17:
348:, the graph formed by removing the longest edge from every triangle in the Delaunay triangulation, was originally proposed as a fast method to compute the relative neighborhood graph. Although the Urquhart graph sometimes differs from the relative neighborhood graph it can be used as an approximation to the relative neighborhood graph.
657:
211:
307:
714:
630:
240:
173:
138:
118:
98:
78:
58:
458:
Katajainen, Jyrki; Nevalainen, Olli; Teuhola, Jukka (1987), "A linear expected-time algorithm for computing planar relative neighbourhood graphs",
269:
Because it is defined only in terms of the distances between points, the relative neighborhood graph can be defined for point sets in any
273:
and for non-Euclidean metrics. Computing the relative neighborhood graph, for higher-dimensional point sets, can be done in time
805:
579:
144:
in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set.
544:
Jaromczyk, J. W.; Kowaluk, M. (1991), "Constructing the relative neighborhood graph in 3-dimensional
Euclidean space",
246:
513:
Lingas, A. (1994), "A linear-time construction of the relative neighborhood graph from the
Delaunay triangulation",
515:
334:
602:
460:
326:
546:
427:
Supowit, K. J. (1983), "The relative neighborhood graph, with an application to minimum spanning trees",
832:
635:
178:
330:
258:
25:
276:
400:
Jaromczyk, J.W.; Toussaint, G.T. (1992), "Relative neighborhood graphs and their relatives",
692:
608:
216:
8:
770:
429:
370:
158:
141:
123:
103:
83:
63:
43:
583:
743:
Urquhart, R. B. (1980), "Algorithms for computation of relative neighborhood graph",
729:
672:
575:
560:
529:
473:
386:
782:
752:
725:
668:
555:
524:
495:
469:
438:
409:
382:
33:
791:
338:
318:
37:
490:
Jaromczyk, J. W.; Kowaluk, M. (1987), "A note on relative neighborhood graphs",
345:
16:
826:
322:
242:
686:
786:
756:
773:(1980), "Comment: Algorithms for computing relative neighborhood graph",
254:
250:
499:
443:
20:
The relative neighborhood graph of 100 random points in a unit square.
413:
373:(1980), "The relative neighborhood graph of a finite planar set",
804:
Andrade, Diogo Vieira; de
Figueiredo, Luiz Henrique (2001),
457:
155:
showed how to construct the relative neighborhood graph of
806:"Good approximations for the relative neighbourhood graph"
605:(1982), "Computing the relative neighborhood graph in the
813:
Proc. 13th
Canadian Conference on Computational Geometry
337:
is a subgraph of it, from which it follows that it is a
140:
than they are to each other. This graph was proposed by
80:
by an edge whenever there does not exist a third point
695:
638:
611:
279:
253:. The relative neighborhood graph can be computed in
219:
181:
161:
126:
106:
86:
66:
46:
803:
317:The relative neighborhood graph is an example of a
708:
651:
624:
584:"Relative neighborhood graphs in three dimensions"
399:
301:
234:
205:
167:
132:
112:
92:
72:
52:
824:
543:
489:
574:
689:(1985), "Relative neighborhood graphs in the
601:
588:Proc. 3rd ACM–SIAM Symp. Discrete Algorithms
494:, New York, NY, USA: ACM, pp. 233–241,
769:
559:
528:
442:
369:
742:
15:
426:
152:
825:
512:
492:Proc. 3rd Symp. Computational Geometry
485:
483:
365:
363:
361:
685:
480:
175:points in the plane efficiently in
13:
644:
358:
264:
36:defined on a set of points in the
14:
844:
312:
30:relative neighborhood graph (RNG
797:
763:
736:
679:
335:Euclidean minimum spanning tree
595:
568:
537:
506:
461:Information Processing Letters
451:
420:
393:
296:
283:
229:
223:
200:
185:
1:
351:
147:
730:10.1016/0031-3203(85)90023-8
673:10.1016/0031-3203(82)90070-X
561:10.1016/0166-218X(91)90069-9
547:Discrete Applied Mathematics
530:10.1016/0925-7721(94)90018-3
474:10.1016/0020-0190(87)90225-0
387:10.1016/0031-3203(80)90066-7
213:time. It can be computed in
7:
652:{\displaystyle L_{\infty }}
245:, for random set of points
10:
849:
206:{\displaystyle O(n\log n)}
40:by connecting two points
302:{\displaystyle O(n^{2})}
402:Proceedings of the IEEE
100:that is closer to both
710:
653:
626:
516:Computational Geometry
331:Delaunay triangulation
303:
259:Delaunay triangulation
236:
207:
169:
134:
114:
94:
74:
54:
26:computational geometry
21:
790:. Reply by Urquhart,
711:
709:{\displaystyle L_{1}}
654:
627:
625:{\displaystyle L_{1}}
304:
247:distributed uniformly
237:
208:
170:
135:
115:
95:
75:
55:
19:
693:
636:
609:
277:
235:{\displaystyle O(n)}
217:
179:
159:
124:
104:
84:
64:
44:
787:10.1049/el:19800611
775:Electronics Letters
757:10.1049/el:19800386
745:Electronics Letters
718:Pattern Recognition
661:Pattern Recognition
500:10.1145/41958.41983
444:10.1145/2402.322386
375:Pattern Recognition
706:
649:
622:
576:Agarwal, Pankaj K.
430:Journal of the ACM
299:
261:of the point set.
232:
203:
165:
142:Godfried Toussaint
130:
110:
90:
70:
50:
22:
168:{\displaystyle n}
133:{\displaystyle q}
113:{\displaystyle p}
93:{\displaystyle r}
73:{\displaystyle q}
53:{\displaystyle p}
840:
833:Geometric graphs
817:
815:
810:
801:
795:
789:
771:Toussaint, G. T.
767:
761:
759:
740:
734:
732:
715:
713:
712:
707:
705:
704:
683:
677:
675:
658:
656:
655:
650:
648:
647:
631:
629:
628:
623:
621:
620:
599:
593:
591:
590:, pp. 58–65
572:
566:
564:
563:
541:
535:
533:
532:
510:
504:
502:
487:
478:
476:
455:
449:
447:
446:
424:
418:
416:
414:10.1109/5.163414
408:(9): 1502–1517,
397:
391:
389:
371:Toussaint, G. T.
367:
308:
306:
305:
300:
295:
294:
272:
241:
239:
238:
233:
212:
210:
209:
204:
174:
172:
171:
166:
139:
137:
136:
131:
119:
117:
116:
111:
99:
97:
96:
91:
79:
77:
76:
71:
59:
57:
56:
51:
34:undirected graph
848:
847:
843:
842:
841:
839:
838:
837:
823:
822:
821:
820:
808:
802:
798:
768:
764:
751:(14): 556–557,
741:
737:
700:
696:
694:
691:
690:
684:
680:
643:
639:
637:
634:
633:
616:
612:
610:
607:
606:
600:
596:
573:
569:
542:
538:
511:
507:
488:
481:
456:
452:
425:
421:
398:
394:
368:
359:
354:
339:connected graph
333:. In turn, the
315:
290:
286:
278:
275:
274:
270:
267:
265:Generalizations
218:
215:
214:
180:
177:
176:
160:
157:
156:
150:
125:
122:
121:
105:
102:
101:
85:
82:
81:
65:
62:
61:
45:
42:
41:
38:Euclidean plane
12:
11:
5:
846:
836:
835:
819:
818:
796:
762:
735:
724:(5): 327–332,
703:
699:
678:
667:(3): 189–192,
646:
642:
619:
615:
594:
580:Mataušek, Jiří
567:
554:(2): 181–191,
536:
523:(4): 199–208,
505:
479:
450:
437:(3): 428–448,
419:
392:
381:(4): 261–268,
356:
355:
353:
350:
346:Urquhart graph
314:
313:Related graphs
311:
298:
293:
289:
285:
282:
266:
263:
231:
228:
225:
222:
202:
199:
196:
193:
190:
187:
184:
164:
153:Supowit (1983)
149:
146:
129:
109:
89:
69:
49:
9:
6:
4:
3:
2:
845:
834:
831:
830:
828:
814:
807:
800:
793:
788:
784:
780:
776:
772:
766:
758:
754:
750:
746:
739:
731:
727:
723:
719:
701:
697:
688:
682:
674:
670:
666:
662:
640:
617:
613:
604:
598:
589:
585:
581:
577:
571:
562:
557:
553:
549:
548:
540:
531:
526:
522:
518:
517:
509:
501:
497:
493:
486:
484:
475:
471:
467:
463:
462:
454:
445:
440:
436:
432:
431:
423:
415:
411:
407:
403:
396:
388:
384:
380:
376:
372:
366:
364:
362:
357:
349:
347:
342:
340:
336:
332:
328:
324:
323:beta skeleton
320:
310:
291:
287:
280:
262:
260:
256:
252:
248:
244:
243:expected time
226:
220:
197:
194:
191:
188:
182:
162:
154:
145:
143:
127:
107:
87:
67:
47:
39:
35:
31:
27:
18:
812:
799:
778:
774:
765:
748:
744:
738:
721:
717:
681:
664:
660:
603:O'Rourke, J.
597:
587:
570:
551:
545:
539:
520:
514:
508:
491:
468:(2): 77–86,
465:
459:
453:
434:
428:
422:
405:
401:
395:
378:
374:
343:
316:
268:
151:
29:
23:
792:pp. 860–861
781:(22): 860,
255:linear time
251:unit square
716:-metric",
687:Lee, D. T.
659:metrics",
352:References
325:. It is a
271:dimension,
148:Algorithms
645:∞
257:from the
195:
827:Category
582:(1992),
327:subgraph
32:) is an
329:of the
321:-based
249:in the
28:, the
809:(PDF)
632:and
344:The
319:lens
120:and
60:and
783:doi
753:doi
726:doi
669:doi
556:doi
525:doi
496:doi
470:doi
439:doi
410:doi
383:doi
192:log
24:In
829::
811:,
779:16
777:,
749:16
747:,
722:18
720:,
665:15
663:,
586:,
578:;
552:31
550:,
519:,
482:^
466:25
464:,
435:30
433:,
406:80
404:,
379:12
377:,
360:^
341:.
309:.
816:.
794:.
785::
760:.
755::
733:.
728::
702:1
698:L
676:.
671::
641:L
618:1
614:L
592:.
565:.
558::
534:.
527::
521:4
503:.
498::
477:.
472::
448:.
441::
417:.
412::
390:.
385::
297:)
292:2
288:n
284:(
281:O
230:)
227:n
224:(
221:O
201:)
198:n
189:n
186:(
183:O
163:n
128:q
108:p
88:r
68:q
48:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
