i)} 1/2 =-; I.:{s(n,j)}l/2 + o(nl/2) m i=l (3.2) nl/2 =-+ o(nl/2). m Squaring (3.2) yields Theorem 1. 4. Equidistribution in the MST, Matching, and Steiner Problems 0 The method just used for the TSP can be applied to the minimum spanning tree, the minimum-length matching, and the rectilinear minimum Steiner tree. If L = L( S) denotes the length associated with any of these, then we can define Pr.(n) = supS:ISI=n L(S) and let s~n) be such that L(S~n)) = PTsP(n). To show that s~n) is asymptotically equidistributed boils down to checking that L satisfies two conditions: 1. Pr.(n) = f3Ln112+o(nlf2), where j3L > 0 is constant; 2. pr.,(n) :::; m- 1 I:Z::1 PL(sL(n, i)) + o(n112), where Sr.( n, i) is the number of points of S~n) contained in the cell Q;. Condition 1 has been proved for the minimum span-ning tree, minimum matching, and rectilinear Steiner tree problems ( cf., Steele and Snyder (1989) and Sny-der (1992)), and Condition 2 can be verified for these problems by the method used in the proof of Lemma 1. For example, if L( S) = MST( S) denotes the to-tal length of a minimum spanning tree of S, we first form minimum spanning trees MST( s~jT n Q;) on the points of sSnjT in the cells Q;, where 1 :::; i :::; m 2 . The trees within cells can then be interconnected at total cost O(m) = o(n112 ) by adding m2 - 1 edges, each costing no more than 51/2/m. This forms a heuris-tic tree on sSnjT. Since the lengths JMST( s8'jT n Q;) I are no greater than the worst-case (within-cell) lengths m-1PMsT(sMsT(n, i)), Condition 2 follows. Checking these conditions for each of the problems yields the following. Theorem 2. If { S~n) : 1 :::; n < oo} is a sequence of worst-case point sets for the function L, where L is the minimum spanning tree, the minimum matching, or the rectilinear minimum Steiner tree, then, for any rectangle R C [0, 1]2, lim .!_ IS~n) n R I= Area(R). (4.1) n-+-oo n 5. Concluding Remarks The asymptotic equidistribution of worst-case point sets for the problems we have considered offers some support to the conjectures of Supowit, Reingold, and Plaisted (1983) that worst-case point sets are approxi-mated by lattices as n - oo. It is still a major open problem to resolve these conjectures. The results also prove that the worst-case travel-ing salesman problem on S(n) is amenable to Karp's probabilistic algorithm for the TSP, even though S(n) is entirely deterministic. For n points selected uniformly from [0, 1)2, Karp's algorithm runs in O(nlogn) time with probability one, and, for all c > 0, the tour formed by the algorithm is within a factor of 1 + E of optimum with probability one, as n - oo. Using Karp's algorithm, given c > 0, we can guar-antee the construction of a tour T of S(n) in O(nlogn) time such that the total length of T is at most 1 + E 466 SNYDER AND STEELE times the optimal length PTsP(n), as n-+ oo. These re- Hence, this problem also remains open. suits are entirely deterministic, and they can be proved by observing that Karp's analysis uses a probabilistic counterpart to the limit theorem expressed by (1.2) along with asymptotic equidistribution of point sets uniformly selected from the unit square. Though the limit (1.2) for the worst-case length ofthe TSP has been known for several years, an equidistribution result for S(n) was required in order to guarantee both the time and performance bounds for Karp's algorithm applied to S(n). Until there are definitive tests for worst-case point sets, however, this result is only of theoretical in-terest; whether it can be used for point sets other than S(n) is an open problem. Several other open problems are motivated by our results. For any finite set of points S, the Steiner prob-lem is to find a minimum-length tree T = (V, E) such that V contains S. The added points Q = V - S are the Steiner points ofT. Though any metric can be used to assign costs to edges, two metrics of interest are the Euclidean and rectilinear (Ll). Let S( n) be a worst-case point set for the rectilin-ear Steiner problem, and let T be a rectilinear minimum Steiner tree of S(n). The asymptotic equidistribution in Theorem 2 applies only to S(n), and not to the set Q of Steiner points ofT; we conjecture that Q is asymptoti-cally equidistributed, as well. For the Euclidean Steiner problem, the limit result for Condition 1 in Section 4 has yet to be established. We believe such a result holds, and it would imply that a worst-case point set for the Euclidean Steiner problem is asymptotically equidistributed. It is also likely that the Steiner points in the Euclidean case are asymptotically equidistributed. A final open problem concerns the greedy matching. Though Condition 1 in Section 4 holds for this problem, the methods used here to verify Condition 2 fail since a greedy matching is not a matching of minimum length. References Fejes-T6th, L.T. (1940). "Uber ein geometrische Satz," Math. Zeit. 46, 299-327. Few, L. (1955). "The shortest path and the shortest road through n points in a region," Mathe-matika 2, 141-144. Goddyn, L. (1990). "Quantizers and the worst-case Eu-clidean traveling salesman problem," J. Comb. Theory, Series B 50, 65-81. Karloff, H.J. (1989). "How long can a Euclidean trav-eling salesman tour be?" SIAM J. Disc. Math. 2, 91-99. Karp, R.M. (1976). "The probabilistic analysis of some combinatorial search algorithms," in Al-gorithms and Complexity: New Directions and Recent Results, J.F. Traub, ed., Academic Press, New York, 1-19. Moran, S. (1984). "On the length of optimal TSP cir-cuits in sets of bounded diameter," J. Comb. Theory, Series B 37, 113-141. Snyder, T.L. (1992). "On minimal rectilinear Steiner trees in all dimensions," to appear in Discrete & Comp. Geom.; also in Proceedings of the Sixth Annual Symposium on Computational Geome-try (1990), Association for Computing Machin-ery. Steele, J.M. and Snyder, T.L. (1989). "Worst-case growth rates of some classical problems of com-

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Equidistribution of Point Sets for the Traveling Salesman and Related Problems

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