[ 299 ] THE SHORTEST PATH THROUGH MANY POINTS


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1 [ 299 ] THE SHORTEST PATH THROUGH MANY POINTS BY JILLIAN BEARDWOOD, J. H. HALTON AND J. M. HAMMERSLEY Received 11 March 1958 ABSTRACT. We prove that the length of the shortest closed path through n points in a bounded plane region of area v is 'almost always' asymptotically proportional to *J(nv) for large n; and we extend this result to bounded Lebesgue sets in ^dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travellingsalesman problem, Steiner's street network problem, and the LobermanWeinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem. The problem of determining the shortest closed path through a given set of n points is often called the 'travellingsalesman problem', particularly in the literature on operational research and linear programming: see for example, (7), (11), (12), (20). A travelling salesman, starting at and finally returning to his depot in a certain town, has to visit (» 1) other towns by the shortest possible route. If n is at all large, it is prohibitively laborious to calculate the total mileages for each of the (n 1)! orders in which the towns may be visited, and to choose the smallest total. There are, however, systematic numerical methods ((21), (23)) of computing the optimum order with a practical amount of labour; though even with these the labour is not light and will usually demand the services of an electronic computer. On the other hand, given a map of the towns, it is not too hard to draw a path which looks pretty much the shortest; and, for practical purposes, such a path could be used with confidence if we knew that its length was nearly that of the actual shortest path. But no explicit formula for the latter exists; and indeed any exact formula, depending as it must on the detailed structure of the distances between pairs of towns, would presumably be too cumbersome to be of any value. As a workable substitute for an exact formula, we shall derive a simple asymptotic formula for the shortest length when n is large. Despite its simplicity, this formula holds under surprisingly general conditions. When handling the problem numerically, one may insert as data the actual road mileages between pairs of towns. However, in an analytical approach, where we are deliberately avoiding the detailed distances between pairs of towns, it is convenient to take as the distance between two towns the length of the straight line joining them. This will not affect the result provided that the road mileage is effectively a known constant multiple of the straightline distance. This proviso is probably met in the large, though in the small it may well be violated. For instance, superficially it might seem that one of the applications of the problem was to the delivery of telegrams in a city. Speed of delivery and operating costs are conflicting factors in a telegraph service. To have to deliver each telegram (not surcharged for special delivery) immediately upon receipt at the delivery office would call for an uneconomically large force of messengers. Accordingly, such telegrams are allowed to accumulate at the
2 300 JILLIAN BBARDWOOD, J. H. HALTON AND J. M. HAMMERSLEY delivery office until a small number nhave arrived; whereupon a messenger delivers all n in a single tour of their destinations. Hence one seeks that value of n which provides the best compromise between the delay caused when n is too large and the delivery expenses when n is very small. The rate of arrival of telegrams at the post office and the delivery time for a batch of n telegrams both affect the answer. This delivery time depends upon the length of the shortest route through the n destinations; but it also depends upon other factors, such as the congestion of certain city streets or restrictions in the number of bridges over rivers and railways (13). Local complications of this kind will probably vitiate application of the theory to small regions, especially if n is not very large. The example just cited raises another complexion of the problem. The points to be visited may be distributed at random over a certain region; and the policy decision on the optimum value of n for a single delivery must rest not on the mileage through n given points but rather on the distribution of mileage through n random points. This random version of the problem also arises in connexion with agricultural surveys ((15), (18)). Previous work on the shortest path through n random points has concentrated on the expected value of the shortest length. We shall show that a much stronger result holds, namely, that this random shortest length is asymptotically equal to a nonrandom function of n with probability one. Thus for large n the distinction between the random and the nonrandom versions of the problem effectively vanishes: indeed it is this fact which ensures the existence of a simple result in the nonrandom case. We shall also mention Steiner's street network problem (2), and an interesting related problem of connecting terminals in electronic equipment using the minimum amount of wire (17). Hereafter for brevity we shall call a shortest closed path a tour, and speak of the tourlength through a set of points. The points will be distributed in ^dimensional Euclidean space 31; and, as already mentioned, we take the distance between two points to be the ordinary Euclidean distance. Except when the contrary is explicitly stated, we assume k ^ 2; the reason being that the case k = 1 is slightly anomalous and, anyway, rather trivial. We use German capitals for point sets in 9t; and, without further mention, we always assume that these point sets (with the sole exception of 91 itself) are bounded. In certain cases (explicitly mentioned) we assume that these sets are Lebesgue sets and sometimes one of the conventions (explained later) automatically implies that they are. We use boldface capitals for sequences of points in 9t: thus P = P 1; P a,... We write P n for the first n points of P: thus P n = P V P 2,...,P n. We write l(~p n ) for the tourlength through P. The tour through points P^P^,...,P 7l in a set will not necessarily be confined if ( is not convex. The closure of a set (S is written (. If is a Lebesgue set, denotes its Lebesgue measure. j...dv is integration with respect to Lebesgue measure. The expected value of a random variable x is written Sx. To state a preliminary result first, we shall prove that there exists an absolute constant a k such that Um sup TC_< fc _ 1)/fc Z( pn) ^ a^g)]* (1)
3 The shortest path through many points 301 whenever P lies within (. This bound is bestpossible in the sense that it can be attained by suitable choice of P in (, provided has a boundary of zero measure. We conjecture that this proviso is unnecessary. Some further statements concerning the uniformity of P and the bestpossible character of (1) appear in Lemma 4 below. The inequality (1) was previously proved by Verblunsky (24) in the particular case k = 2 under the added assumption was an open set with a boundary of zero measure; but Verblunsky did not investigate the conditions under which (1) was bestpossible. Note that, in (1), v(s) exists even when S is not a Lebesgue set. Whilst (1) is bestpossible in the sense aforementioned, our main results (2) and (3) below show that much more can be said if we make the additional assumption that S is a Lebesgue set of strictly positive measure. In this case there exists an absolute constant ft k (independent of P and such that lim n*«* l{ P») = fi k & [>((S)] 1/fc (2) 7l»0O for almost all P in (. The existence of this limit is part of the assertion (2). The phrase 'almost all' can be explained in either of two equivalent senses. In settheoretic language we may represent the sequence P as a point P x x P 2 x... in the infinite product space Ex (Sx..., on which we set up the product measure [#/?>( )] x [w/w(s)] x... in the usual way (9). Then ' almost all P' means that (2) fails only for a set of P of zero product measure. Alternatively, in the language of probability theory, we may regard P v P 2,... as a sample of random points, each independently and uniformly distributed over (. Then l(p n ) is a random variable; and the phrase ' almost all P' means that (2) holds with probability one. More generally, if P lt P 2,... are independently distributed over (5 with a common probability density function p, the density being taken with respect to Lebesgue measure, then with probability one lim»<*«* l(p n ) = P k ki f p^dlk dv. (3) n»oo J< Equation (3) reduces to (2) when p = [u(s)]" 1. Holder's inequality shows that the righthand side of (3) does not exceed that of (2). Hence nonuniformity in the distribution of P tends to decrease l(p"). The extreme case of this arises when P t, P 2,... are independently distributed over a Lebesgue set ( each with same same arbitrary probability law. An arbitrary probability law can be expressed as the sum of a discrete, a singular, and a continuous component; and the continuous component can be represented almost everywhere by a density function, whose integral over (5 will be less than unity when a discrete or a singular component exists. We shall show that (3) remains true provided p is taken to be this density function. Thus singular or discrete components of a probability law contribute nothing to the limit on the lefthand side of (3): they are, as it were, the extreme cases of nonuniformity in distribution. It is natural to ask if (3) remains valid for probability distributions which are not confined to a bounded set (5, that is to say, can we replace (5 by 91 in (3)? We have given some thought to this question; but have not been able to resolve it. It is quite easy to show that if I p*i)/* dv is infinite, then so is the limit on the lefthand side of (3). Jsrc
4 302 JILLIAN BEAEDWOOD, J. H. HALTOK AND J. M. HAMMBRSLEY When p(ki)ik d v j s finite, it appears that a necessary condition for the validity of the Jm. extended version of (3) is that all the absolute kj(k l)th moments of the probability distribution should exist; and we conjecture that this is also a sufficient condition. Unfortunately, we do not possess explicit formulae for the constants a k and fi k, though we have established the following bounds: * (4) where y k is defined by (5). As k * oo, y k > y m = (2ne)i. For the special case k = 2, there is a stronger result QI/4 ^ * n\h IR\ and for k = 3, the stronger result (I) 4 < «8 ( 7 ) The bounds in (6) are due to Verblunsky (24). The lower bounds in (6) and (7) arise from taking P v P 2,..., P n as the centres of equal circles or equal spheres closepacked We should not be surprised if a 2 and a 3 equalled these lower bounds. Numerically, f y 2 = sg /3 2 < , sj a 2 < ; y 3 = < fi 3 s , < a 3 ^ ; y 4 = < /? 4 ^ , ^4 < a 4 < ; (8) y m = < < , /?«, ^ a ro ^ 1. We have not proved that <x k and /3 k tend to limits when k > oo; so that the symbols a m and /? in (8) must be interpreted as lim inf a k, lim sup a k, etc., as appropriate. The k>co fc*oo upper bound for /? 2 in (8) is obtained by a special argument and is stronger than in (5). [For comparative purposes 6~* 12i = ] All other results in (8) come from (4), (5), (6) and (7). We have also obtained a very rough Monte Carlo estimate of /? 2 by a pair of simple sampling experiments. In the first experiment we used random number tables to distribute 202 points uniformly over the interior of a circle. We then drew with pencil and ruler what appeared (partly by eye and partly by measurement and trial and error) to be the tour of these points. We measured the tourlength and estimated /? 2 from (2). The second experiment was similar, except that we distributed 400 points uniformly over the interior of a square. As a precaution against subjective errors in choosing the tour, the two experiments were done independently by different people. Any failure t Since this paper was submitted we have seen a paper by L. Few, 'The shortest path and the shortest road through n points' Mathernatika, 2 (1955) Few proves there that <x 2 ^ 1, and that cc k < &*/{2(&!)}<*«/** for k > 3. The latter inequality gives <z 3 < , a 4 =s= ,..., «< Few also gives the result that <x k > {(kk)v) llk l(nk)l; but this gives the following lower bounds a 2 > , a 3 > , a 4 Js ,...,«,» , and all these (except for the limiting case as k * oo) are weaker then our results. Our result that < a 2 was given previously by L. Fejes Toth, Math. Ze.it. 46 (Z940) 835.
5 The shortest path through many points 303 to pick the actual tour will lead to an overestimate of /? 2. Experience of carrying through this procedure leads us to believe that it is not nearly so untrustworthy and imprecise as it may seem at first sight. The first and second experiments respectively gave estimates /? 2 = and /? 2 = This agreement is gratifying, though possibly coincidental. In the absence of further evidence, we think it probablej" that A =1= ( 9 ) To see how (2), (3), and (9) fared in practice we compared them with some exact results for the tour of Washington, D.C., and fortyeight other towns, one in each State of the U.S.A. (4). The tourlength given in (4) is 12,345 miles. From (2) and (9) we estimate, I = 053(2wF)i, (10) where n = 49 and V is the area of the U.S.A. [Although we only prove (2) for Euclidean space, it could equally well be proved for the surface of a spheroid.] Since (10) gives 9100 miles, the agreement seems indifferent. However, in (4), the distances between towns are road distances. Accordingly, we measured the corresponding straightline distances, using dividers and conical orthomorphic projections of the U.S.A. ((22), Maps 88/89, 90/91, 92/93) on scales of 789 and miles to the inch. On the likely hypothesis that the straightsegment tour will visit the fortynine towns in the same order as the roadmileage tour J, we found a straightsegment tourlength of 10,070 miles (subject to rather less than 1% error for errors of measurement with the dividers). What is responsible for the disagreement between 9100 and 10,070 miles? Presumably the fact that the different States are of different sizes is not to blame; for any allowance for nonuniformity of distribution will decrease the estimated value of I. For instance, if we use a probability density function which is constant over each State and allocates a total probability ^ to each State (regarding the District of Columbia as a State), we find from (3) 49 ^* (11) where V i is the area of the ith State. Equation (11) gives I = 8400 miles. A more likely cause of the discrepancy is that, although (3) allows for nonuniformity of distribution, it pays no attention to where the areas of high population density are within (5. If n is sufficiently large this is j ustified; but the consideration may become important when we impose the condition that there shall only be one point in each area of high density. In the case of the U.S.A. the areas of high population density and the small States are near the seaboard or the Great Lakes; and the fortynine towns, chosen by the authors of (4) because they were sufficiently large centres of traffic to merit appearance in a roadmap table of mileages between towns, lie noticeably near theouter boundary of the U.S.A. No fewer than eleven of the fortynine towns are actually on the coast or lakeside, or t Of course, the Monte Carlo experiment is very slight and gives very little information. With only two experiments, the 95 % confidence limits for /? 2 are and We hope someone will be encouraged to make a more precise estimate. % There is another way of looking at the matter not involving this hypothesis: namely, to the estimate 9100 miles we add 23 % to allow for the empirically determined excess of road over straightline distance, and then compare the resulting figure of 11,200 miles with the correct 12,345 miles.
6 304 JILLIAN BEARD WOOD, J. H. HALTON AND J. M. HAMMERSLEY wide estuaries thereof; and a further 5 he within 10 miles of the coast or Lakes (cf. the approximate diameter of the U.S.A., about 3000 miles). This congregation on the outer boundaries will tend to innate the tourlength above the value estimated by (2) or (3). The example serves as a useful warning of the limitations of our asymptotic formulae. We have proved one result which holds for finite values of n: namely, when ( is a Lebesgue set of positive measure and P lt P 2,... are independently and uniformly distributed over (, then for all n ^ 3 This result is stronger than a previous inequality due to Marks (19) in the case k = 2. (There is a misprint in Marks' paper, which erroneously suggests that S"l(P n ) > [^nw(( )]i; and this misprint is copied by Ghosh(8).) Ghosh also proves for k = 2 limsup«?z(p") ^ [2nv{<$)]l, (13) which is weaker than our result (8). Mahalanobis (18) states without proof that Sl{ P m ) = rth n~b when #(( ) = 1 and k = 2; but his assertion is false in view of (8). The analysis in this paper specifically concentrates on the tourlength l{ P' 1 ). But our arguments apply, with occasional minor modifications, to l*( P n ), the shortest length of wire needed to connect together terminals situated at P lt P 2,..., P n. Consequently, the same results apply to Z* as to I, although the numerical values of the relevant constants a%, /? may differ from those of oc h,fi k.there are two versions of the terminal problem; and an example will serve to distinguish them. Suppose we have four terminals P x, P 2, P 3, P 4 at the vertices ( + \, ± ) of a unit square in the Euclidean plane. It is quite simple to show that Z*(P 4 ) = 1+^/3. For this an appropriate set of straight wires connects the points (+ \, ± \) to the point {\ J^/3,0), the point {\ \4^, 0) to the point ( %+b«j3,0), and the point ( % + i*j3,0) to the points ( \, ± \). This version of the problem is known as Steiner's street network problem ((2), Chapter VII). In the other version of the problem, considered in (l), (16), (17) one is not allowed to join wires together at points which are not terminals; and the resulting shortest length of wire will then be Z**(P 4 ) = 3. In general, in h dimensions one has Z(P M ) ss l*(p n ) s? l**(p n ) s$ l(p n ). (14) To see this, one may regard each segment of wire in the l*(p n ) arrangement as a closed path (going once in each direction along it) and remark that the sum of a number of closed paths with points in common is a closed path, so that 2l*(P n ) ^ l(p n ). The argument used for l( P n ) in Lemma 3 breaks down for l*( P n ); but one may use (12) and (14) instead. Results similar to (1), (2) and (3) hold for l**(p n ) with appropriate a**, fl%*. Some slight simplifications in our argument would result if we used the easily verified fact that l( P n ), and for that matter l*( P n ), are nondecreasing functions of n for every given P. But we eschew these simplifications because l**( P n ) may not have this property: for instance, if we take P 5 = ( ^3,0), P 6 = ( k + hj^, 0) in the above example, we find 1 + ^3 = Z**(P 6 ) < Z**(P 4 ) = 3. The material of this paper raises the following problem, in which we write v k for idimensional Lebesgue measure: Given a region in 9t with known measure v = v k (Gi),
7 The shortest path through many points 305 choose in ( a random k'dimensional subset 6' with prescribedfinitemeasure v' = v k (&), What is the distribution of the measure v" = tv+i( ") of (k' + l)dimensional (not necessarily subsets such that v" is minimal subject to S" being bounded^ by ( '? The present paper considers the special case k' = 0, with the usual convention that the zerodimensional Lebesgue measure of a point set is the number of points it contains (v' = n, v" = I). The case k' = 1, k = 3 would be a randomized version of Plateau's problem ((2), Chapter VII); while, for other values of k', k, we get randomized versions of Douglas' problem ((3), (5)). The main difficulty presented by this generalized problem lies, we believe, in the precise definition of the random nature of ( ' and in the restrictions to be imposed on the connectivity, orientability, and other topological characteristics of the spanning We do not consider the generalized problem in this paper; but we feel intuitively that the basic appeal to subadditivity, which is the key of our present analysis, would still remain appropriate in the general case. That is to say, roughly, that if we take v' proportional to v and regard v" as a function of (5, then v" is a subadditive setfunction of (. The proof of Lemma 5 below should afford a clearer idea of what we mean by this very condensed, and consequently rather cryptic, remark and of how it leads to a solution of the problem. We shall now specify the detailed notation to be used in the proof of our results. After that, we shall give a short outline of the method of proof, together with a few conjectures on results we have not succeeded in proving. The lemmas assembled in the Appendix contain the relevant details of rigorous proof and exact statement; and hence in the short outline we are free to ignore these precise details and to concentrate instead on the broad spirit of the underlying ideas. We have already stated that German capitals denote bounded point sets in 5R, that (5 denotes the closure and that boldface capitals denote sequences of points. These symbols may carry suffices to denote different sets or sequences: e.g. P } = P xi, P 2j>... and Pf = P ip Py,..., P nj. We write (^ + ( 2 for the union of % and ( 2 ; for the union of several sets ( ^; (S^G^ for the intersection x and ( 2 ; (SjC*^ for the (possibly empty) largest subset x not in G 2. Sequences of points can be manipulated in the same way: e.g. P m (5 denotes the subset of P v P 2,..., P n which lies in ; whereas (P( ) n denotes the first n points of that subset of P which lies N( denotes the (possibly infinite) number of points in the sequence and and denote the tour and the tourlength of If P(S is empty, T( PS) is taken to be the empty set of 9ft and l( P(5) = 0. To illustrate a fairly complicated use of this symbolism, which occurs in Lemma 2, denotes the supremum of all tourlengths through not more than v points of the boundary of the set (Bp where v is twice the number of points which are both in (5 3  and amongst the first n points of the sequence P. To avoid tiresome repetition of the phrase ' if ( is a Lebesgue set', we adopt the convention that the mere writing of v(( ) to denote the ^dimensional Lebesgue measure of ( carries with it the automatic postulate that G is in fact a Lebesgue set. Notice, however, that the writing of v(( ) merely confirms f We have only a very vague idea of what we mean here by ' bounded', and part of the problem is to give a suitable definition to this word.
8 306 JnxLiAN BEARD WOOD, J. H. HALTON AND J. M. HAMMERSLBY that a closed set is a Lebesgue set and does not necessarily imply that ( itself is a Lebesgue set. We write #(( ) for the diameter of ( : this is necessarily finite because (5 is bounded. O will denote the empty set of JR. ( will always denote a semiclosed unit hypercube of k dimensions, i.e. a hypercube of unit length side, open on its lowerleft faces and closed on its upperright faces. 1; ( 2,... will denote semiclosed unit hypercubes in different positions in 9t. If is a positive real number, we write ($; for the set of all points with coordinates (ga^, x a,...,,x k ) such that (x 1,x 2,...,x k ) is a point of (. Thus (5 is a fold linear magnification without rotation, which leaves the origin of 9t invariant; and = and u( (S) = E, k v(q,). The symbol E, conventionally operates on all subsequent sets in a product: thus P is the magnification of P(S whereas the intersection of the unmagnified P with the magnified (. However, brackets will override this convention: for instance, in Lemma 5, we have (?? ) to denote the intersection of the tyfold magnification of & with the fold magnification and this contrasts with )/( ($;, which would be the i/fold magnification of the intersection of ( with fold magnification The phrase ' P p((&)' carries the following meaning: (5 is a Lebesgue set of strictly positive measure and P = P V P 2,... is a sample of random points independently distributed over Gc with a common Lebesguemeasurable probability density function p, where pdv = 1. In the special case when this distribution is uniform, i.e. whenp = we write P e it(( ) in place of P e p(( ). The phrase '.P e Wg means that P = P l3 P 2,... is a sample from a Poisson process of density, over Sft; that is to say, for arbitrary disjoint Lebesgue sets prob {N{ *<&,) = N t ; j = 1,2,..., m} = U^ff^ exp { »«,)}. (15) In most of the proof k is fixed and it is convenient to suppress the dependence of constants and functions upon k. In particular we adopt the abbreviations and notation ; q = (k 1)/&; log 2 a; = (logx) z.j < 16) With these notational conventions in mind, the reader will be able to get precise statements of our full results by reading the enunciations of the various lemmas. Suppose that we break a given set into a number of disjoint = 1,2,...,m). Then Lemmas 1 and 2 taken together state that, for any given P n = P 1; P 2,..., P n, the tourlength through all points of P" (namely l( P is roughly equal to the sum of m the corresponding tourlengths in the parts of S (namely 2 Z(P provided n is large. The explanation of this rough statement is that, whereas each of these two quantities is of magnitude n^k 1)lk in view of the ultimate results (2) and (3), their difference is bounded on one side by a fixed quantity, depending on the diameter of (, and on the other side by the sum of tourlengths through points on the boundaries of the ( ^. If, as we shall eventually arrange, these boundaries are sufficiently wellbehaved to be sets of dimensionality k 1, the tourlengths through such boundary points will be of magnitude TJ(*2)/(*I) J which is small in comparison with n <  k ~ r>lk. We therefore seek
9 The shortest path through many points 307 the convolution of the distributions of the several l( However, it is not easy to handle this convolution when the respective Z(P ( ^) are not independent of one another; and clearly they are dependent when n is fixed, inasmuch as an excess of points of P 71 x, say, will tend to reduce the number in ( 2, so that one may expect l( P 71^) and l( P n ( 2 ) to be negatively correlated. To overcome this difficulty we make n a random variable. Specifically, instead of sampling or &(( ), we sample P from W^ for large. Then, by (15), the random point sets 3 , and therefore the random tourlengths Z(P m ($^), are statistically independent for j = 1,2,...,m. This Poisson process has the useful property that n = N( PGr) is a Poisson variable with parameter v(s), and the conditional distribution of given n is identical with the unconditional distribution of P 71 under the assumption that P u((&). When is large, n will be nearly equal to v(<&) with high probability; so that the distribution of Z(Pg) forpejtj should be identifiable with the distribution of J(P x g) for P e u{<&), where x is the integer part of This identification can be made rigorous by a Tauberian argument, in which the Tauberian condition is that l(p n ( ), regarded as a function of n, cannot increase or decrease too rapidly. The foregoing remarks exhibit the essence of our line of attack; but it is convenient to make a further modification to deal with the fact that the limit in (2) is independent of the shape of Gs. To this end we transform the problem so that behaves rather like Lebesgue measure: in brief, we have to remove the power I/A in (2) before we can hope to get additivity in (. Now the distribution of P( when P Wg is, apart from a simple change of scale, the same as the distribution of when PeTTj. Moreover, when PePf 1,«= I(P ( ) will be roughly equal tofl( Gc); and hence the expression n (k ~ 1)lk v llk (E,(&) will behave approximately like, k v(qi), which is additive as required. This motivates the consideration of l( P (g) for P e W 1 in Lemmas 5 and 6. As already stated, l( P Gs) is nearly equal to the sum of the independent random quantities Z(P G%), where the G^ are disjoint parts composing (S. One might therefore expect l( P G?) to be asymptotically normal in view of the central limit theorem. But we have not succeeded in proving this, although we do show that, for real positive t, the cumulant generating function of l( P Gs) satisfies #,(P g) = log(i?exp<z(p ( )~w;(<) M(S) as >oo, (17) where w(t) is a function of A and t only. This in itself does not imply normality, nor even does it provide information about the magnitude of the cumulants of Z(P (S). The following simple counterexample is instructive. Let a; be a random variable which takes the values 0 and with equal probability \. The cumulant generating function of x is K t (x) = log ( + e«) ~ ti as > oo (t > 0); (18) but the asymptotic behaviour of the cumulants of a; cannot be deduced from the coefficients in t regarded as a power series in t: the mean of x is, not + o( ) and the variance of a; is J 2, not o( ). There is every reason to believe that, when P W v <f(p e) = yff ^(( ) + O( & W(*»>) as ^oo, (19) in which case we could deduce results about the magnitude of the error term implicit in (2) and (3). Indeed, by following the type of argument used in Lemma 6, we can
10 308 JDLLIAN BEARDWOOD, J. H. HALTON AND J. M. HAMMERSLEY prove (19) under rather severe restrictions about the nature for instance, (19) holds in the particular case = S. Although we cannot even prove (19) for general (, we believe that the error term in (19) is capable of still further reduction, probably as far as 0( fc1 ). Similarly, we believe that the order of magnitude in Lemma 6 could be reduced. As one may easily see, when k = 1 the corresponding results depend upon the shape of OK, in particular upon how pathologically disconnected (5 is. The analysis for k > 2 fails for k = 1 because is then not small compared with v( S) when is large. There may, however, be some justification for comparing the results for k = 1 with those for k > 1 in the special =. When k = 1 and P W v we find Z(P (S) ~ &(<&) with probability one as, * oo, and varz(p g) = 0(1), but Z(P ) is not asymptotically normal. APPENDIX The appendix is wholly concerned with formalities of proof. LEMMA 1. m+1 = S 1( then m \ m ml H P"S(S,) E(P^) ft+^,)} < S!Z(P re / (S,) a m P B 2,) < XiUP^J + Wi +fy+i)}*s J(P»,) + *IS S % We may suppose P 3  =(= > and take <2 ;  P TC S 3 . The central term of (11) is at least as large as the length of a closed path of straight segments joining Q t, Q 2,..., Q m, <2 X together with digressions at each Qjaround T(P n Qlj). But this path passes through P 2 <&j and so cannot be shorter than the first term of (11). LEMMA 2. // ( m are disjoint, then m \ m ( P ra S % )> S {*( P  2 sup liqi^zp^xi  (g )}. (21) J = l / i=l Qj We may suppose P ^ #= D and m > 1. S,rl P m i;( y) consists of one or more \ i=i 1 connected pieces, some of which may degenerate to points. Discard any such piece which does not contain some point of P n. The closure of any surviving piece will be called ajpiece. The number v of jpieces satisfies v ^ N(P n <Sj). Let Q/ Qij> Q2P > Qzv, j be the set of endpoints of the jpieces, in which there are two coincident Q's for any surviving degenerate _?piece. Let Ij denote the total length of all ^'pieces. Since / m \ m 391 (S 3 , and since li P n 2 31S= S ^> (21) will follow as soon as we have proved \ y=i / j=i (22) We shall establish (22) by constructing a closed path consisting only of the jpieces, each traversed just once, together with a part of a double circuit of T = T(Q 2 j"). Choose a particular (Bj and suppress the suffix j. We may suppose the Q's are labelled so that Q t, Q 2,..., Q Zv, Q x occur in that order when passing around T. We interpret suffices modulo 2v; and write Q r for the point at the other end of the jpiece (denoted by J t )
11 The shortest path through many points 309 terminated by Q t. We construct the closed path in stages. At any particular stage certain of the Q t will have been visited by the path, the remainder unvisited. Suppose that Q a,q b, Q c have been visited, whilst Q a+1, Q a+2,...,q,,^,q b+1,q b+2,...,q e^ have not yet been visited. We say that Q b+1 is an odd neighbour of Q b if c 1 b is odd. Similarly Q b _ 1 is an odd neighbour of Q b if b 1 a is odd. We start the path at Q t and proceed as follows. From Q x pass to Q 2 along T; thence to Q 2. along J 2 ; thence to the (unique) odd neighbour of Q r (say Q x ) along T\ thence to Q x. along J x ; thence to the (unique) odd neighbour of Q x, along T; etc,... Ultimately this procedure of alternate steps along jpieces and along T to the unique odd neighbour will bring the path back to Qi The reason why the required odd neighbours are unique is because 2v is even. If, upon return to Q x, the path has traversed all jpieces, it is the required path. If not, start a fresh similar closed path from any Qpoint not yet visited, the first step of the fresh path being along T to the odd neighbour of the starting point of the fresh path; and continue to construct such closed paths, starting from unvisited points, until all jpieces are traversed. Thereafter we get the required closed path by travelling around T with digressions around each of the alreadyconstructed closed paths whenever such a path is encountered for the first time on T. In following the above argument, it may be helpful to consider some particular case. For instance, when v = 6 and 1' = 6, 2' = 5, 3' = 10, 4' = 7, 8' = 12 and 9' = 11, the ultimately constructed path is LEMMA 3. // P e «(( ) and n ^ 3, then (12) holds. Let x it denote the distance of P i from the ith nearest of P lt P 2,..., P^i, P i+1,, P n Since T( P") does not visit any point twice, the sum of the lengths of the two segments of T( P n ) which terminate at P^ is at least x lf + x^. Hence where S t is the expectation over P jt and S' f is the conditional expectation over P 1,P 2,...,P j _ 1,P j+x,...,p n given P t. Let <B X denote a ^dimensional hypersphere of radius x centred at Pj and write 7 = J =»( x ( )<t;( J = 7rHfc*/(ifc)!. (32) The conditional probability, given P p that x^ exceeds x is the probability that not more than i 1 of the n 1 independently and uniformly random P 1,P 2,..., Pj_ ly P j+1,,p 7I lie in a. Thus tl A, _ 1\ /V\ r I y\nlr prob (^ > x\ Pj ) = S Q (" r *) (f) (lf) (33) Hence #J(x y + x 2j ) = prob (x xj > a; P y ) + prob(a; 2/ > x\pj)dx Jo = r(lv x IV)»*{2 + (nz)v x IV}dx. (34) Jo 20 Camb. Philos. 55, 4
12 310 JELLIAN BEARDWOOD, J. H. HALTON AND J. M. HAMMERSLEY It is an elementary matter to verify that (1 z) n ~ 2 {2 + (% 3)z} is a nonincreasing nonnegative function of z for 0 ^ z ^ 1. Hence we shall not increase the righthand side of (34) if we replace V x by min{«(( ), v(<b x )}. This gives, from w + *w)> [\lx*lp k r%2 + {nz)x k l p K}dx, (35) Jo where p k = F(P)!/7r* fc. (36) Since the righthand side of (35) is independent of Pp (31) provides n ) > in P(l z) n ~ 2 {2 + (n 3) z}dx, (37) Jo where z = x k \p k. The righthand side of (3 7) is the sum of two complete beta functions. Expressing these in terms of gamma functions and simplifying, we get the central term of (12). To prove the rest of (12), we write a n = n\n~ Q l(n q)\ [see (16) for the definition of q\ and combine the wellknown fact that a n » 1 as n > oo with the easily verified fact that a n+l LEMMA 4. There exist an absolute constant a k < 0(k l)* 1 '/ 2 * and a set function e) corresponding to every given e > 0, such that for all P. Consequently Z(P ts : an«[v(( ) + e] 1 <' + a(,e) (a = a k lcl) (41) = lim sup nol{ P"( ) ^ av 1 ^ ). (42) A sequence P is called maximal if equality holds in (42). A necessary and sufficient condition for ( to contain a maximal sequence is should contain one. A sufficient condition to contain a maximal sequence is that the boundary should be of zero Jcdimensional measure. Each maximal sequence (if any) in ( is uniformly dense in the open subsets (if any) // ( is open, then (43) for some P in S. First, take ( = ( ; and for any positive integer m, dissect S into m*" 1 disjoint identical cylinders, each of unit heightand having as crosssection a (k 1)dimensional semiclosed hypercube of side m 1. Each point of P ra E lies not further than %(k 1)4 m~ l from the axis of the cylinder containing it. Consider the closed path starting from one end of an axis nearest to a corner of (J and passing in turn along the axes of each cylinder, digressing at rightangles to and from any point of P ra in the same cylinder as that axis, going from axis to neighbouring axis in an obvious regular fashion on reaching the end of an axis, and finally returning direct to its starting point. The length of this path does not exceed the righthand side of 1.m 1 + l.fti (44)
13 The shortest path through many points 311 Now take m to be the least integer exceeding (k l) 1 l zk n 1/k, substitute in (44), divide through by n Q k^, and let n > oo. This gives JclL(P&) < tfikl)** 1 ***. (45) Define a k to be the supremum of the lefthand side of (45) taken over all P. This establishes (41) for (5 = 6 and also proves the righthand side of (4). In accordance with a standard theorem (see, for example, (25), Theorem 16) we may cover S with a finite number, say m = m((, e), of disjoint semiclosed hypercubes whose measures Vj satisfy m _ ZVj^v( ) + ie. (46) 3=1 be the intersection of the jth hypercube with (, and let n } = N( P Then Making due allowance for a simple change of scale of linear magnitude FJ" 9, we see that the definition of a ensures the existence of a number a*(vp e) such that l( < a»j(tj + iem 1 ) 1 * + a*(v p e), (48) because l( P ( ^) is the tourlength through n^ points contained in a semiclosed hypercube of measure Vj and we have proved (41) = (J. Define m { ii Then Lemma 1, (48), Holder's inequality, (46), (47) and (49) yield m 0 3 } j=\ 3=1 I m \q I m Ua a I S %) ( iem 1 )J + o(, e) m (e,e), (410) which establishes (41) and hence (42). Before proceeding to the remainder of Lemma 4, we make two remarks. First, a > 0; this follows from Lemma 3. Secondly, if we define A(< ) = supz(p(s), (411) p then ( contains a sequence P such that this supremum is attained; that is to say For, by (411), we can find P^ such that = A(( ). (412) A(g) *S nihpjty+j 1 (n = n ip n 2j,...), (413) where n ip n^,... is an increasing sequence of integers for each j = 1,2,...; whereupon we may take for the desired sequence P the sequence MlH'i21' ' " A H * l2>*22> ' 'JB2> ".13» *23' > where ^4,,... is a sufficiently rapidly increasing sequence of the form n xv n T2,...
14 312 JILLIAN BEARDWOOD, J. H. HALTON AND J. M. HAMMERSLEY We now prove (43). Since changes of scale are trivially allowed for, we may suppose that the given open satisfies ( (. By the definition of a, (412) ensures the existence of P such that 2(P ) a. (414) Then, since L(P(f) ^ i(ps), (414) and (42) yield = a. (415) For brevity, write V = v((&). is open, ( is a closed set of measure 1 V. Write n 1 = iv(p»( ), n, = tf[p»(gg)],? 1 = Z(P"( ), and Z 2 = i(p»( ( )]. By Lemma 1 and (41) applied to l 2, Then (415) and (416) yield (416) a < limsup(?i. l /?i) 9 jl(pg) + a(l F + e)*«lim sup (rc 2 /n.) 9. (417) Suppose, in order to obtain a contradiction, that lim sup (njn)* = 0. (418) By (42), (P( ) < av 1 ^ ) < a. Alsow 2 /w < 1. Hence (417) and (418) give a «S a,(lv + e) 1 ^; (419) and because a > 0 and e is arbitrary, (419) is a contradiction unless V = 0, in which case (43) is trivial. Hence we suppose (4*18) is false. By Holder's inequality From (417) and (420), lim sup (njn)* L( P(g) > a{l  (1  F + e) 1 * lim sup (» 2 /w) 9 } 1. (420) >a(fe) 19 limsup(» 1 /») 9. (421) Thereupon (43) follows from (421), the falsity of (418), and the arbitrariness of e. By (42), since S = 6, any maximal sequence in ( is a maximal sequence Conversely, suppose (5 contains a maximal sequence P. Then we can find a 0 in such that the distance between P 3  and Q } does not exceed 2~K Next we can find a closed path n through F x,p %,...,P n of length not exceeding Z(Q ) ~* ^y touring around Q\, Q<L,,Q n an( i making digressions at each Q } to and from P r Thus l(q n ) + 2 > l( P m ); and this implies that O is a maximal sequence in S\ has a boundary of measure zero and *;(( ) > 0, we can find an open set of measurefl(gs), namely, the complement of the closure of the complement of. By (43) this open set contains a maximal sequence. This sequence is a fortiori a maximal sequence
15 The shortest path through many points 313 Finally, suppose that ( contains a maximal sequence P; and suppose that ( contains an open interval Q; of (necessarily positive) measure V. Let Since Q also has measure V, Lemma 1 and (42) give lim sup {a^/ ) 9 F 1 "* + a(?i 2 /n)«[d((1)  F] 1 *}. (422) n»oo By Holder's inequality, the reverse inequality (and therefore equality) holds in (422). But equality holds in Holder's inequality only if the relevant terms are proportional. ThuS limsup» 1 /n= F/»((g). (423) n>oo Since (423) holds for every open interval y P is uniformly dense in the open subsets LEMMA 5. There exists an absolute constant f! k such that for every Lebesgue set (5 <?Z(P g)~/? fc &M ( ) = /? %(< ) as g^oo (51) when P ew ±. Moreover for t ^ 0, the cumulantgenerating function K*m) = log<?{exp[fl(pgg)]} ~ w(0g fc v(e) as g^ oo, (52) where w(t) is a function of k and t only, such that w(t) = o(t k ) as t > oo. In the proof of Lemma 5 we assume without further mention that all sets are Lebesgue sets. It is convenient to prove (51) and (52) simultaneously by denning #?*( < ) = *M(P G), (53) and writing K for K* and K** in all statements true for both K* and K**. Since e x is a convex increasing function of x, we have 0 < K**m) < K*m). (54) Let $!, 2,..., m be disjoint sets; and write m+1 = x. The disjointness together with the hypothesis P e W 1 implies that P j, P 2,..., P m are stochastically independent. Hence ^P!), J(P 2 ),...,Z(P m ) are independent. Hence, by Lemma 1, A similar result holds for Kf*; and so we have \j 1 / / 1 If n = N{P ),l(p&) *s n5(g). Hence log S c^^^s^e" 1^) = i>(<5) (e M <  1). (57) n=0 n 
16 314 JILLIAN BEARDWOOD, J. H. HALTON AND J. M. HAMMBRSLBY By (54) and (57), 0 s K t {$) <»( ) (e w <« 1). (58) Define &(< )= lim sup *if ( ( (S), (59) so that xi(a&) = lim sup  k K t (gag) = a fc lim sup 1 a) k Ki(g<te) = a fc &(( ). (510) This definition is valid provided that we work with extended real numbers (i.e. including + oo) as far as (514), where we prove that Xt is actually finite. From (55) and wnere mllk Choose a such that (5 a(. Cover a & with disjoint )/S 1, i/ 2 > > ^m' the integer part of l+ag/rj; and arrange for t\^i and i/s^+1 to abut facially for j = 1, 2,...,?» 1. From (56), we have, with S m+1 = S^ is = lim sup  fc if, S (7 j) (g ) < lim sup g~ fc I From (58) m < (lim sup * S ^[(^ y) + a fc (fc + 3)*^**. (512) m S *i[fo,) ( )] ^ 2 «[(^e,) (g )] {exp («[( 7,) (g )])  1} 3=1 j=l,) (g )] = (e'^fc l)»(g ). (513) From (512) and (513) 0 < Xt( ) < (e'"^  1) + «fc (i + 3)* I?**«, ' (514) which shows that Xt is finite. Now consider the special = a. We have because Z[P(5/ ^) ( ( )] ^ Z(Pi/ j). (This is also true for I** in place of I, because i) (g ) i s a convex set.) Hence (512) and (515) give Let t] * 00 in (516), and apply (59) and (510), with the result (516) Xi(aE). (517) This implies the existence of h'mg fc^(gae) = a*x(( ) = a k w(t), say. (518) Of course, w(t) is one function when K = if* and is another function when K = K**.
17 The shortest path through many points 315 Next let ( be once more an arbitrary Lebesgue set. Prescribe e such that 0 < e < 1. Because (Sis a Lebesgue set we can find disjoint a 1^. 1,a 2 (E i,...a^s^such that By (511), (518), (514) and (519) a&a < e. (519) \ i=i / [>(( ) + e] + e* + (2aifc4«) fc (  log ef k, (520) upon choosing y = ^~^i~ 1 loge. Let e ^ 0 in (520), with the result By (56) K^ad) < X,( ) + Z,[(o )  ( ] + 2te^fc.. (522) Hence, from (522), (518) and (521), liminf *Z,(g ) > lim,"z,(oge)  urn sup *,[ {(«)  }] f»oo »oo J»a3 Thus, by (59), (521), and (523) 5= am0^(0<(as) ] = w(t)v(<&). (523) lim *,( ( ) = w{t) «(g). (524) To prove w>( ) = o(< fc ) as t + oo, put ( = S in (514) and choose rj = Ar^Mogf. The required result (52) follows from (524) with K* in place of K. Similarly (51) follows from (524) with K** in place of K and t = 1. LEMMA 6. // P W t and ( is a Lebesgue set G, then uniformly 0( 2*2/(*l)) (i > lo(»log«) (* = 2)J By Lemma 4, there exists an absolute constant c fc such that for any sequence P whatsoever; and hence oo (61) Z(P W (S) ^ c fc n«= c lc n< k  r >" c, (62) f (63) (The conclusion l**(p n &) ^ cf*n^ follows from (63) and (14).) The inequality (63) also holds trivially when k = 1 if we take c x = 1. Now let P e T^. By (63), n=o n\ n=0
18 316 JELLIAN BEARDWOOD, J. H. HALTON AND J. M. HAMMERSLEY Hence we may define the finite function w{ ) = sup [varz(p e)]i (65) Cover ( with 2 k equal disjoint hypercubes & 1; ( 2,..., G 2 *; and write 7 = 1 / 7=1 We have var L ± = var cov (L 2, Z x L 2 ) + var (Z^ L 2 ) (66) *k {[vari, 2 ]*+ [var {L^L^f ^ {[varl 2 ]* + \S{L X L 2 ) 2 ]*} 2. (67) The lengths l( ( 3 ) are stochastically independent because P e W v Thus varl 2 = I varz(p ( S,) = ^ S [^(M)] 2 = 2 fc [^(^)] 2, (68) 4=1 4=1 i=i because 2(( ^( 3 ) is a subset of some semiclosed unit hypercube. By Lemma 1 we have 2* fc i (69) To get an inequality in the opposite direction to (69), we write n = N(P (g) and consider the situation conditional upon n beingfixed.we may then number the points of P so that the first n of them are the ones in (. In this case Applying Lemma 2 to (610), we have p^ro^)(*,)] ^ 2*+!supZ[Q 2m (ife)(jr 4=1 Qi Q (611) Now l[q 2n (U&) (0t is the length of a path through 2ra points on the boundary of ( ; and is therefore, by Lemma 1, at most equal to 2k[l' + #( ( )], where V is the maximum length of a path through 2m points on one of the 2k faces of. Since each face is a (k 1)dimensional hypercube of side \i, we have V < c k _ L \E,(2n) <  k  2)Kk  v> by (63). Thus L x Li> 2 fc + 1.2fc[c ft _ 1^(2w)< &  2 W*«+ p4g]. (612) By (69) and (612) (L x L 2 f < 2 2fc + 2 P P[i*c fc _ 1 (2n)<* 2 '«fc » + I] 2. (613) Now if/? and e are positive constants, we have a standard result ((10), Theorem 137 (6)) n=0 as 1/ > oo. Hence (613) and (614) give 2 e'^w" = [(1 + e) yy + 0{ei^") (614) n t  L 2 ) 2 ^ S e"«l VS, 2 2fc + 2 i 3 g 2 [)fcic fc _ 1 (2»)< fc  2 )«fc  1 > + I] 2»=o m '
19 The shortest path through many points 317 where A and B are absolute constants (depending only on k) since we may by & after summation without decreasing the sum. From (67), (68) and (615), we get In (616) the righthand side is independent and hence taking the supremum of the lefthand side for all ( <=, 6, we get Now define x(g) = w{h)\ k l ^ v >. (618) Then (617) and (618) give >1). (619) If k > 2, (619) gives *( ) < *( ) + C; (620) and hence a(2* ) < C + \x{2 n Tg) < C(l+ ) + ia;(2" 2 g) <... ^ 2C + 2 n x(,). (621) Thus *(9/) ^ 2C + ([log 2 i/]) 1 sup a;( ) f (622) 1S «2 where [z] is the integer part of z and 7} = 2 n. Thus, in the case k > 2, (61) follows from (618), (64) and (622). When k = 2, (619) gives *( ) < z( ) + C, (623) in which we make the substitution y(g)=*(s)aog, (624) with the result [*/( ) */( )]log \i > y{i)log2c. (625) We shall show that (61) follows from (625), inasmuch as the assumption that y( ) is unbounded leads to a contradiction in the following way. If y{,) is unbounded there exists 0 > 2 such that, p * */( )> max r^, sup y(i). (626) Substitution of (626) into the righthand side of (625) gives ^ (627) since log 0 > 0. If 0 > 2, we may repeat the process with = f 0 in (625); and so on until we get y(2_^^ ^ ^ 3 ^ ^ ^ ^ y(^ > sup y(g)> (6. 28) 2<J<4 where is the integer part of Iog 2 0. However (628) is a contradiction because 2 <2" 0 < 4. LEMMA 7. // P e «(( ), (2) /wms with probability one. Since we may make a simple change of scale, there is no essential loss of generality in assuming (5 c (. Let P o e W x and P x e W g k. Since the distribution of Z(P o g(s) is the same as the distribution of l( P t ( ), Lemmas 5 and 6 give as g>c» (71)
20 318 JILLIAN BBABDWOOD, J. H. HALTON AND J. M. HAMMERSLEY and varz(p 1 (g) =^ 2 var?(p o ge) = C>(^2(fc  1)  {2/(&  1))+2e ) as >oo, (72) for any prescribed e > 0. Hence for all sufficiently large and therefore, by Chebyshev's inequality, provided is sufficiently large and 0 < e < l/k(k 1). Next, writing k v(qi) = rj, we have robtl^g)^^ )! < e fc M )] Ss l^g 2 " 5, (74) prob[(le)j; < iv(pjg) <(l+e)5/] = (l = 1  O(ei^) ^ l  JgW* (75) for large enough. Since the pair of relations it follows from (74) and (75) that the probability of (77) is at least \g~ m enough. Next define/(n) by for large in which Pe u((&). Since the conditional distribution of l( P x ( ) given that N( P x ( ) = n is the unconditional distribution of l(p n ), we have Since 0^1 /(w) < 1, (79) gives e'^[l /(n)] = prob[(77) holds] > / fc. (79) n=0 where ^4 is a constant (depending upon e and (S) and and i/ are sufficiently large. Now let J m denote the set of integers n satisfying f=j) <w< (l3i) (~l,v>. (711) Then (710) gives, for large enough m, sup [1/(»)]> 14 (^t. (712)
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