Proc. of the 23rd Intl. Conf. on Parallel Processing, St. Charles, Illinois, August 1994, vol. 3, pp. 227{ Hanan Samet

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1 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 1 DT-PRE SPTI JOI GORITHMS Ek G. Hl y Gy Dv B C W, D.C H S C S D C R I v C S Uvy Myl Cll Pk, Myl { E -lll l j l R-, l,. T l l (.., B C TIGER/ l). Plll l l y. T l l SM (S-- M-) l lll y C M. ITRODUCTIO T l j l. T lly j w ll y [9]. I x, j w l vy l w y l. Tlly, vlv - w. I l v j, lly (.., w l j) w l v l. I w lll l l v l j ll l z l l. T ll. T v l w k ll l v y l y. T v lx. I k ll l l w v l y. T l y. I l kw z GIS [22], l [19]. F xl, w v U S Cl. l j y k ll Cl, wl l j y k ll l w 10 l Cl. W l l l [20, 21]. I T wk w y l S F G IRI y l w C R Uvy Myl. w w. T l vlv. T w w (.., w-l l) ll k. O kw R- [13] k ( l) l. I, l (lly y xy), - [8]. T wk R- l j, l l y vl. Evlly, l y lly vl l, y ly w l. T l y y vl l k l l. T j ll. E l j l l w ll. T v. Ty ly y ly l y wy w ll. T j v y l l, w v v ll ll. T w y kw PMR [17]. I vly l lk l. T l y. W SM (S--M-) l lll [3]. I y lly w llw l-w -w. [4, 16] k v,v [ 0 ; 1 ;;,1 ], v [ 0 ; ( 0 1 ); ;( 0 1,1 )]. B w w lly, y l. w w lly w lly. B -l [5], SM l k, llw

2 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 2 O(l ). T l wy l l j. I w w y l l l. W v - w wk w lll l w j (.., l ). O, -w [6] l l v ; w lv wk. T l l k vy l vy l w. Illy, wwl lk k v ly w l w v l ly. T y w PMR v ly j lk. Mv, y y lk w ly, w v k l ly. Hwv, l R- l j. I, R- y w l vl w l. T ll wl. T l l R-' ly l ly w l, v l w. Uly, xly ly l l j l ly. T l R- l v PMR. T z llw. S 2 l w PMR. I l lll v wll l l. T llw y l w l j l. S 3 w R- z wy S 2. S 4 w lll w l j l Tk M CM-5 lll. S 5 l k wll. Sl PMR Q R- T l w PMR. I ly vly l k vy z. T -- w k lk wl w l w. T l [20]. Ty y vx- -. T l k. ll l y ly l ly k ll v (k lyl ) lk l z ( l) l ly l z y. T PMR [17, 18] -. I k l l l. lk vl l. T PMR y l -y- lly y lk. E l ll lk y. D, y lk k x l l. I l l x, lk l, ly, lk l z. T l v l y w j w vy l l lk. I, w v llly. F l, [17]. l l PMR y v ll lk y. D, y lk l ( w w w l) k l l l l l l. I l l x y lk l, y vly l l lk l. yy w l l. 1 l 1 F 1: Il PMR. D-Plll PMR Q T l l lll PMR llw. Illy, l l, l PMR l F 1. V ww, l w l (9 xl). I l w x l l ( xl, l l 2),

3 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 3 l l. w. T v w - l j w lll R- l S 3.1. l y w, l l l l ll ( l [3]). F xl, l F 2 w W E. T l v w F l F 2: Rl v, l l -. C w v, l (.., l ) l,. F xl, F 2, l 1, l w 2,. E w l x l l. I l l x, wll v l wll w l y. I F 2, W SE wll v, l F l F 3: Rl v. T v v wll l ll lll PMR v l l l l l, xl l (.., z 1 1)., v xl l, l l w x l l. F l l l (.., v), xly wll y l - l v l l w v l F 4: Rl PMR l. T l l v xl w F 4. ( 9) ll l l x. T l l, wll v. T -lll PMR l lxy O(l ), w O(l ) v O(1) ( -). PMR Q M I I llw l, w w w -lll PMR ;,. T l z (.., y 2 2 ). T wll l w (.., y), l w wll j (.., y). Gv -lll PMR, w l w. T wll l l w w w l. Wl l y ( ), ly. T llw l. T lly l l, l yl z x (.., 2 2 ). T l w ll l ( xl, F 5). T ( w

4 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 4 l l l ( 4 F 6) l F 7, w l l. 1 l l l F 5: Exl PMR ( w l l 2) v l () l () l l () l l F 6: () M l v w l. () M l v. lly ly ) ly v l w l, l. Elly, v l --, --y, y-- l l w. F xl w F 5, l w, x, l. T l l w F 6. C w, 1, 2, 3 F 6 v l. M 4 v w v v. T l 4 F 6 l w F 6. T () F 7: () M l v w --, y--, -y w. () Q l --. O v l ( F 7 w l ), x --, y--, --y w. T y l -- l w w. F, w l ( F 7), (w wll ) (w l l v). T wll l -- w ( F 7). l, y. T l l w ll l (.., ll l ) l. F 8 l l xl ( l v w y l l w ). T l y l w v (w v l )

5 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 5 l. I F 8, l w w w. T l ll l v. l l F 8: S l l. S l y. E v l lly w l l. I l l, l k. C, ll l w v v., l ll l v, l lll w l. O ll v l v l, l ll l y l v. PMR Q Sl R Qy T -lll PMR l y l l l w ly xz lll. I llw, w -lll PMR z. W wll l x (l x ), l w w x. l l F 9: O-- y x x. T l y l w l ly l. O -- l ( F 9 xl xl x w y y ), ll l y l x x v. l l F 10: v l z (w l v ) l. T l y l l z (.., ll z 2 2 w ), w ll z. Gv ll z (

6 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 6 F 10), lll, ll l (.., ll l ) l. T l l w w l, w l w v l. T ll l v. E v l lly ll El w l l. I l x, l k ly w y x. C, l w v v., l ll l v, l ll w l l. O ll v l v l, ly v z l. F xl, ly l x. F 11: Exl w l y l, l y l. F v x, l ' x l ly B. B (.., w v z v), ly v l. z l l, ll l ly v w lk x l. I l l l x ' lk, l l. T l' B x ( w x l) y l w lk l' (.., l y l vy l j ). F xl, F 11 w l lk. Gv xl (w x ), y ly l x ly y lk, wl y l ' x lk (..,lk B). l y l ll l k w l lll. T vl l. O, l, y l z. T l l l xl F 12. l w l x lk l. l l F 12: v l z (w l v l ) l. l w k x v. O ll ly v v, w v, k ll w z ly v v (.., w l lvl w v w ). O lvl, l y l, w ll l x k. l l l k y vl. Hwv, ly l l. T

7 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 7 v l l w [1]. F 13 ly l l. l l F 13: v ly l. l v vly k x. lly, l,,, w l x ly lk. R- T R- v z ll y l j ( ly w-l l) y - l l. E ll -l l l. l j,. T j y ll l l. I, j l. T R- l -. ll l lvl. E y l 2-l (R; O) R ll l l l O (w O l l ). E y y (-l) 2-l (R; P ), w R l l l l l y P. R- (; M), w x, w M=2 M. T l 2 l l l. T l l R- l ly w l B-. l. T l R1 R4 R5 R3 R2 R6 R3 R4 R1 R2 R5 R6 F 14: Exl (1; 3) R-. -w y v R- w l wll v l l y l. O l, k l wll l vw. I l vw, l M l w l. Sl y R-. l R- wll l. I lll v [14], ll l, l lll R- wll lkly l. C 0 0 #: : F 15: Il R-. D-Plll R- T lll R- l l llw ( l, [15]). Illy, l, l R- l F 15. O xl (1; 3) R-. I, ll l, C 0 R-, w l w R- (.., l 0 ). W ll l w l R-. W l, l w w w l. ww (.., w v [( 0 1,1 );( 1 2,1 );;,1 ]) l

8 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 8 l w l R-. T w F 15 l l. T l l R-. I l x M, R- l w l ( lly w l R-). T w w l R-, w / w w l. T w l vw j l R- (.., R*- [2] R- v l l vl R- [13]). F lll R-, l l ll l l x. lll w (.., w v [ 0 ; ( 0 1 ); ;( 0 1,1 )]), x x y ll l l. ww wll lly x ll llw l. F ll ll l (.., w w l v l =M l ), x vl ll, w l l vl l x-x. l ly y-x y-x l vl. O w l vl, l x vl l. I v, l w l x l y ly. T l l lxy O(l ) l w ly w O(l ). C 0 0 F 16: U-. O l x vl, - [3] (w w x y w j v w ) l w w, w wll w R- l F 16. F xl, ll l w v l l vl lly w l, wl w l vl lly w l. T l - l F 16 w F 17. R- w w l (.., (; ; ; ) (; ; ; ; )), l v l. C 0 0 F 17: Rl -. T, l l, w wll w w w l, w R- C 0 C 1 ( R-), w F 18. C 1 0 C F 18: Cl l. T l wll w vly, w l, R-. I l x M, ( R- ) wll v (.., l). v y l l R- l lv (w l ly R-). T l wll w ll R- v M l ( l R- l ), w F 19 xl. T R- l C 2, l C 0, ll l l l l 3 (ll w l w (1; 3) R- xl).

9 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 9 C 2 0 C C F 19: Cl -lll R- l. T -lll R- l lxy O(l 2 ), w O(l ) O(l ) ( l w w x ). R- M I Gv -lll R-, w R- l w wll lll. Blly, R- l, w l. y l l l. F xl, F 20, l ( B), l ( ) B. I l, l l. I F 20, l l ( ) B. D B C R- K R- M l l l B C D K M l F 20: Exl -lll R- l F 1. l w (, B, C, D) (K,, M, ) R-. Il R- x ly. F, w l w. I, wll l l y l (.., l F 20). T wll ll l v. O l kw l, lll, l w. I, x l ' l. T F 20, w w l ( K) l ( ) w. T w l ly l ll l. D B R- K R- M C l l l l B C D K K M K M l F 21: S R- ll l v. F 21 ll l v l l, l l l. T l l ll x (.., B' l K ). O ll / v, l wll ll l ll l l. E l w y l. Ulk lll PMR, -j l y R- l l. I-

10 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{ D B R- K R- M C ll v lll w -- vll PMR, R- l l v. T w ll l (.., F 21, B, C, D ). T l l l C Ix l [12] w l y v, w w v. T v, w x lyl l l y [11]. I lv l l, -l l w y., l l (.., F 22, l K). T x l vly l y (.., F 22, B l ). C, ll llw l l vly l y. O ll l v, l l v ( F 22,, B, D). U l ly PMR l, v ll l (.., ll l R- ) l R-. T l y l w v l R-. I F 22, l (.., l,, ) w w w. O l v l R-, y ll l v. E v l lly w R- l. I l l, l k. C, ll l w v R- (.., F 22, l,, ) ' l l l l B C D K K M K M l F 22: S R- w l l (v l ). v., l ll l v, l lll w l. O ll v l v l, l, ll v l w l. D B R- K R- M C l l B C D K l l K M l F 23: S R- w l l. l v llw F 10. lly, l k x w w v l (.., ). C w, l. Fllw l l,

11 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{ k l. E llw ( y l l ) w y ' l y. I, l, llw k l (v l l vll). I F 23, l w l l l. D B R- K R- M C l l B C D l l K M l F 24: S R- w - l l l. ly D w. O ll v l l w ll l, ll l v k y l, l. F xl, l w F 24, w ly D l l. R- Sl R Qy T l l y -lll R- vy l ly w vly. T w ll y R- l l y l. T vlv w w. I l, ll /. I l x y x ll ll, y y l l vlk. ' F 25: Exl w l R- l y x ( l ') y. T l B R- l. T l R- l l l ll. R w w l, wl ll w l. I l x, l wll lly x l l k. O w ll R- l, wll l l y l. P C I v w lll l w l j l, l w l C* Tk M CM-5 (32 l). T w l l B C TIGER/ l [7]. O x P G Cy Myl xl. I l j, l l w ; w l wk y. I, l j w y w w w l l. T x v w 0 ( y) 50. lly, l l PMR v w 8 32, wl R I F 26, ( ) PMR v l j, w vl x l, w x x l l. Ov, w w y, l l ly ll x. x B

12 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{ CPU x k y F 26: Ex PMR l j l xl (P G Cy, Myl TIGER/ l). w 50, l l x l v. T w wk x l l. F, w l, w l l (ll F 11), l l l. S, l l x x, w w l y. T l l llw (w wll l w l j l), wll k l l l. 32 w vl x l, w x x y (l l l PMR ). T lly R- w ll (.., 10 15) x x l l (.., 45 50). T l ll l ; ll wll ll. W ly, y lll w l. ly, x x y x. Sl w w v w lll PMR, l / ( l x w F 25) ly x. W x PMR R- l j l, PMR v. T ly PMR yl l j w l lll w R-. T R-' -j, l -lll v. O k x w l j. I w l l v (.., v l l lwy), lll R- wl ly, lvl l ly y lll PMR. T l l j l. 400 CPU Cl Rk x y F 27: Ex R- l j l xl. I F 27, ( ) R- v l j, D lll l l j PMR R- l v. T l v vl y PMR. T v PMR yl j wl R-. T l w w w vl w v l j. T l w -. I l l v y -w. T j wk.

13 P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{ R [1] W. G. H. S, Uly l j: y l. P F Il Sy Sl D Hl, Cl, SC, 1992, 178{189. [2]. Bk, H. P. Kl, R. S, B. S, T R*-: l, P SIGMOD C, l Cy, J, J 1990, 322{331. [3] T. Bl, Plll l, P.D., CS-TR-2897, C R, C S D, Uvy Myl, Cll Pk, MD, [4] G. E. Blll J. J. l, Plll l l l, P Il C Plll P, S. Cl, I, 1988, 218{ 222. [5] G. E. Blll, S v lll, IEEE T C, C-38, 1989, 1526{1538. [6] T. Bk, H. P. Kl, B. S, E l j R-, P 1993 CM SIGMOD Il C M D, W, DC, My, 1993, 237{246. [7] B C, TIGER/ C Fl, 1990 Tl D, W, [14] W. D. Hll G.. Sl, D lll l, C CM, 29, 12 (D 1986), 1170{1183. [15] E. G. Hl H. S, D-lll R- l, P Il C Plll P, S. Cl, I, 1993, 49{53. [16] F. T., I Plll l, M K, S M, C, [17] R. C. l H. S, l v, C G 20, 4 ( 1986), 197{206 (l P SIGGRPH'86 C, Dll, 1986). [18] R. C. l H. S, l ly l, P SIGMOD C, S F, My 1987, 270{277. [19]. Rl. C. Kk, Dl P P, P, Y, [20] H. S, T D ly Sl D S, -Wly, R, M, [21] H. S, l Sl D S: C G, I P, GIS, -Wly, R, M, [22] C. D. Tl, G I Sy C Mll, P Hll, Elw Cl, J, [8] D. C, T B-, CM C Svy 11, 2 (J 1979), 121{137. [9] R. El S. B. v, Fl D Sy, Bj/C, Rw Cy, C, [10] C. Fl, T. Sll,. Rl, ly j l, P SIGMOD C, S F, My 1987, 426{439. [11] H.. Gw, U El l l", Il Jl Cl I S, 5, 345{355. [12] M. R. Gy D. S. J, C Ily: G Ty P- Cl, F, Y, [13]. G, -: y x l, P SIGMOD C, B, J 1984, 47{57.

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