Interval Stabbing Problems in Small Integer Ranges. Jens M. Schmidt

Transcription

1 Intv Stng Pom n Sm Intg Rng Jn M. Scmdt

2 Outn 1. Pom Dfnton 2. Dt Stuctu

3 Intv Stng I = t of n ntv [, ] wt Stng quy on vu q: A fo ntv n I tt contn q. Wntd: Dt tuctu tt uot qu ffcnty.

4 Intv Stng Pom Intv Stng Pom Intv Intcton Pom: Gvn quy ntv [q 1,q 2 ], ot ntv n I tt ntct [q 1,q 2 ].

5 Intv Stng Pom Intv Cov Pom: Gvn quy ntv [q 1,q 2 ] n I, ot ntv n I tt contn [q 1,q 2 ]. Mut Quy Pom: Gvn mut qu otd n xcogc od, xtnd c o om to ot ntv ng contnd n t unon of outut.

6 Intv Stng Pom Wot c unnng tm fo quy O(n). outut-ntv comxty W wnt otm tm O(1+) fo ntv n t outut.

7 Wy Sm Intg Rng? Lt ntv ndont n {1,...,N}. Tm (Bm nd Ftc 1999): Fo ty N, vy dt tuctu ung n O(1) mmoy c nd og n /og og n tm fo tng quy.

8 Wy Sm Intg Rng? To cv contnt tm w v to mo tcton: W um tt ndont nd q n {1,...,O(n)}. W..o.g. ndont w dtnct.

9 Ovvw Wntd: Dt tuctu fo Intv Stng Pom Intv Intcton Pom Intv Cov Pom Mut Quy Pom wt O(n) ocng Stng qu n otm tm O(1+), outut-n. Outut otd y ft ndont

10 Ovvw Wntd: Dt tuctu fo Intv Stng Pom Intv Intcton Pom Intv Cov Pom Mut Quy Pom 1986 Cz: Ftng Sc wt O(n) ocng Stng qu n otm tm O(1+), outut-n. Outut otd y ft ndont

11 Ovvw Wntd: Dt tuctu fo Intv Stng Pom Intv Intcton Pom Intv Cov Pom Mut Quy Pom 1986 Cz: Ftng Sc 2000 Atu t.: 3-dd ng qu, O(1+), ut nvovd wt O(n) ocng Stng qu n otm tm O(1+), outut-n. Outut otd y ft ndont

12 Ovvw Wntd: Dt tuctu fo Intv Stng Pom Intv Intcton Pom Intv Cov Pom Mut Quy Pom 1986 Cz: Ftng Sc 2000 Atu t.: 3-dd ng qu, O(1+), ut nvovd wt O(n) ocng Stng qu n otm tm O(1+), outut-n. Outut otd y ft ndont W w focu on t ft om.

13 Outn 1. Pom Dfnton 2. Dt Stuctu

14 Dt Stuctu An ntv n ut of I gtmot f t t on wt mxmum ft ndont. Fo n ntv : Pnt() := gtmot ntv tt contn

15 Dt Stuctu oot A B oot 1 oot 2 m n o D C

16 Dt Stuctu oot A B oot 1 oot 2 m n o D C A Pnt cn comutd n O(n) y w n g. Pnt ud fot Dt Stuctu: T fot + vtu oot (t odd) W nd quy on q y tvng t fot fom t (comutd) gtmot ntv contnng q.

17 Pocng Quy oot A B oot 1 oot 2 m n o D C

18 Pocng Quy oot A B oot 1 oot 2 m n o D C A ntv n A td. No ntv n C td. Std ng dcnt. Std ntv n B cn comutd ffcnty. Ony D mn.

19 Pocng Quy oot A B oot 1 oot 2 m n o D C Lmm 1: Evy td vtx v D (td) ncto n B.

20 Pocng Quy oot A B oot 1 oot 2 m n o D C Lmm 2: T ng w to t gt of td vtx v D td w, f t xt. z B td z' B A td v D td w D

21 Pocng Quy oot A B oot 1 oot 2 m n o D C It foow tt vy td vtx v D cn cd fom td vtx n B y zg-zg-t contng of td vtc. Ony 3 dcton to cc on ng td: to t gtmot cd, to t ft, nd u (ony n A).

22 Exm ft ndont m n o

23 Exm tt tv t t gtmot ntv contnng q m n o

24 Exm ncto of td ntv td m n o

25 Exm C do not contn td ntv m n o

26 Exm cc ft ng uccvy n O(1)... m n o

27 Exm...nd gt B m n o

28 Exm ty ft to go to t gtmot cd, tn to go to t ft m n o

29 Exm ty ft to go to t gtmot cd, tn to go to t ft m n o

30 Exm ty ft to go to t gtmot cd, tn to go to t ft m n o

31 Exm ty ft to go to t gtmot cd, tn to go to t ft m n o

32 Exm ty ft to go to t gtmot cd, tn to go to t ft m n o

33 Exm ty ft to go to t gtmot cd, tn to go to t ft m n o

34 Pom Vnt Intv Intcton Pom: Pfom quy on t gt ndont of quy ntv, ut cng t tong condton of t tnv. Intv Cov Pom: Lmm 1 nd 2 t od wn q n ntv. Mut Quy Pom: Stt wt t gtmot quy nd coo dtvy t nxt ow quy vu.