343:
510:
716:
256:
621:
385:
551:
650:
435:
405:
195:
175:
139:
115:
95:
75:
52:
157:, which explored theoretical and algorithmic problems associated with tree spanners. Some of the conclusions from that paper are listed below.
261:
796:
652:, it is NP-complete to determine whether a weighted graph contains a tree t-spanner, even if all edge weights are positive integers.
781:
Graph-Theoretic
Concepts in Computer Science: 26th International Workshop, WG 2000 Konstanz, Germany, June 15–17, 2000, Proceedings
443:
820:
32:
664:
204:
55:
562:
118:
815:
351:
779:
Handke, Dagmar; Kortsarz, Guy (2000), "Tree spanners for subgraphs and related tree covering problems",
345:. Furthermore, every tree 1-spanner admissible weighted graph contains a unique minimum spanning tree.
515:
554:
792:
733:
784:
762:
629:
414:
149:
There are several papers written on the subject of tree spanners. One of these was entitled
408:
390:
180:
160:
124:
100:
80:
60:
37:
809:
728:
154:
655:
A tree spanner (or a minimum tree spanner) of a digraph can be found in linear time.
201:
A tree 1-spanner, if it exists, is a minimum spanning tree and can be found in
766:
788:
338:{\displaystyle \beta (m,n)=\min \left\{i\mid \log ^{i}n\leq m/n\right\}}
783:, Lecture Notes in Computer Science, vol. 1928, pp. 206–217,
440:
The complexity for finding a minimum tree spanner in a digraph is
97:
in which the distance between every pair of vertices is at most
661:
The quasi-tree spanner of a weighted digraph can be found in
670:
568:
449:
357:
210:
559:
The minimum 1-spanner of a weighted graph can be found in
258:
time (in terms of complexity) for a weighted graph, where
753:
Cai, Leizhen; Corneil, Derek G. (1995). "Tree
Spanners".
505:{\displaystyle {\mathcal {O}}((m+n)\cdot \alpha (m+n,n))}
667:
632:
565:
518:
446:
417:
393:
354:
264:
207:
183:
163:
127:
103:
83:
63:
40:
177:is always the number of vertices of the graph, and
710:
644:
615:
545:
504:
429:
399:
379:
337:
250:
189:
169:
133:
109:
89:
69:
46:
711:{\displaystyle {\mathcal {O}}(m\log \beta (m,n))}
251:{\displaystyle {\mathcal {O}}(m\log \beta (m,n))}
286:
616:{\displaystyle {\mathcal {O}}(mn+n^{2}\log(n))}
8:
658:A digraph contains at most one tree spanner.
153:written by mathematicians Leizhen Cai and
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631:
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322:
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209:
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126:
102:
82:
62:
39:
745:
348:A tree 2-spanner can be constructed in
7:
755:SIAM Journal on Discrete Mathematics
380:{\displaystyle {\mathcal {O}}(m+n)}
14:
553:is a functional inverse of the
705:
702:
690:
675:
626:For any fixed rational number
610:
607:
601:
573:
546:{\displaystyle \alpha (m+n,n)}
540:
522:
499:
496:
478:
469:
457:
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280:
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215:
1:
837:
767:10.1137/S0895480192237403
789:10.1007/3-540-40064-8_20
197:is its number of edges.
712:
646:
645:{\displaystyle t>1}
617:
547:
506:
431:
430:{\displaystyle t>3}
411:for any fixed integer
401:
381:
339:
252:
191:
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135:
111:
91:
71:
48:
713:
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548:
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432:
402:
382:
340:
253:
192:
172:
136:
112:
92:
72:
49:
821:NP-complete problems
665:
630:
563:
516:
444:
415:
407:-spanner problem is
391:
352:
262:
205:
181:
161:
125:
101:
81:
61:
38:
387:time, and the tree
708:
642:
613:
555:Ackermann function
543:
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427:
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335:
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187:
167:
131:
107:
87:
67:
44:
798:978-3-540-41183-3
734:Geometric spanner
400:{\displaystyle t}
190:{\displaystyle m}
170:{\displaystyle n}
134:{\displaystyle G}
110:{\displaystyle k}
90:{\displaystyle G}
70:{\displaystyle T}
47:{\displaystyle G}
828:
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116:
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96:
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68:
56:spanning subtree
53:
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50:
45:
836:
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831:
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747:
742:
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663:
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628:
627:
585:
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560:
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513:
442:
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389:
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350:
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289:
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203:
202:
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123:
122:
99:
98:
79:
78:
59:
58:
36:
35:
12:
11:
5:
834:
832:
824:
823:
818:
808:
807:
804:
803:
797:
773:
772:
761:(3): 359–387.
744:
743:
741:
738:
737:
736:
731:
724:
721:
720:
719:
707:
704:
701:
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680:
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672:
659:
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641:
638:
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609:
606:
603:
600:
597:
592:
588:
584:
581:
578:
575:
570:
557:
542:
539:
536:
533:
530:
527:
524:
521:
501:
498:
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480:
477:
474:
471:
468:
465:
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459:
456:
451:
438:
426:
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420:
396:
376:
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370:
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359:
346:
333:
329:
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186:
166:
146:
143:
130:
106:
86:
66:
43:
13:
10:
9:
6:
4:
3:
2:
833:
822:
819:
817:
816:Spanning tree
814:
813:
811:
800:
794:
790:
786:
782:
777:
776:
768:
764:
760:
756:
749:
746:
739:
735:
732:
730:
729:Graph spanner
727:
726:
722:
699:
696:
693:
687:
684:
681:
678:
660:
657:
654:
639:
636:
633:
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595:
590:
586:
582:
579:
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558:
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534:
531:
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519:
493:
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487:
484:
481:
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472:
466:
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460:
439:
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410:
394:
371:
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347:
331:
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319:
316:
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305:
301:
297:
294:
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283:
277:
274:
271:
265:
239:
236:
233:
227:
224:
221:
218:
200:
199:
198:
184:
164:
156:
155:Derek Corneil
152:
151:Tree Spanners
145:Known Results
144:
142:
128:
120:
104:
84:
64:
57:
41:
34:
30:
28:
23:
21:
780:
758:
754:
748:
150:
148:
117:times their
26:
25:
19:
17:
15:
409:NP-complete
24:(or simply
810:Categories
740:References
688:β
685:
599:
520:α
476:α
473:⋅
317:≤
311:
298:∣
266:β
228:β
225:
723:See also
512:, where
119:distance
29:-spanner
22:-spanner
31:) of a
795:
718:time.
623:time.
54:is a
33:graph
18:tree
793:ISBN
637:>
422:>
785:doi
763:doi
682:log
596:log
302:log
287:min
222:log
121:in
77:of
812::
791:,
757:.
141:.
16:A
802:.
787::
769:.
765::
759:8
706:)
703:)
700:n
697:,
694:m
691:(
679:m
676:(
671:O
640:1
634:t
611:)
608:)
605:n
602:(
591:2
587:n
583:+
580:n
577:m
574:(
569:O
541:)
538:n
535:,
532:n
529:+
526:m
523:(
500:)
497:)
494:n
491:,
488:n
485:+
482:m
479:(
470:)
467:n
464:+
461:m
458:(
455:(
450:O
437:.
425:3
419:t
395:t
375:)
372:n
369:+
366:m
363:(
358:O
332:}
328:n
324:/
320:m
314:n
306:i
295:i
291:{
284:=
281:)
278:n
275:,
272:m
269:(
246:)
243:)
240:n
237:,
234:m
231:(
219:m
216:(
211:O
185:m
165:n
129:G
105:k
85:G
65:T
42:G
27:k
20:k
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