Abstract
We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood‐Halton‐Hammersley theorem to this setting.
Highlights
The classical Beardwood-Halton-Hammersley theorem [4] concerns the minimum cost Traveling Salesperson Tour through n random points in Euclidean space
We suppose that only a subset of the pairs of points are joined by traversable connections, independent of the geometry of the point set
The distance between a typical pair of vertices in Xn is arbitrarily large (Figure 1). (We write logk x fork and A B in place of A = o(B), for expressions A, B.) In particular, this means that when logd n n p < 1 − ε, the supremum in the stretch factor theorem of Mehrabian and Wormald is due just to pairs of vertices which are very close together
Summary
The classical Beardwood-Halton-Hammersley theorem [4] (see Steele [23] and Yukich [26]) concerns the minimum cost Traveling Salesperson Tour through n random points in Euclidean space. It guarantees the existence of an absolute (though still unknown) constant βd such that if X1, X2 . Is a random sequence of points, uniformly distributed in the d-dimensional cube [0, 1]d, the length T (Xn,1) of a minimum tour through X1, . The present paper is concerned still with the problem of traveling among random points in Euclidean space. We study random embeddings of the Erdos-Renyi-Gilbert random graph Gn,p into the dTan-hddiismwiseentdhsieeonnaouatltehcbouyrbmeX[na0,np,u1ts]hdce.riWrpatneadlceoctmeXpgtnerdadpefhnorowtpehuoabslueicnvaietfoirotrenmxalysnedrtahinsadsXoumnnadsneedtrgowofhnpoeosfienutlplsapXirese1,roXfre2vv,ei.re.twi.c,eXbsunatr∈e [0, 1]d, joined hbyasendogtesbeeeanchthwroituhghintdheepecnopdeynetdiptirnogb,atbyiplietsyetpt.inEgd, pgeasgianraetiwoneigahntdedprboyoftrheeadEinugclpidreoacnesdsi,swtahnicche between mthaeyirlepaodinttos,daifnfderwenecaersebienttwereeesntetdhisnvtehresitoontaalneddgthee-wVeeigrhsitorneoquf iRredcotrod.tPralevaelseabciotuetththiseagrrtaicplhe
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