.S. 252 Pro. Rorto Taassa oputatoal otry S., 1992 1993 Ltur 9 at: ar 8, 1993 Sr: a Q ol aro Prorty Sar Trs - Part 1 trouto t last ltur, w loo at trval trs. or trval pot losur prols, ty us lar spa a optal t. Toay, w sall stuy prorty sar trs, usul or prols tat volv trvals trst otr trvals. Our xapls wll as o a st o 13 trvals a trou alo a l l, sow. 1. T trvals av raw urat t l so as to stus ovrlapp stos. l a ur 1: Trt trvals or trval trsto prols. W start o y ar o o l t pots o a trval. T, w xt t two pots o a trval aoally tll ty t (s. 2). W ow av a o-to-o a oto app tw pots aov t l l a trvals o l. W sall us sall lttrs or as o trvals, a orrspo aptal lttrs or as o pots. 2 Ra Qurs W av loo at ra qurs arlr lturs. Ty aswr qustos o t ollow typ: v a st o pots S a a ra R, all pots ro S w all wt R. ra s sp y prov o or or o t ollow: (1) a uppr ou o X, (2) a lowr ou o X, (3) a uppr ou o Y, (4) a lowr ou o Y. ow, lt us osr t ollow tr qustos o a x st o trvals: 1
l a ur 2: app trvals o l to pots aov l. 1. v a trval X, w trvals ar ota X? 2. v a trval X, w trvals trst t? 3. v a trval X, w trvals ota X? W wll sow tat all ts qustos a rprst as ra qurs o t st o pots rat as aov. or ts purpos, w rst wat to rotat. 2 y 45 rs, ota. 3. T xt tr aras (s. 4, 5, 6) sow ow t aov tr qurs o trvals ar rprst as qurs o t orrspo pots. a as, t qury s a quaral lat y two rays, a t aswr to t qury s t st o trvals w orrspo to t pots tat all wt t quaral (ty ar t t aras). a rular ra qury, t ra s a rtal (.., all our o t ous ar sp). W av alray s tat ala sar trs ar t st ata strutur or ts qurs. owvr, as w a s ro t aras aov, t ra qurs us or trval otat prols, t ra s always uou o two ss. Ts l to a sar or a astr alort tat woul soow t ro ts at. 2
a ur 3: Rotat app o trvals o l to pots aov l. a ur 4: W trvals ar ota? 3
a ur 5: W trvals trst? a ur 6: W trvals ota? 4
3 Prorty Sar Trs: t arou W a otr xapls o ra sar o a o-rtaular ra. or xapl, all pots tw two v vrtal ls (s. 7 (a)). T st approa or ts prol woul to us a ala ary sar tr. (a) () ur 7: Splr ass o ra sar prols otr xapl woul to all pots aov a v orzotal l (s. 7 ()). Tou w oul also us a ala sar tr r, a ttr approa ts as s a ap. prorty sar tr s a yr o a ap a a ala sar tr. Ty wr sovr rtly [1], to us or ra qurs wr at last o o t ss o t ra s uou. T rst o t papr assus tat w ar al wt a prorty sar tr wr t uppr ou o Y s ss (.., t ra or t qurs wll oly spy a lowr ou o Y). t soul asy to oy t alorts to apply to ass wr a rt ou s ss. 4 rat a Prorty Sar Tr or a v st o pots S, w rat a prorty sar tr as ollows: S s pty, w rtur a ULL potr a o ot otu. T pot P S wt t t ratst Y-oorat os t root R. S P s pty, R s also a la; w rtur R a o ot otu. Lt X(P ) a valu su tat al o pots S P av X-oorat lowr ta X(P ), a al av r. Rursvly rat a prorty sar tr o t lowr al o S P, lt ts root t lt l o R. Rursvly rat a prorty sar tr o t uppr al o S P, lt ts root t rt l o R. 5
X() X() X() 1st lvl 2 lvl (-1,9):8 X() X() X() X() 3r lvl (1,8):3 (-2,5) (2,3) (4,0) (15,7):11 (7,6):5 (6,4) (16,2):13 (12,1) (10,-2) (14,-1) (9,-3) oplt tr ur 8: rat a Prorty Sar Tr. 8 llustrats t ostruto o a prorty sar tr o our st o 13 pots. T pots alray t tr ar sol; t pots os xt ar sa; ull potrs ar sow as potrs to lttl oxs. t al ptur, a tr o P s lal y ts oorats, ollow y X(P) t s rt ro t X-oorat. Ts lals wll us t xt aptr. 6
5 Qury a Prorty Sar Tr W sall ot or wt upat t prorty sar tr yaally; ts top wll arss t xt ltur. qury o a prorty sar tr s as ollows: v x, x, a y, wat ar all t pots t tr wos X-oorat s tw x a x, a wos Y-oorat s ratr ta y? T ollow s t alort or pror ts qury: t tr s ULL, w rtur wtout ay pots. Lt R t root o t tr, x ts X-oorat, y ts Y-oorat, a X(R) t valu o t axs sparat t X-ras o R s l sutrs. opar y to y. y < y, w rtur wtout ay pots (all otr os t tr wll av a v sallr Y-oorat). x x x, rport t root pot. x < X(R), t X-ra o t lt sutr ust ovrlap wt t X-ra o t qury. Rursvly sar t lt sutr o R. X(R) < x, t X-ra o t rt sutr ust ovrlap wt t X-ra o t qury. Rursvly sar t rt sutr o R. xapl o a prorty sar tr qury, wt x = 0, x = 11, y = 4.5, s sow. 9. os tat ar vst ut ot rport ar sa; os tat ar vst a rport ar sol. 6 T a Spa alyss t s lar tat a prorty sar tr o pots tas up spa O(), s tr s o o or a pot. T t o t tr s O(lo ), s t os ar partto al at a lvl. t ras to sow tat or a qury wt a aswr o sz, qury t wll O( + lo ). T qury t s proportoal to t ur o os vst. Lt us atorz all t vst os as ollows (rr to. 9): 1. o s vst a rport. os, ar ts atory. T ur o rport os s y to. 2. o s vst a ot rport, ut ts X-oorat alls wt t [x, x ] ra. os,,, ar ts atory. Ts o ust av a a Y-oorat. 7
(-1,9):8 x = 0 y = 4.5 x = 11 (1,8):3 (15,7):11 (-2,5) (7,6):5 (2,3) (4,0) (6,4) (10,-2) (12,1) (14,-1) (16,2):13 (9,-3) o ; x = -1, y = 9, X() = 8 y >= 4.5, otu x ot [0..11], o ot rport x < 8, o lt sutr x > 8, o rt sut o ; x = 1, y = 8, X() = 3 y >= 4.5, otu x [0..11], rport x < 3, o lt sutr x > 3, o rt sutr o ; x = 15, y = 7, X() = 11 y >= 4.5, otu x ot [0..11], o ot rport x < 11, o lt sutr x <= 11, o t o rt sutr o ; x = -2, y = 5, X() = -2 y >= 4.5, otu x ot [0..11], o ot rport x >= -2, o t o lt sutr x > -2, o t rt sutr o, x = 7, y = 6, X() = 5 y >= 4.5, otu x [0..11], rport x < 5, o lt sutr x > 6, o rt sutr o ; x = 10, y = -2, X() = 10 y < 4.5, so rtur o ; x = 2, y = 3, X() = 2 y < 4.5, so rtur o ; x = 4, y = 0, X() = 4 y < 4.5, so rtur o ; x = 6, y = 4, X() = 6 y < 4.5, so rtur ur 9: Sar a Prorty Sar Tr 8
ts as, ts part a to vst, a av a oo Y-oorat, all to o o t otr ators. Tror, t total ur o ts os s at ost 2 * (all otr vst os). 3. o s vst a ot rport aus ts X-oorat s a. os,, ar ts atory., or a lvl o t tr, w lst t os orr ro lt to rt, ts lst wll sort y as X-oorat. xt, w ot t ollow: two aat os t lst ar ot to t lt (rt) o t [x, x ] ra, t ltost (rtost) o t aot possly vst y t sar alort. Tror, tr wll at ost 2 os pr lvl w all outs t [x, x ] ra (o o t rt a o o t lt) w a vst. S tr ar O(lo ) lvls t tr, tr ar at ost O(lo ) os ts atory. Su up t total ur o vst os, w ota a qury t o O( + lo ), as s sr. Rrs [1] war. rt, Prorty Sar Trs, S. oput., Vol. 14, o. 2, pp. 257 276, ay 1985. 9