Quantizers and the worst-case Euclidean traveling salesman problem

JOURNAL

OF COMBINATORIAL

Series B 50, 65-81

THEORY,

(1990)

Quantizers and the Worst-Case Euclidean Traveling Salesman Problem Lurs A. Department University

of Combinatorics and Optimization, of Waterloo, Waterloo, Ontario. Canada

Communicated

by the Managing

Received

For k problem n points possible.

GODDYN*

Editors

June 13, 1988

> 2 we consider a worst-case instance of the Euclidean traveling salesman through n points in the unit k-cube [0, Ilk. That is, for each n we arrange in the k-cube so that the shortest circuit containing them is as long as Such optimum circuits are known to have length asymptotic to

as n -+ to, where ak depends only on k and is bounded. We are concerned with the problem of determining a, for small values of k and also the asymptotic behavior of ak as k + co. It was shown by Few (Mathematika 2 (1955), 141-144) that 3-““3, (r(l

+k/2))‘~k(~k)-1’2~t(k~k1’2(2-“2(k-1)-’~2)’k-’)ik

so that, for large k, 0.2419 <,ak(0.7071. Later, Moran (f. Combin. Theory Ser. B 37 (1984), 113-141) improved the upper bound, for large k, to a,50.6136. Few’s upper bound is better than Moran’s for small values of k. Here, we sharpen each of these upper bounds by a factor of about 4. By generalizing the classic “strip” algorithm used by Few, we obtain

where yk is the mean deviation per dimension of the best quantizer of unit density in Ek. yk can be roughly defined as follows: Let R be a very large k-dimensional cube with integer volume M, and let z be a uniformly distributed random point in R. Then yk is the least possible expected value of I/k l/z - Q(z)11 2, over all functions Q from R onto an M-subset of R (Q is called a quantizer for R). Numerically, we have yI = 0.25, yk <0.271/G and yk & + 0.2420... (not monotonically) as k+ co. We also review Moran’s asymptotic upper bound on ak. By using recent estimates on the maximum sphere-packing density, we improve Moran’s argument to give ak 5 0.4051. Again, the latter bound is the better one only when k is large. 0 1990 Academic

* Support

from

Press, Inc.

NSERC

is gratefully

acknowledged.

65 0095-8956190

$3.00

Copyright KI 1990 by Academic Press, Inc. All rights of reproduction m any form reserved.

66

LUIS

A. GODDYN

1. INTRODUCTION The purpose of this paper is to give improved upper bounds on the asymptotic worst case behavior of the Euclidean traveling salesman problem. Given a finite subset X of Euclidean k-space [Ek,let the length of X, E(X), be the minimum arc length over all circuits which contain X. The problem of determining I(X) is the k-dimensional Euclidean traveling salesmanproblem (k-TSP). Although, the k-TSP is known to be NP-hard for k = 2 and hence for all k> 2 [27], there are polynomial-time algorithms [19, 20, 23, 24, 321 which find nearly optimal solutions with high probability. Unfortunately, such heuristics do not lend themselves well to statistical and asymptotic analysis. The statistical behavior of 1(X) is studied by letting X” be a collection of n independent identically distributed points with some fixed probability distribution. This makes 1(Y) a random variable whose stochastic properties have been studied since the early 1940s (see [2, 16, and references therein]). In a classic paper [2], Beardwood, Halton, and Hammersley showed that, under quite general conditions, I(Y) behaves very predictably when n is large. We illustrate this result by describing the special case of a uniform distribution on some compact measurable region R in [Ek with k-measure (or volume) p(R) > 0 and with a “nice” boundary (say, a (k - 1 )-manifold). Let X” denote the first n points of an infinite sequence of independent random points in R. Then, with probability tending to one as n -+ 00, f(P) - Bk ,,h p(R)‘/” nCkP‘Ilk,

where Pk depends only on k and is bounded. It is not surprising that fik is independent of the “shape” of R since “boundary effects” tend to diminish as n becomes large. Subsequent works [19, 20, 24, 303 have improved some of the results, proofs, and estimates for Pk. It is known, for example, that

-
Xr R, 1x1 =n}.

(2)

Since the above supremum is taken over a compact subset of Rk” we may, if we wish, replace the “sup” with “max.” An n-set is a longest set in R if

EUCLIDEAN

TRAVELING

SALESMAN

PROBLEM

67

Z(X) = Z,(n). As in the random case, Beardwood et al. showed that lim supn _ o. Z,(n)n” ~ k)/k is bounded and depends only on the volume, not the shape, of R. Recently, Steele and Snyder [31] have shown that, as in (1 ), the limit exists, I,(n) - cxk $

p(R)‘lk

rick- “lk,

(3)

where elk depends only on k, when R is a unit cube. The two results imply that (3) holds for all compact regions R of positive k-volume, with a “nice” boundary. For convenience, one usually takes R to be the closed unit k-cube [0, 11”. Some authors (for example [25]) take R to be a k-dimensional sphere or a region of bounded diameter. We turn to the problem of determining elk. As for lower bounds, it is clear that

A much better lower bound was noticed by Fejes-Toth [ 111 for k = 2 and by Few [ 141 for general k. By setting X to be the set of centers of n nonoverlapping congruent k-spheres which are densely packed into R, Few showed that, for all k k 1, cI

where Bk is the maximum into Ek. and where

k,

>2,l-Wv-‘lk

k

SW k

9

(4)

density to which equal spheres may be packed

(5)

is the volume of a unit k-sphere. By Stirling’s formula, large k,

we note that, for

Good lower bounds for 8k are known for small k (see Table I). For larger k, we have the lower and upper bounds of Roger [29, pp. 4-131 K(k) et1 -e-k)2k-”


where i(k) is the Riemann zeta function. Asymptotically, that 0.5 Nkw* < 6 Ilk < 0 6602 .

it is known [21]

(7)

cc

1 2 3 4 5 6 7 8 9 10 11 12 13 14

k

0

32x5/(98555 $) #/19440 0.001168 0.000629

.&“J$

~“1384

rr3/105

1 x/(3 $1 7s $1 x2/16 n2/(15 &I ~‘(48 fi)

Lower

6,

0

1.0000 0.9069 0.7405 0.7500 0.6187 0.5000 0.3978 0.3125 0.243 1 0.1875 0.1436 0.1094 0.0826 0.0625

Upper

0.2420

0.2500 0.2659 0.2686 0.2683 0.2673 0.2661 0.2649 0.2637 0.2627 0.2617 0.2609 0.2601 0.2594 0.2587

Lower

and upper

0.2420

(rand)

CD:,) (DA)

(K,2)

G-1) (AZ) (W (D4*) (D:) (0,’ 1 W) V’,‘) CL’,’ ) P:o) (D:, 1

Upper (9) 0.2500 0.2667 0.2708 0.2707 0.2706 0.2698 0.2674 0.2658 0.265 1 0.2649 0.2647 0.2648 0.2652 0.2654

Yk Ji;

Lower

TABLE

I

314

(2ae)-“’

3-W

2-l 21’23 - 1 (4/.3) I/Z0 5-l”

2311O3 - 112 3-W 2X/77 - l/2

2

3-W 21/63-W

bounds

N/A

Lower

0.2420

0.7598 0.6481 0.5946 0.5506 0.5268 0.5087 0.5000 0.4714 0.4537 0.4390 0.4387 0.3324 0.3275

UP

0.4840

1.0000 0.9042 0.8363 0.7861 0.7510 0.7234 0.6982 0.6789 0.6646 0.6536 0.6443 0.6369 0.6312

N/A

Thm. 2

Upper

0.4051

1.9268 1.2326 1.0555 9.9239 0.8416 0.7852 0.7436 0.7117 0.6863 0.6656 0.6483 0.6336 0.6211

N/A

Thm.

1

2

8

8

?

EUCLIDEAN TRAVELINGSALESMAN

PROBLEM

69

Details on sphere packing density can be found in [9, 12, 291. For large k, (7) gives the asymptotic lower bound n&&0.24197. The remainder of this paper deals with upper bounds on elk. In general, one finds upper bounds on elk by analyzing specific TSP heuristic algorithms. Two different approaches have been proposed. One is the so-called “strip” algorithm which was used most successfully by Few [14] and the other is a recursive algorithm introduced by Moran [25]. The bound on c(~ obtained by Moran’s algorithm is better than that resulting from Few’s algorithm, although only when k is large. We shall review these two algorithms and show that both upper bounds on uk can be improved by a factor of about $. These improved upper bounds are illustrated in Fig. 2. In the sequel, ilx112 denotes the Euclidean norm while d(x, y) = 11x- yllz is the distance from x to 4’. 2. MORAN'S

BOUND

For large values of k, the best known upper bound on elk is that found by Moran [25]. Like Few’s lower bound (4), (8), Moran’s upper bound depends on the sphere packing density bk. In terms of dk, Moran’s bound determines elk to within a factor of 3 - fix 1.268, for large k. We outline here the proof of Moran’s bound. THEOREM

1. For all dimensionsk 2 2, uk 6 2(3 - fi)(,,‘%/(k

- 1)) V;‘lk ~5;‘~.

Proof: Let R be a unit k-cube in lEk and let X be a set of n points in R. Pick x, y E X so that d(x, y) is minimum. Consider a shortest route

through the (n- l)-set X’= (X\{x, y})u {z}, where z is the midpoint of x and y. We may replace the portion a, z, b of this route which contains z with either a, x, y, b or u, y, x, b (whichever is shorter) to obtain a route through X. Using the minimality of d(x, y) it is not hard to show [22; 25, p. 1261 that the resulting route through X is longer than that through X’ by at most (3 - fi) d(x, y). (The extremal case occurs when Waxy and A bxy are both equilateral triangles.) Thus, by induction on n, Z(X)<(3-fi)(d(2)+d(3)+

... +d(n)),

(9)

where d(r) is the greatest possible value of min{d(x, y) 1x # y, x, y E X} over all r-sets XG R. Ignoring boundary effects, d(r) equals twice the radius

70

LUIS A. GODDYN

of r congruent 6, = rV,(d(r)/2)k have

k-spheres which are densely packed into R. That is, + o(l) so d(r) = 2V,‘lk 8L’krP1’k + o(r-‘jk). From (9) we

This completes the proof.

1

Moran [25] used the trivial estimate 6, < 1 to obtain CrkSo.6136. However, one does much better by using the more sophisticated bounds of (6), (7), and Table I. COROLLARY

1.

and asymptotically, a < 2(3 - $)(0.6602)

x o 4o5 1

km JG?

.

....

(When simplifying the first inequality, a factor of ((k + 2)/2)‘lk was incorporated into the gamma function.) It is natural to try to improve Moran’s argument. Other than the obvious improvement that a better upper bound on 6, would effect, there are a couple of strategies that might prove fruitful. It is plausible, for example, that if the distance between a closest pair of points in a set X is only slightly less than the upper bound d(n) provided by the densest sphere packing, then there must be many (say cn for some constant c) disjoint pairs of points in X at distance at most only slightly more than d(n) (say d(n) + E.) Instead of replacing the nearest pair of points in X with a single point between them, one would replace each of the cn pairs of nearly nearest pairs with the point halfway between them. The right-hand side of (9) would then look something like

which is better than (9) provided that E can be shown to be small. A related possibility is to attempt to replace a relatively “dense clump”

EUCLIDEAN TRAVELING

SALESMAN PROBLEM

71

of points S s X by a single point s at the “centroid” of S. The problem here is to find appropriate definitions for dense clump and centroid, and then to show that any set X has a subset S which is dense enough that modifying a route through (X-S) u (s} so that it becomes a route through X increases its length by less than (3 - &)( 1SI - 1). Such an argument could result in the factor 3 - 4 of Theorem 1 being replaced by a smaller factor.

3. "STRIP"

ALGORITHMS

For small values of k, the best upper bounds for elk come from analyzing variations of what has been called the “strip” algorithm. In 1951, Verblunski [34] showed that CQ< 0. Later, Few [14] showed that cl* $1 and that ak < &(2(k - 1 ))‘I -k)‘(2k) for k > 3. Independently, Beardwood et al. [2] obtained the weaker bound tlk <,/% (k- 1)‘1-k”(2k’. Supowit, Reingold, and Plaisted [32] have rediscovered Few’s upper and lower bounds for q. Recently, Karloff [22] has found a sharp improvement a2 < 0.9840, and Goldstein and Reingold [ 171 have sharpened Few’s upper bound for k > 3 by about 5 %. All of these upper bounds on elk were found by using strip algorithms. Strip algorithms are also important components of fast heuristics described in [19, 22-24, 353 for solving k-TSP problems. Upper bounds on the Euclidean matching problem [28] and the Euclidean Steiner tree problem [43 have also been obtained using a variation of a strip algorithm. We now describe what is meant by a strip algorithm. Given n points p,, .... pn in the unit k-cube R = {(x,, .... 2,): 0
I YI + 2 i i=

IWp,II2 + 4 Y)

(10)

I

since each <, has length 1 and since the horizontal

line segments have total

72

LUIS A. GODDYN

length equal to that of a (k - 1)-dimensional traveling salesman tour through Y. A strip algorithm is any method of choosing a set Y, given the set of points pi, ideally so that the length of G?(Y) is not too large. In the strip algorithm of Few [14], Y was chosen to be the better of two candidates, Y’n F and Y” n F, where Y’ is the (scaled) cubic lattice isomorphic to rZkP ’ and where Y” is its “body-centered” translation, Y’ + (r/2, .... r/2). (The scalar r depends on n and k.) Our improvement results from choosing Y’ more sensibly and then defining Y to be the best of an infinite number of candidates of the form Y’ + t (mod l), where t is any translation vector in the face F. Intuitively, we try to chose Yin such a way that the detour segments $S,,, are, on average, as short as possible. In some sense, we want to spread the points in Y throughout the (k - l)-dimensional face F as “evenly” as possible. For this, we need a concept from Information Theory.

4. QUANTIZERS Imagine, if you will, a large wall covered with a large fixed number of small dots. You throw a dart z which lands, sticking, with a uniform probability distribution. The 2-dimensional quantization problem is to arrange the dots so that the expected distance from z to the nearest dot is minimized. We now formalize and generalize this notion. Let R be a bounded region of positive volume in Ek. A k-dimensional quantizer of size M in R is a subset Y of R consisting of M points, together with a function Q: R + Y which maps z E R to a point in Y which is nearest to z (with respect to Euclidean distance.) That is, Q and Y satisfy d(z, Q(z)) = TGi; d(z, Y), Quantizers are studied as a means of “making discrete” or “quantizing” blocks of data into a finite number of “output levels” (see, for example, [ 1, 5-9, 15, 26, 381). Where z is a random point with a uniform probability distribution in R, we are interested in minimizing the expected error (also called expected deviation or first absolute moment) of Q

u,(Y) := E llz- Q(z)llz

(11)

over all quantizers Q of size A4. A quantizer is optimal for R if it minimizes UR( n Most of the literature on quantizers [l, 5-9, 36, 373 concentrates on

EUCLIDEAN

TRAVELING

SALESMAN

PROBLEM

73

minimizing the mean squared error R //z - Q(z)11:, mainly because the integrals involved are easier to evaluate [S]. Unfortunately, our application calls for minizing the expected error. However, as observed in [38], quantizers which minimize the mean squared error tend also to give optimal or nearly optimal expected error. The expected error is usually normalized to a dimensionless quantity which is independent of R and of scalar expansion of Y. One way to do this is to let R be a fixed region of unit volume and to study the optimum quantization

error

U(M)

:= min{ U,(Y):

YE R, 1YI = M}

(12)

as M becomes large. Zador [38] proved that, as A4 -+ co, the “shape” of R becomes irrelevant, and that ; U(M)M”k

-+ yk,

where the constant yk depends only on the dimension k of R. (In fact, he even considered using (13) to define the dimension k of a set R.) The factor l/k in (13) facilitates the comparison of two quantizers of different dimension. One might describe yk as the optimum normalized quantization error per dimension. However, we shall call yk the optimum quantization error coefficient. Further, Zador [38] showed that

where V, is defined in (5). The lower bound on yk, called the “sphere bound,” is obtained by assuming that each Voronoi region of Y (i.e., preimage of Q) is a perfect k-sphere. Recently, Conway and Sloane [S] have proposed (without giving a proof) a sharper lower bound on yk. The upper bound in (14) is obtained non-constructively by letting Y be a set of points independently chosen with uniform probability from R. Surprisingly, the two bounds in (14) agree asymptotically, giving yk -+ &

% 0.24197/$.

(15)

For small values of k, Zador’s upper bound (14) is poor and one can do much better by calculating the expected error U,(y) for specific lattices 3. In [S-9] Conway and Sloane have shown that for k < 8 certain root

74

LUIS

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lattices have very small mean squared errors and that certain other lattices, such as the Leech lattice, do very well in higher dimensions. In two and three dimensions, the lattice quantizers which minimize the mean squared error are known to be the hexagonal lattice, A,, [ 18,261 and the bodycentered cubic lattice, DC, [ 11, respectively. (D: is sometimes denoted as A; or 2: .) Conway and Sloane have an interesting conjecture [6] that the best quantizer with respect to mean squared error is the lattice which is dual to the lattice which gives the densest sphere packing. One can expect the expected error of these lattices to give very good upper bounds On

Yk.

Where R is the Voronoi region containing the origin in a k-dimensional lattice 2, we can express the quantization error coefficient of .?& as

Note that y(Y) is invariant over scalar expansion of 2 and that Yk < y(Z). For k = 1, 2, the optimum quantizers (with respect to both the expected error and the mean square error) are Z and the hexagonal lattice A, [ 18, 261. In these cases, one can evaluate (16) analytically to obtain y1 = 0.25 4/3 + ln(3) Y2=

2512 .33/4

M 0.188598368.

The presence of a square root in the integrand makes the exact determination of y(Y) extremely difficult when k 2 3. For such k, we resort to Monte Carlo experiments similar to those described in [8] in order to estimate y(JE’) for the good lattice quantizers of Conway and Sloane, and obtain good upper bounds on Yk. In fact there is no reason to restrict ourselves to lattices 2 when estimating (16). Any configuration of points whose Voronoi regions are congruent do just as well. For example, we have found that the configuration, denoted 0: in [9], which is the union of two cosets of the root lattice D, is the best known quantizer for 7
5.

THE

STRIP

BOUND

Using a strip algorithm, we can bound elk in terms of the optimum quantization error coefficient yk.

EUCLIDEAN

TRAVELING

SALESMAN

THEOREM 2. For all dimensions k > 2, tlk < &

75

PROBLEM

(2y, _ I)‘k-

“lk.

Proof. We use the notation of Section 3. Let P = { piI i = 1, .... a} be n points in the unit k-cube. Let Y’ be the points of an optimum quantizer of size M in the (k - l)-dimensional face F of R. For any point t in this face denote by Y’ + t c_ R the “translated” point set { y + t(mod 1): y E Y’ }. For t E F, consider a tour V ( Y’ + t) as described in Section 3. For each pi E P the expected length of the detour BPl over all choices of t is U,( Y’) = U(M). (See (ll), (12).) Thus, from (lo), the expected length of U(Y’+t) is

h!r+ 2nU(M)

+ E(I( y’ + t)).

Assuming that M grows with, but is bounded by, n, we have from (3) and (13) that the expected length of U( Y’ + t) is

This is approximately

minimized

by setting

M= (2ny,- ,)(k- lVk giving an expected length of ,?& :=k(2yk~,)‘k~“lkn(k-“lk+o(n’k~“/k).

Since EO is an average,%( Y’ + to) must have length at most E,, for some t, E F. Thus E, is an upper bound on the length of P. This completes the proof. 1 From (15) we get the asymptotic

bound (17)

which is worse than Moran’s bound (Corollary 1) by about 19%. Few’s original strip algorithm gave an asymptotic upper bound of l/$=0.7071 for ak. For 3
76

LUIS

A. GODDYN

by replacing close pairs by their midpoints, a strip heuristic is applied, and the resulting circuit is modified to include the original set of points. Such “hybrid” heuristics certainly warrant further attention.

6. IMPROVING THE “STRIP"

ALGORITHM

One might try to improve the strip algorithm even further. For example, one could try quantizing on fewer than k - 1 dimensions. In the above version, a (k - 1)-dimensional quantizer of size M is used to map the k-cube onto M copies of the l-cube (the vertical lines yV.) Instead, one might try to use a (k - s)-dimensional quantizer Q to map the k-cube onto M copies of the s-cube, for some s > 1. One would then construct a traveling salesman tour through points pi analogously to the case s = 1. We first construct A4 optimal tours through the images Q(pJ, one tour for each of the A4 s-cubes which form the range of Q. After linking the resulting M tours into a single tour through all of the Q(pi), we add each of the detour segments gP, from Q(pi) to pi and back to form a tour through the points pi. One can analyze such a heuristic as was done in the case s = 1 to obtain the recursion,

c(k< ,,%(2y,-,)‘k-““k

(18)

For example, setting k = 2s gives clzs,< 23J46. These inequalities might be useful if we knew that ~1,was small for some particular value of s. If we chooses,l~s~k,suchthats~oo,k-s~cO,andsJk~~ask-,co,and take the lim sup, _ o. of both sides of (18), we obtain the asymptotic bound (19) Unfortunately, this bound is strictly greater than the bound (17) if 1> 0, and reduces to (17) if il = 0. Another possible improvement is to study the effect of “straightening” the paths from each point pi to its successor pj along the tour. Can we shorten the tour significantly by replacing each segment pi -+ Q(pi) + Q(pi) + pi with the straight line pi + pi.7 The answer to this is probably yes, but one would probably need to know more about the distribution of the lengths of the lines in the tour. Such straightening occurs to a limited extent in Karloffs “hybrid” heuristic [22]. In some sense, this accounts for his improved upper bound on ~1~.For general k, it might prove helpful to

EUCLIDEAN TRAVELING

SALESMAN PROBLEM

consider using the generalized heuristic of the previous paragraph growing slowly with k.

7. NUMERICAL

77

with s

RESULTS

In Table I are presented the best known upper and lower bounds on the sphere packing density dk, the quantization error coefficient yk, and the corresponding lower and upper bounds on tlk from (4) and Theorems 1 and 2. All values have been rounded to four decimal places of precision. The bounds on 6, come from [12; 29, p. 4; 331. For yk we actually list bounds for yk & since this is nearly constant. The lower bound on yk is the sphere bound of (14). For the upper bound on yk we calculate the normalized expected error y(z) of various lattices and configurations LY listed in [S-9] as being good quantizers. For each k, Table I lists y(T) only for that k-dimensional configuration 9 which gave the best upper bound of all those tested. The word “rand” in the column labeled “(Y),, indicates that Zador’s “random” upper bound (14) is used instead of a hxed configuration Y. This upper bound listed is sharp for k = 1, 2 and is probably sharp for k = 3. The values of y(P) (when k > 2) were estimated for the various configurations 2 by running Monte Carlo experiments similar to those in [S]. Specifically, we used the fast quantizing algorithms described in [7] to find Q(Z) for about one million random points z. The resulting values

2

4

6

8

10

1’2

1’4

FIG. 1. Bounds on the quantization coefficient yk & versus k. Sphere and Random are Zadofs lower and upper bounds on yk ,,k. The remaining lines give y( 2’) & for four important families of quantizers, Z,, D,, Dz, and 0:. Each of these is an upper bound on yk ,,/%.

78

LUIS A. GODDYN

of jlz- Q(z)112 were used to estimate the integral in (16). The random points were determined by a non-linear additive feedback pseudo-random number generator with a period of approximately 16(231 - 1). Accuracy was checked by also computing the mean squared error during the experiments, and comparing it with the known exact values derived analytically in [S-9, 36, 373. The graph in Fig. 1 summarizes the result of the experiments by plotting y(Y) versus k for various classes of lattices and configurations .Y. Zador’s upper and lower bounds on yk are also plotted here. Zk denotes the k-dimensional cubic lattice consisting of all integer points in IE“. Dk is the lattice of all integer points in [Ekwhose coordinates sum to an even integer. For any lattice 9, Y* denotes the dual of Y, while L? + denotes the union of the lattice P’ and the coset L!’ + (i, .... 4). $P + need not be a lattice, even if SC is. For example, DS+ is the well-known root lattice usually denoted by E,, while 0: is a non-lattice which is a better quantizer than the best known lattice quantizer, ET. The lattice Dz is isomorphic to Zl . The quantization coefficients listed for the lattices K,,, Ar6, and AZ4 are only upper bounds. These bounds are derived from their known mean squared errors by noting the inequality Ed < E(z’). This upper bound is quite good when k > 12. For k = 6 the best known quantizer is Ez. However, the lattice 0: is used instead since it is only slightly worse, yet substantially simpler to program.

Few

Theorem

2-c2

4I

6

i

1’0

1’2

Ib

16

18

20

22

2

24

I

26

k

FIG. 2. Bounds on the worst-case Euclidean TSP problem coefficient a, versus k. Sphere is Few’s sphere-packing lower bound (4). Few and Moran are the previously best known upper bounds on 0~~which appear in [14, 251. Corollary 1 and Theorem 2 are the improved bounds of this paper.

ELJCLIDEANTRAVELINGSALESMANPROBLEM

79

The lower bound on ak comes from (4) and the lower bounds on 6,. Two upper bounds on ak are listed in Table I. The column labeled “Thm. 2” is the bound of Theorem 2 and depends on the upper bound of yk. The column labeled “Thm. 1” is the bound of Theorem 1 and depends on the upper bound of ak. The last row of Table I gives the best known asymptotic bounds on yk and the corresponding bounds on ak, as k + co. Figure 2 plots the upper bounds on ak given in Theorems 1 and 2. Also plotted are the previous best known upper bounds of Few [14] and Moran [25]. Few’s lower bound on ak is also plotted here. 8. CONCLUDING

REMARKS

The study of quantizers is relatively new although the ideas are simple and natural. Roughly speaking, quantizers are useful any time one needs to approximate points in Euclidean space by discrete structures (such as lines). Although this paper deals only with the worst-case Euclidean traveling salesman problem, the method of Theorem 2 can be used equally well for finding upper bounds for the coefficient fik of the random traveling salesman problem. Heuristics based on quantizers could also prove useful for other related Euclidean optimization problems such as minimumweight matching [lo, 28, 351 and shortest Steiner tree [2, 4, 14, 31-j. There are a few natural questions and conjectures regarding the behavior of ak and Pk. What is the rate of convergence of I,(n)/& as n + co? Is Few’s lower bound (4) on ak tight for all k? It does seem reasonable to conjecture that the centers of a densest sphere packing always form a worst-case instance of the Euclidean k-TSP problem. Does ak or /Ik converge as k + co? All of these questions have been asked by previous authors [2, 25, 311. Regarding the optimal quantization error coefficient Yk, one might ask whether yk & decreases monotonically with k for k> 3. Are quantizers which are optimal with respect to expected error also optimal with respect to mean squared error? Conway and Sloane [6] have conjectured that the lattices which are dual to those which give the densest sphere packing are optimal quantizers (with respect to mean squared error.) This conjecture appears to hold true for k < 8. Optimization problems can be studied in other metric spaces (such as spherical, hyperbolic, and Hamming geometries), and one might expect analogues of Moran’s argument and the strip algorithm to be useful in these settings, too. Although sphere-packing problems in other metric spaces have been studied (see, for example, [ 12]), very little seems to be known about good non-Euclidean quantizers.

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