GEOMETRIC PROBABILITY MODELS TO ANALYZE STRATEGIES FOR FINDING A BURIED CABLE

Transcription

1 Journ of the Opertions Reserch Society of Jpn Vo. 6, No. 3, Juy 17, pp c The Opertions Reserch Society of Jpn GEOMETRIC PROBABILITY MODELS TO ANALYZE STRATEGIES FOR FINDING A BURIED CABLE Ken-ichi Tnk Kn Shiin Keio University (Received Mrch 31, 16; Revised Jnury 1, 17) Abstrct Different strtegies cn be used to find stright cbe buried underground. The origin probem considered by Fber et. focused on teephone compny tht wishes to dig trench to octe stright cbe. The cbe is known to pss within given distnce,, from mrker erected bove the puttive oction of the cbe. Fber et. showed tht the shortest simpy connected curve gurnteed to find the cbe is U-shped curve whose ength is bout 18% ess thn tht of circur trench of rdius. This probem cn be regrded s minimizing the mximum ength tht trench digger must dig. In reity, once the cbe is found, digging cn stop. So fr, however, no ttempt hs been mde to evute the trench shpe on chrcteristics other thn the mximum trench ength. In this pper, we present geometric probbiity modes to nyticy derive the distribution of trench ength nd ccute the expected vue nd vrince for both the short-ength (U-shped) trench nd circur trench. Our min resut is tht the expected digging ength is bout 5% ess for the circur trench thn for the U-shped trench. Keywords: Appied probbiity, bem detector, geometric probbiity, digging ength distribution, urbn infrstructure, urbn opertions reserch 1. Introduction Fber et. [4, 5] considered strtegies for finding stright cbe buried underground by digging trench, finding the mximum ength necessry to gurntee discovery of stright cbe known to be buried within units from given mrker. In this pper, geometric probbiity modes re presented to nyze expected ength nd vrince of vrious shpes of trench. In the origin formution, the probem considers the ength of the trench tht gurntees octing cbe whie digging ner it. In other words, it focuses on the mximum trench ength tht trench digger woud need to dig before finding the cbe. Of course, circur trench of rdius centered t the mker woud gurntee this, but it is known tht more efficient shpes of trench exist. One such exmpe is the U-shped trench shown in Figure 1 (b), composed of semicirce nd two ine segments of ength, where is the rdius of the semicirce. Fber et. [4] proved the very interesting fct tht, mong simpy connected curves tht ensure finding cbe, the U-shped trench is the best in terms of minimizing the mximum tot trench ength. This U-shped trench, with ength (π ) 5.14, is bout 18% shorter thn the circur trench, with ength π 6.8. Focusing on this spect, the existing itertures rgue tht U-shped trench is 18% better thn the circur trench. Whie the bove resut is interesting mthemticy, the mximum trench ength is ony one of mny indices tht re importnt in re ppictions. So fr however, to our knowedge, no ttempt hs been mde to evute the trench shpe on the bsis of mesures other thn the mximum necessry trench ength. Since trench digger cn stop digging 4

2 Strtegies for Finding Buried Cbe 41 once cbe is found, the expected trench ength before trench digger octes cbe my be more importnt. In this pper, we focus on two strtegies for finding cbe: digging ong the circumference of the circur trench, nd digging the U-shped trench described bove. We nyticy derive the distributions of trench ength with stopping on discovery for the two modes nd compute the expected vues of trench ength nd their vrinces. This pproch ows us to evute the two strtegies more brody thn when ony the mximum trench engths re vibe. To derive the trench ength distribution, probbiistic pproch is necessry. We deveop geometric probbiity modes tht describe how ech possibe position of cbe occurs retive to the mrker, using rndom ines. One of our min resuts is tht, in terms of the expected trench ength, the circur trench is superior to the U-shped trench. The reminder of this pper is orgnized s foows. In the next section, we introduce some reted iterture. In Section 3, we discuss gener ssumptions nd describe the probem. We then focus on the circur trench (O-mode) nd derive the cumutive distribution function (cdf) of trench ength in Section 4. From the cdf, we nyticy obtin the probbiity density function (pdf) of trench ength nd the corresponding expected vue nd vrince. We show tht the pdf is obtined s the cosine curve, nd this simpicity ows us to nyticy derive sever other indices. Next, in Section 5, we investigte the U-shped trench (U-mode). Athough the nysis is much more compicted thn tht for the O- mode, we nyticy obtin the cdf nd pdf of the U-mode. In Section 6, we compre chrcteristics of the two strtegies for finding cbe. One of our min resuts is tht, in terms of expected trench ength, the circur trench is bout 5% superior to the U-shped trench. In Section 7, we concude the pper nd present sever importnt directions for future reserch.. Literture Review Croft [1] considered the probem of finding the shortest continuous curve in the pne tht intercepts every chord of fixed disk of unit rdius. There re sever vrints of this probem, differing ccording to which of the foowing curves (or sets of curves) re owed: () singe continuous (rectifibe) curve; (b) countbe coection of rectifibe curves somehow connected to ech other; nd (c) countbe coection of rectifibe curves not necessriy connected to ech other []. Croft conjectured tht for cses () nd (b), the U-shpe is the shortest curve, with ength of π 5.14 [1]. For cse (), this conjecture ws proven by Fber et. [4]. For cse (c), which rexes the connectivity ssumption, the ength cn be reduced substntiy. The best known - nd 3-curve connected sets hve engths bout nd bout 4.8, respectivey (see for exmpe [6]). Simir probems cn be defined for n rbitrry convex region K insted of the unit circe. Fber et. [5] give short curves in cses (), (b), nd (c) for vrious convex sets K, in prticur, for regur n-gons with n = 3, 4, 5, nd 6. Probems of this type, whie simpe nd esiy understndbe, re often very hrd to sove. For exmpe, when K is unit squre, the engths of the shortest curve for () nd (b) re known to be 3 nd , respectivey. However, these two cses correspond to prticur instnces of we-known cssic probems with nodes tken from four vertices of unit squre: the former is n instnce of the minimum spnning tree probem, nd the tter is the geometric Steiner tree probem. In contrst, for cse (c), the best known -curve connected set hs ength However, proof of the optimity of

3 4 K. Tnk & K. Shiin this resut hs not yet been given. A curve tht intercepts ines intersecting given convex poygon is so ced bem detector [6]. This is becuse the ines cn be considered s bems, nd curves tht intersect every such ine cn be seen s detector for bems. The word brrier is so used with the sme mening [3]. Opque set is nother term with simir mening. Agorithms for finding short opque set hve been deveoped, such s tht in [3]. There re sever vrints of the bem detection probem. Exmpes incude the probems of finding nd for swimmer ost in dense fog in river or the se. A simir exmpe is finding stright rod when ost in forest. These nd other types of vrints cn be found in [, 6 8, 1]. As introduced in the previous section, the probem cn be seen s finding trench tht ensures finding n underground cbe during digging of the trench [5]. A simir sitution ws treted by Stewrt [14], in which Sherock Homes hs to find n underground tunne s quicky s possibe. The ony wy to find the underground tunne tht runs through stright in court ws to dig trench. The U-shped trench ws introduced s the most efficient shpe to find the tunne. The trench-digging ppiction hs n ddition spect not considered in bem detector ppictions. In digging trench, the time to find cbe depends on the oction in which the cbe is found. In the iterture, however, this importnt spect hs not yet been sufficienty exmined. Erier studies evute the efficiency of curves excusivey on the bsis of the mximum ength trench digger might need to dig [5]. Since the probem invoves geometric nd spti spects, the proposed probbiistic modes use some bsic concepts of geometric probbiity. Geometric probbiity hs been ppied to vriety of probems tht tret distribution of objects in spce [11 13]. These pproches re usefu in vrious fieds, incuding stronomy, bioogy, crystogrphy, forestry, nd serch theory. In this pper, we empoy rndom ines to mode the reiztion of rndom cbe retive to the mrker. In the fied of geometric probbiity, probems deing with rndom chords in circe hve been extensivey studied. Distributions of chord ength nd their expected vues hve been ccuted under sever different genertion rues for rndom chords (see for exmpe [11, 13]). Whie the sitution we ssume in modeing rndom cbe is simir to those ssumed in the bove studies, distributions of trench ength tht trench digger hs to serch before identifying cbe hve not been treted so fr. Geometric probbiity hs so been ppied to evuting urbn service systems [9, 15, 16]. For exmpe, the distribution of distnce from uniformy distributed points in given region to fciity fixed t given point in the region hs been derived. Aso, by ssuming trve ptterns in n urbn re, the distribution of distnces between vrious pirs of points within region hs been ccuted. These resuts hve been summrized in [9, 15, 16]. The proposed modes cn be seen s new geometric probbiity modes rising in urbn ppictions. 3. Probem Description This section expins the gener sitution nd presents bsic ssumptions. We ssume tht the retive position of the cbe from the mrker is unknown, but it is wys within units of the mrker. This pper focuses on two methods of finding cbe: one is to dig ong the circumference of the circe of rdius centered t the mrker (the O-method), the other is to dig ong the U-shped curve with the center of the semicirce t the mrker

4 Strtegies for Finding Buried Cbe 43 (the U-method). From the iterture it is known tht the U-shped curve, with tot ength (π ), is the shortest simpy connected curve tht ensures finding cbe. This cn be seen s sying tht, in terms of the mximum trench ength, the U-shped curve is optim. Our focus of this pper is to compre the two methods from different viewpoint, by deriving the pdfs of the trench engths for both modes. In this pper, the pth of the trench must be simpy connected t times. In other words, trench digger continuousy digs ong given curve without interruption. For the U-method, this ssumption ows trench digger to strt finding the cbe ony from one endpoint of the U, moving towrd the other endpoint. Rextion of this ssumption is n interesting future topic, which we discuss in the fin section. As shown in Figure 1, we introduce n x-y coordinte system with the mrker t the origin for both modes. The position of cbe is represented by the distnce p between the mrker nd the cbe, nd the nge θ tht the perpendicur to the cbe mkes with the x-xis. A probbiistic pproch is necessry to derive the pdf of the trench ength, unike in previous studies tht focus soey on the mximum vue needed to ensure discovery. We construct probbiistic modes by ssuming tht p nd θ re rndom vribes. Throughout the pper, we ssume tht p nd θ re independent nd tht θ is uniformy distributed in the rnge [, π]. With this ssumption, we cn consider, without oss of generity, tht trench digger strts checking cbe from (, ) in the counter-cockwise direction ong the circumference of the circe of rdius. Aso for the U-shpe, we cn focus on the sitution where trench digger strts checking from (, ) nd moves towrd the other end of the U-shpe t (, ). The joint pdf of θ nd p is denoted by f Θ,P (θ, p). In this pper, we formute method to derive the cdfs nd the pdfs of the trench ength for the O-mode nd U-mode, ssuming tht θ nd p re independent nd θ is uniformy distributed in the rnge θ [, π]: f Θ,P (θ, p) = 1 π f P (p), (θ, p) R, (3.1) where R = [, π] [, ]. Bsed on the method, we expicity derive the cdfs nd the pdfs of the trench ength for both modes in the most importnt cse where θ nd p re independent nd uniformy distributed in the rnges θ [, π] nd p [, ]: f Θ,P (θ, p) = 1, (θ, p) R. (3.) π 4. Trench Length Distribution: O-mode This section focuses on the O-mode. We derive the cumutive distribution function (cdf) of the trench ength L O, which gives importnt mesures such s the pdf nd the expected trench ength. The nysis proceeds s foows. (i) Divide the points in (θ, p) R into sub-regions, ech hving the sme intersection pttern between cbe nd the circumference of the circe; (ii) For ech sub-region specified in (i), express the trench ength s function of θ nd p; (iii) For n rbitrry trench ength [, π], identify the set of points Ω O () where the trench ength L O is ess thn or equ to ; (iv) Integrte f Θ,P (θ, p), the joint density function of θ nd p, over Ω O () to obtin F LO (), the cdf of L O ;

5 44 K. Tnk & K. Shiin () O-mode (b) U-mode Figure 1: The sitution for O- nd U-modes (v) Obtin f LO () (the pdf of L O ) nd the expected trench ength nd its vrince. Figure shows the possibe ptterns of intersections between cbe nd the circumference of the circe. There re three cses: (A), (B), nd (C). In the cse of the O-mode, the cbe wys intersects the circe t two points, except the cse of mesure zero (i.e., when the cbe is units from the origin), where the cbe ies tngent to the circe. To express L O s function of θ nd p, we need to identify which of the two intersection points the trench digger first octes whie digging trench strting from (, ). In Figure, the two intersection points re indicted s bck dots nd the pth of circur rc ong which the trench digger proceeds is iustrted by thick bck rc. Let us denote the nge corresponding to the thick bck rc for the three cses s φ A for (A), φ B for (B), nd φ C for (C). These three nges cn be written s foows: φ A = θ rccos p, (4.1) φ B = θ rccos p, (4.) φ C = rccos p (π θ). (4.3) Figure 3 iustrtes how to identify ech sub-region over R = [, π] [, ] in which the three ptterns (A), (B), nd (C) re reized. Figures 3 (), 3 (b) nd 3 (c) correspond to the cse for θ π, π θ 3π nd 3π θ π, respectivey. The dshed ine shows the cbe tht psses through (, ). The bck dot is the intersection point found by the trench digger, nd the gry dot is the other intersection point. As cn be seen from Figures 3 () nd 3 (c), the expression for L O chnges depending on whether the vue of p is rger thn or smer thn cos θ. Figure 3 () shows the threshod vue of p between cse (A) nd cse (B); Figure 3 (c) shows the threshod vue of p between cse (C) nd cse (B). As cn be seen in Figure 3 (b), when π θ 3π the intersection pttern is wys the sme: cse (B). We cn summrize the division of R into set of sub-regions ccording to intersection pttern s shown in Figure 4. By Equtions (4.1) (4.3), the trench ength L O cn be expressed s foows:

6 Strtegies for Finding Buried Cbe 45 () region (A) (b) region (B) (c) region (C) Figure : Intersection ptterns of cbe nd the circe (O-mode) () θ π (b) π θ 3π (c) 3π θ π Figure 3: Iustrtion of chnges of ptterns (i) (A) (ii) (B) (iii) (C) ( L O = θ rccos p ), (4.4) ( L O = θ rccos p ), (4.5) L O = (rccos p ) (π θ). (4.6) To derive the cdf of the trench ength, F LO (), we hve to identify the region Ω O () in R tht stisfies the condition L O. Using the nottion, F LO () cn be expressed s foows: F LO () = f Θ,P (θ, p)dθdp. (4.7) Ω O ()

7 46 K. Tnk & K. Shiin Figure 4: Division of R into sub-regions by intersection pttern (O-mode) It shoud be noted tht the shpe of Ω O () chnges depending on the vue of, where [, π]. For more detied process for deriving the cdf, see the ppendix A. Under the ssumption of Eqution (3.), the cdf cn be derived s foows: F LO () = π 1 π sin ( ), ( π). (4.8) As Eqution (4.8) shows, the cdf cn be expressed in the sme form for vues of [, π]. By differentiting Eqution (4.8) with respect to, the pdf of trench ength f LO () cn be obtined s foows: f LO () = 1 π 1 ( ) π cos, ( π). (4.9) As cn be seen from Eqution (4.9), f LO () is very simpe cosine curve pus constnt vue. Becuse of this simpicity, we cn esiy ccute importnt mesures from the pdf. From Eqution (4.9), the expected vue of the trench ength nd the vrince cn be derived s foows: E[L O ] = Vr[L O ] = π π f LO ()d = π , (4.1) π ( E[L O ]) f LO ()d = π4 48 3π (4.11) Figure 5 iustrtes the trench ength corresponding to the expected ength. The nge from the mrker is bout 17. It is worthwhie to point out tht this vue is ess thn one third of the mximum vue of Trench Length Distribution: U-mode This section focuses on the U-mode nd derives the cdf of the trench ength. The procedure for deriving the cdf is simir to tht for the O-mode, except tht the process is much more compicted due to the ower symmetry of the U-mode. The nysis proceeds s foows. (i) Divide the points in (θ, p) R into sub-regions with ech hving the sme intersection pttern between cbe nd the U-shpe; (ii) For ech sub-region specified in (i), express the trench ength s function of θ nd p;

8 Strtegies for Finding Buried Cbe 47 Figure 5: Expected vue of trench ength for O-mode (iii) For n rbitrry trench ength [, (π )], identify the set of points Ω U () where the trench ength L U is ess thn or equ to ; (iv) Integrte f Θ,P (θ, p), the joint density function of θ nd p, over Ω U () to obtin F LU (), the cdf of L U ; (v) Obtin f LU () (the pdf of L U ) nd the expected trench ength nd its vrince. Unike in the O-mode, there re some cses in which ony one intersection point occurs between cbe nd the U-shped trench. It shoud be noted tht though we need to know ony the point where the trench digger finds the cbe in order to derive the cdf of the trench ength, it is vube to identify both intersection points (when there re two such points) to formute vrints of the proposed mode. Figure 6 iustrtes the intersection ptterns between cbe nd the U-shpe. In the figure, bck dots iustrte intersection points, nd thick ines indicte the pth of trench digging. For the U-mode, 11 different intersection ptterns occur, beed (A) through (K). For cses (B), (D), (H), nd (J), there is ony one intersection point; other cses hve two intersection points. Figure 7 shows 11 sub-regions in the region R. Sub-regions chnge t the foowing four curve sections: p = sin θ, (5.1) p = sin θ, (5.) p = (cos θ sin θ), (5.3) p = (cos θ sin θ). (5.4) For ech sub-region, the intersection point(s) re beed by S 1 (initi segment), S (termin segment), C(semicirce) or two of these three. For exmpe, sub-region (E) hs two bes S 1 nd C, which indictes cbe intersects the U-shped trench t S 1 nd C. Aso the point trench digger finds is the one shown in the upper oction, S 1 in the cse of sub-region (E). To express L U s function of (θ, p), coordintes for intersection points tht trench digger finds whie digging from (, ) ong the U-shpe hve to be identified. There re four different expressions for the coordinte of intersection points: two for the semicirce, one for the initi segment S 1 nd one for the termin segment S. For the cse of the semicirce prt, the nge φ from the x-xis cn be expressed s φ = θ ± rccos p. (5.5)

9 48 K. Tnk & K. Shiin () region (A) (b) region (B) (c) region (C) (d) region (D) (e) region (E) (f) region (F) (g) region (G) (h) region (H) (i) region (I) (j) region (J) (k) region (K) Figure 6: Intersection ptterns of cbe nd the U-shpe

10 Strtegies for Finding Buried Cbe 49 Figure 7: Division of region R into sub-regions by intersection pttern (U-mode) For the cse of segments S 1 : (y = ) ( x ) nd S : (y = ) ( x ), we cn obtin expressions for the coordintes s ( ) p sin θ S 1 :,, (5.6) cos θ ( ) p sin θ S :, (5.7) cos θ by focusing on the fct tht the cbe cn be chrcterized by p = x cos θ y sin θ. (5.8) Since the intersection point tht trench digger finds is one of the bove four cses, the trench ength L U cn be obtined s foows: (i) (A), (B), (E), (I) nd (K) L U = p sin θ, (5.9) cos θ (ii) (C) nd (D) (iii) (F), (G) nd (H) ( L U = θ rccos p π ), (5.1) ( L U = θ rccos p π ), (5.11)

11 41 K. Tnk & K. Shiin (iv) (J) L U = π p sin θ. (5.1) cos θ To derive F LU (), the cdf of the trench ength, we hve to identify the region Ω U () in the region R tht stisfies the condition L U. F LU () cn be expressed s foows: F LU () = f Θ,P (θ, p)dθdp. (5.13) Ω U () It shoud be noted tht the shpe of Ω U () chnges depending on the vue of, where [, (π )]. For more detied process for deriving the cdf, see the ppendix B. Under the uniformity ssumption of Eqution (3.), the cdf cn be derived s foows: (i) ( ) F LU () = 1 π rccos, (5.14) (ii) (π 1) { F LU () = 1 π π (iii) (π 1) (π ) sin ( { F LU () = 1 3π 4 π ( ) ( π rccos } ) ( cos ) 3, (5.15) ) (π) (π π) (π1) }. (5.16) By differentiting Equtions (5.14) (5.16) with respect to, the pdf f LU () of trench ength cn be derived s foows: (i) f LU () = (ii) (π 1) f LU () = 1 π π{ ( ) }, (5.17) (iii) (π 1) (π ) { f LU () = 1 π sin ( ) ( cos ) ) ( cos sin ( { (π 1)} ) 1 3, (5.18) (π ) 4 {(π ) } 1 }. (5.19)

12 Strtegies for Finding Buried Cbe 411 Since nytic derivtion of the expected vue of the trench ength E[L U ] nd the vrince Vr[L U ], is difficut, we evute these vues by numeric integrtion. The foowing re pproximte vues of E[L U ] nd Vr[L U ] obtined by Wofrm Mthemtic 1 ( E[L U ] = Vr[L U ] = (π) (π) f LU ()d 1.966, (5.) ( E[L U ]) f LU ()d (5.1) This resut iustrtes n interesting fct: The expected vue of the trench ength for the U-mode is bout 5% onger thn tht for the O-mode s derived in Eqution (4.1). 6. Comprison of Two Modes We compre the two modes on the bsis of the resuts derived in the two preceding sections. Figure 8 compres the cdfs of the O-mode (thinner ine) nd the U-mode (thicker ine) for = 1. Figure 9 shows the pdfs of the two modes. We so present summry vues (expected vues, mximum vues, nd vrinces) in Tbe 1. As cn be seen from Figure 8, the cdf of the O-mode (thinner ine) is rger thn tht of the U-mode (thicker ine) for smer vues of. This fct cn so be seen from Figure 9. As presented in Eqution (4.9), the pdf of the O-mode is given by the simpe cosine curve, which decreses monotonicy. In contrst, the pdf of the U-mode hs pek t. For smer, the probbiity of finding trench in the O-mode is much rger thn in the U-mode. In fct, it cn be esiy shown tht the pdf for the O-mode t = is twice tht of the U-mode t = : f LO () = 1 π, (6.1) f LU () = 1 π. (6.) Note tht the expected vue of the trench ength of the O-mode is bout 5% ess thn tht of the U-mode. Erier studies hve focused on ony the mximum trench ength (i.e., the worst-cse scenrio) nd hve concuded tht the U-method is the best method mong those empoying singe simpy connected curve. Using our resuts, it is interesting to compre O- nd U-modes from simir pessimistic viewpoint, s in the iterture [5]. It cn be esiy derived tht the probbiity of trench digger hving to dig onger trench under the O-method thn the mximum trench ength of the U-method is given s foows: Pr{L O (π )} = π (π) f LO ()d = π cos(1) π.97. (6.3) Interestingy, this probbiity is ess thn 1%. This resut suggests tht the circur trench my be better choice thn the U-shped trench, except when it is cruci to void the worst-cse trench ength. Lsty, in terms of the vrince of the trench ength, the U-method, which hs smer vue thn the other, is superior to tht of the O-method.

13 41 K. Tnk & K. Shiin Figure 8: Cumutive distribution functions of O-mode nd U-mode ( = 1) Figure 9: Probbiity density functions of O-mode nd U-mode ( = 1) Tbe 1: Summry vues of O-mode nd U-mode O-mode U-mode Expected vue Mximum vue Vrince Concusions nd Future Reserch Perspectives This pper focused on mthemtic modes proposed by Fber et. [5] tht de with efficient methods for finding buried cbe. The probem is to find the shortest simpy connected curve tht ensures finding cbe when trench digger serches ong it to octe the intersection point between cbe nd the curve. Fber et. [4] proved the very interesting fct tht the U-shpe is the shortest singe-curve simpy connected trench. This probem cn be seen s finding curve tht minimizes the mximum trench ength tht trench digger must serch. Whie the bove resuts re very interesting mthemticy, the mximum vue is ony one of the mny importnt indices in re ppictions. In reity n importnt fct is tht once cbe is found, trench digger cn stop serching. Thus, in this pper, we constructed probbiistic mode tht describes rndom reiztion of cbe. Using the probbiistic mode, we succeeded in obtining nytic expressions of the cdfs nd the pdfs of the O- nd U-modes. One of our min resuts is tht though the U-mode is superior to the O-mode in terms of the mximum trench ength, s proved by Fber et. [4], the O-mode is more thn 5% better thn U-mode in terms of the expected vue of trench ength. To our knowedge, our resuts re the first to nyticy derive mesures other thn the mximum necessry trench. The pproch tken in this pper cn be extended in wide vriety of directions. We mention some of them in the foowing. This pper modeed the reiztion of cbe using rndom ines. Whie this is the most importnt bsic cse, other possibiities my be considered. One importnt cse is tht the probbiity of finding cbe octions coser to the mrker is higher thn tht t more remote octions. This sitution my be described by ssuming tht f P (p) is monotonicy decresing function of p. This cse cn be rediy det with using formutions presented in the Appendix. In urbn ppictions, some ine-shped network fciities such s wter nd sewer networks re buried under rod segment. Thus, nother possibiity is to consider the sitution where the probbiity of finding cbe running in prticur direction my

14 Strtegies for Finding Buried Cbe 413 be higher thn finding one running in other directions. Next, rexing the ssumption tht trench digger moves continuousy ong the predefined continuous curve my be n interesting topic to expore. In reity, trench digger my evute the probbiity of finding cbe for ech prt of the curve nd my strt serch for the cbe from the prt with the highest probbiity. Since this pproch is more fexibe, trench digger my find the cbe in smer mount of time thn suggested by the pproch tken in this pper. However, in such situtions, the costs of stopping nd restrting digging cnnot be ignored. Therefore, we cn extend the origin probem by incorporting this cost into the time required to find cbe. We focused on the pdfs nd expected trench engths obtined for two strtegies of finding cbe. The probem formution ntury eds to the foowing interesting, but chenging, question. Wht trench shpe is best in terms of the minimum expected ength to find cbe? Acknowedgements The uthors woud ike to thnk two nonymous referees for their constructive comments nd suggestions. This work ws supported by JSPS KAKENHI Grnt Number JP549. References [1] H.T. Croft: Curves intersecting certin sets of gret-circes on the sphere. Journ of the London Mthemtic Society, (1969), [] H.T. Croft, K.J. Fconer, nd R.K. Guy: Unsoved Probems in Geometry (Springer, New York, 1991). [3] A. Dumitrescu, M. Jing, nd J. Pch: Opque sets. Agorithmic, 69 (14), [4] V. Fber, J. Mycieski, nd P. Pedersen: On the shortest curve which meets ines which meet circe. Annes Poonici Mthemtici, 44 (1984), [5] V. Fber nd J. Mycieski: The shortest curve tht meets ines tht meet convex body. Americn Mthemtic Monthy, 93 (1986), [6] S.R. Finch: Bem detection constnt. In Mthemtic Constnts (Cmbridge University Press, Cmbridge, 3), [7] S.R. Finch nd J.E. Wetze: Lost in forest. Americn Mthemtic Monthy, 111 (4), [8] B. Kwoh: Some nonconvex shpe optimiztion probems. In A. Cein nd A. Ornes (eds.): Optim Shpe Design, Lecture Notes in Mthemtics (Springer, New York, ), [9] T. Koshizuk: Distnce distribution in n urbn re. In H. Tnimur, H. Kji, S. Iked, nd T. Koshizuk (eds.): Mthemtic Methods in Urbn Pnning (Askur Pubishing, Tokyo, 1986), 1 55 (in Jpnese). [1] P.J. Nhin: When Lest is Best: How Mthemticins Discovered Mny Cever Wys to Mke Things s Sm (or s Lrge) s Possibe (Princeton University Press, New Jersey, 7). [11] A.M. Mthi: An Introduction to Geometric Probbiity: Distribution Aspects with Appictions (Gordon nd Brech Science Pubishers, Amsterdm, 1999). [1] L.A. Snto: Integr Geometry nd Geometric Probbiity, nd Edition (Cmbridge University Press, Cmbridge, 4). [13] H. Soomon: Geometric Probbiity (SIAM, Phidephi, 1978).

15 414 K. Tnk & K. Shiin [14] I. Stewrt: The gret drin robbery. Scientific Americn, 73 (1995), 6 7. [15] R.C. Lrson nd A.R. Odoni: Urbn Opertions Reserch (Prentice H, New Jersey, 1981). [16] R.J. Vughn: Urbn Spti Trffic Ptterns (Pion, London, 1987). Appendix A: Derivtion of F LO () To derive the cdf of the trench ength F LO (), we hve to identify how Ω O () chnges ccording to the vue of. As shown in Figure 1, there re three different shpes of Ω O (): (I), (II) nd (III). In the Figures, the vertic htched res, the horizont htched res nd the meshed res correspond to the set of points (θ, p) in which the trench engths re given by Eqution (4.4), Eqution (4.5) nd Eqution (4.6), respectivey. () cse (I) (b) cse (II) (c) cse (III) Figure 1: Three different shpes of Ω O () For cses (I), (II) nd (III), F LO () cn be expressed s foows:

16 Strtegies for Finding Buried Cbe 415 (I) π F LO () = π (II) π π 3π 3π π F LO () = π cos(θ ) f Θ,P (θ, p)dpdθ cos(θ ) f Θ,P (θ, p)dpdθ cos θ cos θ cos θ π (III) π π F LO () = f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ π π π π π 3π cos(θ ) f Θ,P (θ, p)dpdθ π cos θ π f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ. π cos θ f Θ,P (θ, p)dpdθ cos(θ ) f Θ,P (θ, p)dpdθ cos θ cos(θ ) f Θ,P (θ, p)dpdθ, cos(θ ) f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ π cos θ 3π f Θ,P (θ, p)dpdθ, cos(θ ) f Θ,P (θ, p)dpdθ (A.1) (A.) (A.3) Appendix B: Derivtion of F LU () The process of deriving the cdf, F LU (), is bsicy the sme s tht of F LO (). However, identifying Ω U () for possibe cses is much more compicted thn tht of Ω O () due to the ower symmetry of the U-shpe. As shown in Figure 11, there re five different shpes of Ω U (): (I), (II), (III), (IV) nd (V). In the Figures, the vertic htched res, the horizont htched res, the meshed res nd the gry res correspond to the set of points (θ, p) in which the trench engths re given by Eqution (5.9), Eqution (5.1), Eqution (5.11) nd Eqution (5.1), respectivey. For cses (I), (II), (III), (IV) nd (V), F LU () cn be expressed s foows: (I) F LU () = π rctn(1 ) 3π 4 π { } ( ) rccos ( ) π rctn(1 ) sin θ( ) cos θ (cos θsin θ) sin θ( ) cos θ sin θ( ) cos θ f Θ,P (θ, p)dpdθ 3π 4 π rctn(1 ) f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ, 7π 4 π sin θ( ) cos θ (cos θsin θ) (cos θsin θ) f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ (B.1)

17 416 K. Tnk & K. Shiin () cse (I) (b) cse (II) (c) cse (III) (d) cse (IV) (e) cse (V) Figure 11: Five different shpes of Ω U ()

18 (II) ( π 4 1) F LU () = 1 π 1 { f Θ,P (θ, p)dpdθ 1 sin ( 1 ) π rctn 1 cos(1 ) (sin θcos θ) π 1 sin(θ 1) 3π 4 } (III) ( π 4 1) ( 3π 4 1) F LU () = 1 f Θ,P (θ, p)dpdθ Strtegies for Finding Buried Cbe 417 { } 1 sin(1 π rctn ) 1 cos ( 1 ) 1 f Θ,P (θ, p)dpdθ 1 3π 4 { } 1 sin(1 ) π rctn 1 cos(1 ) (sin θcos θ) 1 sin(θ 1) π π 1 (IV) ( 3π 4 1) (π 1) π F LU () = π 1 π rctn { f Θ,P (θ, p)dpdθ 1 sin(1 ) 1 cos(1 ) } f Θ,P (θ, p)dpdθ 3π 4 sin(θ 1) f Θ,P (θ, p)dpdθ π 1 π (sin θcos θ) 7π 4 { } 1 sin(1 π rctn ) 1 cos(1 ) f Θ,P (θ, p)dpdθ π 1 (sin θcos θ) f Θ,P (θ, p)dpdθ sin(θ 1) (sin θcos θ) f Θ,P (θ, p)dpdθ, f Θ,P (θ, p)dpdθ sin(θ 1) f Θ,P (θ, p)dpdθ π 1 3π 4 π (sin θcos θ) 7π 4 { } 1 sin π rctn ( 1 ) 1 cos ( 1 ) f Θ,P (θ, p)dpdθ, f Θ,P (θ, p)dpdθ f Θ,P (θ, p)dpdθ, sin(θ 1) (B.) (B.3) f Θ,P (θ, p)dpdθ (B.4) (V) (π 1) (π ) F LU () = { } (π1) π rccos ( (π1)) π rctn{ (π) } { } (π1) π rccos ( (π1)) π π rctn{ (π) } f Θ,P (θ, p)dpdθ sin θ( (π1)) cos θ (sin θcos θ) f Θ,P (θ, p)dpdθ. f Θ,P (θ, p)dpdθ (B.5) Ken-ichi Tnk Keio University Hiyoshi, Kohoku-ku, Yokohm Kngw 3-85, Jpn E-mi: ken1tnk@e.keio.c.jp