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Computing the Diameter of a Point Set

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(1)Computing the Diameter of a Point Set Grégoire Malandain, Jean-Daniel Boissonnat. To cite this version: Grégoire Malandain, Jean-Daniel Boissonnat. Computing the Diameter of a Point Set. RR-4233, INRIA. 2001. �inria-00072354�. HAL Id: inria-00072354 https://hal.inria.fr/inria-00072354 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Computing the Diameter of a Point Set Grégoire Malandain — Jean-Daniel Boissonnat. N° 4233 July 27, 2001. ISSN 0249-6399. ISRN INRIA/RR--4233--FR+ENG. THÈMES 3 et 2. apport de recherche.

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(9) “f™¹¸ dbQU‚Pw‡V`_nV]cedf[L_fuºmfR$dfQURV`^]yblPo‡ce_fQURV]R$^$ª. Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65.

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(249) 65º °db^@V]NPR,SzR o‡cqSzV]^`ck|PwPV]cedf[LScqS@uq_n^`mfR^V`NL_n[ °df^ V`NPRg|L_nueu_f[Lo o‡R,a^`R$_bSzR,S ¨NPR$[ V]NLR[{wPQ|>R$^df †šfR^]V]cqaR,Sdf V]NPRg‚0dfuey.mbdf[LSck[>a^`R$_fS]R$S$ª <‡ckQUceue_f^]ueyfl V`NPRadfQU‚PueR¥‡cV€ydn =dfwL^H_nuemfdb^]ckV]NPQ  °df^­S]wLajN‚0dfce[bVto‡ceSzV]^`ce|Pw‡V]cedf[>S†o‡R$a^]R,_fS]R$S@¨NPR$[V]NPR[{wPQ|0R^†df  šbR^]V]cqaR,S§ce[La^]R,_fS]R$S@_n[LoV`R[LoPSV`d'|>RvV`NPR S`_nQURƒ_fS °df^‚0dfce[.V`S@ce[*_|>_nueu3ª    G H¶ ’ — ”Ÿ“ ² ”,“ ’ — ¢ ¤ Y\[ ›LmfwP^`RtŒ‡lb¨R†‚P^`d—š{ceo‡RHR¥‡‚>R$^]ceQUR[.V`_fuL^`R$S]wPukV`S§ck[Ÿ¢=¤I_n[>o'Rš—_nuewL_nV]RV`NPRtadfQU‚PueR¥‡ckV€yI_bSB_v °wP[>aEV]cedf[ df ©V`NPR o‡ceQIR$[LS]ckdb[  ª . <. oporq.sutvxwxw. .

(250) ŽD. 6. , .     $ '.. 10. 10. 6000. 9. Rmin / Rmed = 0.2 Rmin / Rmed = 0.596 Rmin / Rmed = 1.0. 9. 5000. 8 4000. 7. 3000. 6. 8. 7. 5 2000 6. 4 1000. 3. 0 1. 2 1. 5. 0.8. 1 0.6. 4. 0.8. 1 0.6. 0.8 0.6. 0.8 3. 0.6. ®©šbR^j_nmbRH[{wPQ|0R^Hdn ^`RQ'_fck[Pce[PmU‚0dfce[bVjS ®©šbR^j_nmbR­[{wPQ|>R$^dn 1[<~ S`a_f[LS[PRR,o‡R$oV]d adfQU‚Pw‡V`R dfwPV`S]ceo‡RtV`NPRƒ|L_nueu[-4

(251) 65 _Uo‡dfwL|PukRv[Pdb^]Q'_fu [

(252) cemfwP^`R _:<‡dfQUR­SzV`_nV]cqS€V`cea$S„ °db^S]RVjSBdf  %s(s^s st‚0dfce[bVjSdf['V`NPRtS]wP^] F_faR†dn ©_IT Rueukce‚LS]dfcqoºª@Ž$bvV]^`ce_fueS ¨R^`Rt‚>R$^z °db^]QUR$oŸ °df^R,_fajNSzRV†dn <_—¥‡R$S$ª 0.4. 0.4. 0.2. 0. 0.4. 0.2. 0.2. 0. Rmin / Rmed. 0.4. Rmin / Rmed. Rmed / Rmax. 25. 0.2. 0. 2 0.2. 0. 0.3. 0.4. 0.5. Rmed / Rmax. 200. 0.6 Rmed / Rmax. 0.7. 0.8. 0.9. 2500. 180. 160. 15. 10. 5. 2000 average number of dot products per point. average number of dot products per point. average number of dot products per point. 20. 140. 120. 100. 80. 60. 40. 1500. 1000. 500. 20. Ž,f‚>dbck[.VjS Ž$fb ‚0dfce[.V`S Ž$bff‚>dbck[.VjS [

(253) cemfwP^`R u_=W o‡cqS€V`^]ce|Pw‡V`ckdb[LS„ce[LS]ceo‡R­o‡dfQ'_nce[LS=df xadf[>S€Vj_n[.V„¨ceo‡V]Nxª=M@NPR›>mfwP^`RHS]NPd—¨†S

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(256) šbR^]V]cqaR,S@dn <V]NPRƒ‚0dfuey{mfdb[LSª §uewPRƒueck[LDR _„¨ckV]N*[Pd'^`R$oPwLaEV`ckdb[gdf  ª= R$oŸukce[P8R _=¨cV`N^`R$o‡w>aEV]cedf[Ÿdn  ª ­_bSzNukce[PR,S _

(257) V]NLRvadfQU‚PueR¥‡ckV€y °db^_IW 8oPceSzV]^`ck|Lw‡V]cedf[ck[LS]cqo‡Rt_ |>_nueu3ª . 0. 0. 5. 10. 15. 20 #points. 25. 30. 35. 0. 40. 0. 5. 10. 15. 20 #points. 25. 30. 35. 40. 0. 0. 5. 10. 15. 20 #points. 25. 30. 35. 40. . Ÿ” 1o[Ÿ¢¡. 1.

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(264) ~ 8V]ceQUR N°^`ckmbN.V  _n¥{cqScPEª [. . . oporq.sutvxwxw.

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(269) cemfwP^`R$ S MŸ_f[Lo6SzNLd—¨†S†V`NPRIadfQU‚PueR¥‡ckV€ygdf „V]NLRQURV`NPd‡o ¨ckV]Nœ_n[Lo ¨cV`NPdfw‡Vƒ^`R$o‡w>aEV]cedf[6dn   °df^ š—_f^]cedfwLS@T ˆ‚>dbck[.V†o‡cqS€V`^]ce|Pw‡V`ckdb[LS$ª„M@NPR$S]Rt›LmfwP^`R$S@N>_ šfR­V]d'|0R adfQU‚L_f^]R,o'V`d›>mfwP^`R$STU_n[>oŸŠ>ª . . . "%. &. &)*. . *#/. ". . #"/. . šfdfuewPQURv|L_fS]R$ogo‡cqS€V`^]ce|Pw‡V`ckdb[LS 10. 600. 9 500 average number of dot products per point. average number of dot products per point. 8. 7. 6. 5. 4. 3. 400. 300. 200. 2 100. 1. 0. 1. 2. 3. ck[_'awP|0R 4. 5. 6. 7. 8. 9. 0. 10. #points. 1. 2. ck[*_I|L_fuku. 3. 4. 5. 6. 7. 8. 9. 10. #points. 4. x 10. 4. x 10. S]wP^] F_faRv|L_bSzR,ooPceSzV]^`ck|Lw‡V]cedf[LS 10. 15. 5. 9. 4.5. 7. 6. 5. 4. 3. average number of dot products per point. 4. average number of dot products per point. average number of dot products per point. 8. 10. 5. 3.5. 3. 2.5. 2. 1.5. 2. 1. 1. 0.5. 0. 1. df[_^`RmbwPuq_n^@Rueukce‚LS]dfcqo 2. 3. 4. 5. 6. #points. 7. 8. 9. 10 4. x 10. 0. 1. 2. db[*_n[gRueukce‚LS]dfcqo 3. 4. 5. 6. #points. 7. 8. 9. 0. 10. 1. 2. 3. df[_'Sz‚LNPR^`R 4. 5. 6. 7. #points. 4. x 10. 8. 9. 10 4. x 10. [

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