Temple University CIS Dept CIS 616 Principles of

Temple University – CIS Dept. CIS 616– Principles of Data Management V. Megalooikonomou Spatial Access Methods (SAMs) (based on notes by Silberchatz, Korth, and Sudarshan and notes by C. Faloutsos at CMU)

General Overview n Multimedia Indexing n Spatial Access Methods (SAMs) n n n k-d trees Point Quadtrees MX-Quadtree z-ordering R-trees

SAMs - Detailed outline n spatial access methods n n n problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

Spatial Access Methods - problem n n Given a collection of geometric objects (points, lines, polygons, . . . ) organize them on disk, to answer spatial queries (like? ? )

Spatial Access Methods - problem n n Given a collection of geometric objects (points, lines, polygons, . . . ) organize them on disk, to answer n n point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

Spatial Access Methods - problem n n Given a collection of geometric objects (points, lines, polygons, . . . ) organize them on disk, to answer n n point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

Spatial Access Methods - problem n n Given a collection of geometric objects (points, lines, polygons, . . . ) organize them on disk, to answer n n point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

Spatial Access Methods - problem n n Given a collection of geometric objects (points, lines, polygons, . . . ) organize them on disk, to answer n n point queries range queries k-nn queries spatial joins (‘all pairs’ queries)

Spatial Access Methods - problem n n Given a collection of geometric objects (points, lines, polygons, . . . ) organize them on disk, to answer n n point queries range queries k-nn queries spatial joins (‘all pairs’ within ε)

SAMs - motivation n Q: applications?

SAMs - motivation traditional DB age salary GIS

SAMs - motivation traditional DB age salary GIS

SAMs - motivation CAD/CAM find elements too close to each other

SAMs - motivation CAD/CAM

SAMs - motivation eg, . std S 1 F(S 1) 1 365 day Sn F(Sn) eg, avg 1 365 day

SAMs: solutions n n n K-d trees point quadtrees MX-quadtrees z-ordering R-trees (grid files) Q: how would you organize, e. g. , n-dim points, on disk? (C points per disk page)

SAMs - Detailed outline n spatial access methods n n n problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

k-d trees n n n Used to store k dimensional point data It is not used to store region data A 2 -d tree (i. e. , for k=2) stores 2 -dimensional point data while a 3 -d tree stores 3 dimensional point data, etc.

2 -d trees – node structure n n n Binary trees Info: information field Xval, Yval: coordinates of a point associated with the node Llink, Rlink: pointers to children Properties (N: node): n If level N even -> n n n for all nodes M in the subtree rooted at N. Llink: M. Xval < N. Xval for all nodes P in the subtree rooted at N. Rlink: P. Xval >= N. Xval If level N odd -> n Similarly use Yvals

2 -d trees – Example

2 -d trees: Insertion/Search n To insert a node N into the tree pointed by T n n n If N and T agree on Xval, Yval then overwrite T Else, branch left if N. Xval < T. xval, right otherwise (even levels) Similarly for odd levels (branching on Yvals)

2 -d trees – Example of Insertion City (Xval, Yval) Banja Luka (19, 45) Derventa (40, 50) Toslic (38, 38) Tuzla (54, 35) Sinj (4, 4) Splitting of region by Banja Luka Splitting of region by Toslic Splitting of region by Derventa Splitting of region by Sinj

2 -d trees: Deletion n Deletion of point (x, y) from T n n If N is a leaf node easy Otherwise either Tl (left subtree) or Tr (right subtree) is non-empty n n Find a “candidate replacement” node R in Tl or Tr Replace all of N’s non-link fields by those of R Recursively delete R from Ti Recursion guaranteed to terminate - Why?

2 -d trees: Deletion n Finding candidate replacement nodes for deletion n Replacement node R must bear same spatial relation to all nodes in Tl and Tr as node N

2 -d trees: Range Queries n n Q: Given a point (xc, yc) and a distance r find all points in the 2 -d tree that lie within the circle A: Each node N in a 2 -d tree implicitly represents a region RN – If the circle (specified by the query) has no intersection with RN then there is no point in searching the subtree rooted at node N

SAMs - Detailed outline n spatial access methods n n n problem dfn k-d trees point quadtrees z-ordering R-trees

Point Quadtrees n n n Represent point data Always split regions into 4 parts 2 -d tree: a node N splits a region into two by drawing one line through the point (N. xval, N. yval) Point quadtree: a node N splits a region by drawing a horizontal and a vertical line through the point (N. xval, N. yval) Four parts: NW, SW, NE, and SE quadrants Q: Quadtree nodes have 4 children?

Point Quadtrees n Nodes in point quadtrees represent regions

Point quadtrees - Insertion City (Xval, Yval) Banja Luka (19, 45) Derventa (40, 50) Toslic (38, 38) Tuzla (54, 35) Sinj (4, 4) Splitting of region by Toslic Splitting of region by Banja Luka Splitting of region by Tuzla Splitting of region by Derventa Splitting of region by Sinj

Point Quadtrees - Insertion

Point quadtrees: Deletion n Deletion of point (x, y) from T n n If N is a leaf node easy Otherwise a subtree (N. NW, N. SW, N. NE. N. SE) is nonempty n Find a “candidate replacement” node R in one of the subtrees such that: n n n n Every other node R 1 in N. NW is to the NW of R Every other node R 2 in N. SW is to the SW of R etc… Replace all of N’s non-link fields by those of R Recursively delete R from Ti In general, it may not always be possible to find such as replacement node Q: What happens in the worst case?

Point quadtrees: Deletion n Deletion of point (x, y) from T n n If N is a leaf node easy Otherwise a subtree (N. NW, N. SW, N. NE. N. SE) is nonempty n Find a “candidate replacement” node R in one of the subtrees such that: n n n n Every other node R 1 in N. NW is to the NW of R Every other node R 2 in N. SW is to the SW of R etc… Replace all of N’s non-link fields by those of R Recursively delete R from Ti In general, it may not always be possible to find such as replacement node Q: What happens in the worst case? May require all nodes to be reinserted

Point quadtrees: Range Searches n n Each node in a point quadtree represents a region Do not search regions that do not intersect the circle defined by the query

SAMs - Detailed outline n spatial access methods n n n problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

MX-Quadtrees n Drawbacks of 2 -d trees, point quadtrees: n n n shape of tree depends upon the order in which objects are inserted into the tree splits may be uneven depending upon where the point (N. xval, N. yval) is located inside the region (represented by N) MX-quadtrees: shape (and height) of tree independent of number of nodes and order of insertion

MX-Quadtrees n n Assumption: the map is represented as a grid of size (2 k x 2 k) for some k When a region gets “split” it splits down the middle

MX-Quadtrees - Insertion After insertion of A, B, C, and D respectively

MX-Quadtrees - Insertion After insertion of A, B, C, and D respectively

MX-Quadtrees - Deletion n Fairly easy – why? All point are represented at the leaf level Total time for deletion: O(k)

MX-Quadtrees –Range Queries n n Same as in point quadtrees One difference: n Checking to see if a point is in the circle defined by the range query needs to be performed at the leaf level (points are stored at the leaf level)

SAMs - Detailed outline n spatial access methods n n n problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

z-ordering Q: how would you organize, e. g. , n-dim points, on disk? (C points per disk page) Hint: reduce the problem to 1 -d points(!!) Q 1: why? A: Q 2: how?

z-ordering Q: how would you organize, e. g. , n-dim points, on disk? (C points per disk page) Hint: reduce the problem to 1 -d points (!!) Q 1: why? A: B-trees! Q 2: how?

z-ordering Q 2: how? A: assume finite granularity; z-ordering = bit-shuffling = N-trees = Morton keys = geo-coding =. . .

z-ordering Q 2: how? A: assume finite granularity (e. g. , 232 x 232 ; 4 x 4 here) Q 2. 1: how to map n-d cells to 1 -d cells?

z-ordering Q 2. 1: how to map n-d cells to 1 -d cells?

z-ordering Q 2. 1: how to map n-d cells to 1 -d cells? A: row-wise Q: is it good?

z-ordering Q: is it good? A: great for ‘x’ axis; bad for ‘y’ axis

z-ordering Q: How about the ‘snake’ curve?

z-ordering Q: How about the ‘snake’ curve? A: still problems: 2^32

z-ordering Q: Why are those curves ‘bad’? A: no distance preservation (~ clustering) Q: solution? 2^32

z-ordering Q: solution? (w/ good clustering, and easy to compute, for 2 -d and n-d? )

z-ordering Q: solution? (w/ good clustering, and easy to compute, for 2 -d and n-d? ) A: z-ordering/bit-shuffling/linearquadtrees ‘looks’ better: • few long jumps; • scoops out the whole quadrant before leaving it • a. k. a. space filling curves

z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x, y) )? A: 3 (equivalent) answers!

z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x, y))? A 1: ‘z’ (or ‘N’) shapes, RECURSIVELY order-1 order-2 . . . order (n+1)

z-ordering Notice: n self similar (we’ll see about fractals, soon) n method is hard to use: z =? f(x, y) order-1 order-2 . . . order (n+1)

z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x, y) )? A: 3 (equivalent) answers! Method #2?

z-ordering bit-shuffling x 00 y 11 10 01 00 y 11 z =( 0 1 )2 = 5 00 01 10 11 x

z-ordering bit-shuffling x 00 y 11 10 01 00 y 11 z =( 0 1 )2 = 5 How about the reverse: 00 01 10 11 x (x, y) = g(z) ?

z-ordering bit-shuffling x 00 y 11 10 01 00 y 11 z =( 0 1 )2 = 5 How about n-d spaces? 00 01 10 11 x

z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x, y) )? A: 3 (equivalent) answers! Method #3?

z-ordering linear-quadtrees : assign N->1, S->0 e. t. c. W E 1 0 0 1 N 01. . . 11. . . S 00. . . 10. . .

z-ordering. . . and repeat recursively. Eg. : zgray-cell = WN; WN = (0101)2 = 5 W E 00 1 0 0 1 N 01. . . 11. . . S 00. . . 10. . . 11

z-ordering Drill: z-value of grey cell, with the three methods? W E 1 N 0 S 0 1

z-ordering Drill: z-value of grey cell, with the three methods? W E 1 N 0 S 0 1 method#1: 14 method#2: shuffle(11; 10)= (1110)2 = 14

z-ordering Drill: z-value of grey cell, with the three methods? W E 1 N 0 S 0 1 method#1: 14 method#2: shuffle(11; 10)= (1110)2 = 14 method#3: EN; ES =. . . = 14

z-ordering - Detailed outline n spatial access methods n z-ordering n n n main idea - 3 methods use w/ B-trees; algorithms (range, knn queries. . . ) non-point (eg. , region) data analysis; variations R-trees

z-ordering - usage & algo’s Q 1: How to store on disk? A: Q 2: How to answer range queries etc

z-ordering - usage & algo’s Q 1: How to store on disk? A: treat z-value as primary key; feed to B-tree PGH SF

z-ordering - usage & algo’s MAJOR ADVANTAGES w/ B-tree: n already inside commercial systems (no coding /debugging!) n concurrency & recovery is ready PGH SF

z-ordering - Detailed outline n spatial access methods n z-ordering n n n main idea - 3 methods use w/ B-trees; algorithms (range, knn queries. . . ) non-point (eg. , region) data analysis; variations R-trees

z-ordering - variations Q: is z-ordering the best we can do?

z-ordering - variations Q: is z-ordering the best we can do? A: probably not - occasional long ‘jumps’ Q: then?

z-ordering - variations Q: is z-ordering the best we can do? A: probably not - occasional long ‘jumps’ Q: then? A 1: Gray codes

z-ordering - variations A 2: Hilbert curve! (a. k. a. Hilbert-Peano curve)

z-ordering - variations ‘Looks’ better (never long jumps). How to derive it?

z-ordering - variations ‘Looks’ better (never long jumps). How to derive it? order-1 order-2 . . . order (n+1)

z-ordering - variations Q: function for the Hilbert curve ( h = f(x, y) )? A: bit-shuffling, followed by post-processing, to account for rotations. Linear on # bits. See textbook, for pointers to code/algorithms (eg. , [Jagadish, 90])

z-ordering - variations Q: how about Hilbert curve in 3 -d? n-d? A: Exists (and is not unique!). Eg. , 3 -d, order-1 Hilbert curves (Hamiltonian paths on cube) #1 #2

z-ordering - Detailed outline n spatial access methods n z-ordering n n n main idea - 3 methods use w/ B-trees; algorithms (range, knn queries. . . ) non-point (eg. , region) data analysis; variations R-trees. . .

z-ordering - analysis Q: How many pieces (‘quad-tree blocks’) per region? A: proportional to perimeter (surface etc)

z-ordering - analysis (How long is the coastline, say, of England? Paradox: The answer changes with the yardstick -> fractals. . . )

z-ordering - analysis Q: Should we decompose a region to full detail (and store in B-tree)?

z-ordering - analysis Q: Should we decompose a region to full detail (and store in B-tree)? A: NO! approximation with 1 -3 pieces/zvalues is best [Orenstein 90]

z-ordering - analysis Q: how to measure the ‘goodness’ of a curve?

z-ordering - analysis Q: how to measure the ‘goodness’ of a curve? A: e. g. , avg. # of runs, for range queries 4 runs 3 runs (#runs ~ #disk accesses on B-tree)

z-ordering - analysis Q: So, is Hilbert really better? A: 27% fewer runs, for 2 -d (similar for 3 -d) Q: are there formulas for #runs, #of quadtree blocks etc? A: Yes ([Jagadish; Moon+ etc] see textbook)

z-ordering - fun observations Hilbert and z-ordering curves: “space filling curves”: eventually, they visit every point in n-d space - therefore: order-1 order-2 . . . order (n+1)

z-ordering - fun observations. . . they show that the plane has as many points as a line (-> headaches for 1900’s mathematics/topology). (fractals, again!) order-1 order-2 . . . order (n+1)

z-ordering - fun observations Observation #2: Hilbert (like) curve for video encoding [Y. Matias+, CRYPTO ‘ 87]: Given a frame, visit its pixels in randomized hilbert order; compress; and transmit

z-ordering - fun observations In general, Hilbert curve is great for preserving distances, clustering, vector quantization etc

Conclusions n n n z-ordering is a great idea (n-d points -> 1 -d points; feed to B-trees) used by TIGER system and (most probably) by other GIS products works great with low-dim points

SAMs – Detailed Outline n spatial access methods n n n problem dfn k-d trees point quadtrees MX-quadtrees z-ordering R-trees

SAMs - more detailed outline n R-trees n n n main idea; file structure (algorithms: insertion/split) (deletion) (search: range, nn, spatial joins) variations (packed; hilbert; . . . )

R-trees n n n z-ordering: cuts regions to pieces -> dup. elim. how could we avoid that? Idea: Minimum Bounding Rectangles

R-trees n [Guttman 84] Main idea: allow parents to overlap! n n n => guaranteed 50% utilization => easier insertion/split algorithms. (only deal with Minimum Bounding Rectangles - MBRs)

R-trees n eg. , w/ fanout 4: group nearby rectangles to parent MBRs; each group -> disk page I AC F B D E G H J

R-trees n eg. , w/ fanout 4: P 1 P 3 AC F B P 2 D E I G H P 4 J A B C D E H I F G J

R-trees n eg. , w/ fanout 4: P 1 P 3 AC F B P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E H I F G J

R-trees - format of nodes n {(MBR; obj-ptr)} for leaf nodes P 1 P 2 P 3 P 4 x-low; x-high obj y-low; y-high ptr. . . A B C

R-trees - format of nodes n {(MBR; node-ptr)} for non-leaf nodes x-low; x-high node y-low; y-high ptr. . . P 1 P 2 P 3 P 4 A B C

R-trees - range search? P 1 P 3 AC F B P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E H I F G J

R-trees - range search? P 1 P 3 AC F B P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E H I F G J

R-trees - range search Observations: n every parent node completely covers its ‘children’ n a child MBR may be covered by more than one parent - it is stored under ONLY ONE of them. (i. e. , no need for dup. elim. ) n a point query may follow multiple branches. n everything works for any dimensionality

SAMs - more detailed outline n R-trees n n n main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert; . . . )

R-trees - insertion n eg. , rectangle ‘X’ P 1 P 3 AC F B X P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E H I F G J

R-trees - insertion n eg. , rectangle ‘X’ P 1 P 3 AC F B X P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E X H I F G J

R-trees - insertion n eg. , rectangle ‘Y’ P 1 P 3 AC F B Y P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E H I F G J

R-trees - insertion n P 1 eg. , rectangle ‘Y’: extend suitable parent. P 3 AC F B Y P 2 D E I G P 1 P 2 P 3 P 4 H P 4 J A B C D E Y H I F G J

R-trees - insertion n n eg. , rectangle ‘Y’: extend suitable parent. Q: how to measure ‘suitability’?

R-trees - insertion n n eg. , rectangle ‘Y’: extend suitable parent. Q: how to measure ‘suitability’? A: by increase in area (volume) (more details: later, under ‘performance analysis’) Q: what if there is no room? how to split?

R-trees - insertion n P 1 eg. , rectangle ‘W’ P 3 K AC W B P 2 D E F I G P 1 P 2 P 3 P 4 H P 4 J A B C K H I D E F G J

R-trees - insertion n P 1 eg. , rectangle ‘W’ - focus on ‘P 1’ - how to split? K AC B W

R-trees - insertion n P 1 eg. , rectangle ‘W’ - focus on ‘P 1’ - how to split? • (A 1: plane sweep, K AC B W until 50% of rectangles) • A 2: ‘linear’ split • A 3: quadratic split • A 4: exponential split

R-trees - insertion & split n n pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed 2 R seed 1

R-trees - insertion & split n n n pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ Q: how to measure ‘closeness’?

R-trees - insertion & split n n pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ Q: how to measure ‘closeness’? A: by increase of area (volume)

R-trees - insertion & split n n pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed 2 R seed 1

R-trees - insertion & split n n pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed 2 R seed 1

R-trees - insertion & split n n n pick two rectangles as ‘seeds’; assign each rectangle ‘R’ to the ‘closest’ ‘seed’ smart idea: pre-sort rectangles according to delta of closeness (ie. , schedule easiest choices first!)

R-trees - insertion - pseudocode n n decide which parent to put new rectangle into (‘closest’ parent) if overflow, split to two, using (say, ) the quadratic split algorithm n n propagate the split upwards, if necessary update the MBRs of the affected parents.

R-trees - insertion - observations n many more split algorithms exist (next!)

SAMs - more detailed outline n R-trees n n n main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert; . . . )

R-trees - deletion n n delete rectangle if underflow n ? ?

R-trees - deletion n n delete rectangle if underflow n n n temporarily delete all siblings (!); delete the parent node and re-insert them

SAMs - more detailed outline n R-trees n n n main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert; . . . )

R-trees - range search pseudocode: check the root for each branch, if its MBR intersects the query rectangle apply range-search (or print out, if this is a leaf)

R-trees - nn search P 1 P 3 AC F B q P 2 D E I G H P 4 J

R-trees - nn search n Q: How? (find near neighbor; refine. . . ) P 1 P 3 AC F B q P 2 D E I G H P 4 J

R-trees - nn search n A 1: depth-first search; then, range query P 1 P 3 I AC F B q P 2 D E G H P 4 J

R-trees - nn search n A 1: depth-first search; then, range query P 1 P 3 I AC F B q P 2 D E G H P 4 J

R-trees - nn search n A 1: depth-first search; then, range query P 1 P 3 AC F B q P 2 D E I G H P 4 J

R-trees - nn search n A 2: [Roussopoulos+, sigmod 95]: n n priority queue, with promising MBRs, and their best and worst-case distance main idea:

R-trees - nn search consider only P 2 and P 4, for illustration P 1 P 3 AC F B q P 2 D E I G H P 4 J

R-trees - nn search best of P 4 => P 4 is useless for 1 -nn worst of P 2 H q P 2 D E P 4 J

R-trees - nn search n what is really the worst of, say, P 2? worst of P 2 q P 2 D E

R-trees - nn search n n what is really the worst of, say, P 2? A: the smallest of the two red segments! q P 2

R-trees - nn search n variations: [Hjaltason & Samet] incremental nn: n n build a priority queue scan enough of the tree, to make sure you have the k nn to find the (k+1)-th, check the queue, and scan some more of the tree ‘optimal’ (but, may need too much memory)

SAMs - more detailed outline n R-trees n n n main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins performance analysis variations (packed; hilbert; . . . )

R-trees - spatial joins Spatial joins: find (quickly) all counties intersecting lakes

R-trees - spatial joins Assume that they are both organized in Rtrees:

R-trees - spatial joins for each parent P 1 of tree T 1 for each parent P 2 of tree T 2 if their MBRs intersect, process them recursively (ie. , check their children)

R-trees - spatial joins Improvements - variations: - [Seeger+, sigmod 92]: do some pre-filtering; do plane-sweeping to avoid N 1 * N 2 tests for intersection - [Lo & Ravishankar, sigmod 94]: ‘seeded’ R-trees (FYI, many more papers on spatial joins, without R-trees: [Koudas+ Sevcik], e. t. c. )

SAMs - more detailed outline n R-trees n n n main idea; file structure algorithms: insertion/split deletion search: range, nn, spatial joins variations (packed; hilbert; . . . )

R-trees - variations Guttman’s R-trees sparked much follow-up work n can we do better splits? n what about static datasets (no ins/del/upd)? n what about other bounding shapes?

R-trees - variations Guttman’s R-trees sparked much follow-up work n can we do better splits? n i. e, defer splits?

R-trees - variations A: R*-trees [Kriegel+, SIGMOD 90] n defer splits, by forced-reinsert, i. e. : instead of splitting, temporarily delete some entries, shrink overflowing MBR, and re-insert those entries n Which ones to re-insert? n How many?

R-trees - variations A: R*-trees [Kriegel+, SIGMOD 90] n defer splits, by forced-reinsert, i. e. : instead of splitting, temporarily delete some entries, shrink overflowing MBR, and re-insert those entries n Which ones to re-insert? n How many? A: 30%

R-trees - variations Q: Other ways to defer splits?

R-trees - variations Q: Other ways to defer splits? A: Push a few keys to the closest sibling node (closest = ? ? )

R-trees - variations R*-trees: Also try to minimize area AND perimeter, in their split. Performance: higher space utilization; faster than plain R-trees. One of the most successful R-tree variants.

R-trees - variations Guttman’s R-trees sparked much follow-up work n can we do better splits? n what about static datasets (no ins/del/upd)? n n Hilbert R-trees what about other bounding shapes?

R-trees - variations n n what about static datasets (no ins/del/upd)? Q: Best way to pack points?

R-trees - variations n n n what about static datasets (no ins/del/upd)? Q: Best way to pack points? A 1: plane-sweep great for queries on ‘x’; terrible for ‘y’

R-trees - variations n n n what about static datasets (no ins/del/upd)? Q: Best way to pack points? A 1: plane-sweep great for queries on ‘x’; bad for ‘y’

R-trees - variations n n what about static datasets (no ins/del/upd)? Q: Best way to pack points? A 1: plane-sweep great for queries on ‘x’; terrible for ‘y’ Q: how to improve?

R-trees - variations n A: plane-sweep on HILBERT curve!

R-trees - variations n n n A: plane-sweep on HILBERT curve! In fact, it can be made dynamic (how? ), as well as to handle regions (how? ) A: [Kamel+, VLDB 94]

R-trees - variations Guttman’s R-trees sparked much follow-up work n can we do better splits? n what about static datasets (no ins/del/upd)? n what about other bounding shapes?

R-trees - variations n n n what about other bounding shapes? (and why? ) A 1: arbitrary-orientation lines (cell-tree, [Guenther] A 2: P-trees (polygon trees) (MB polygon: 0, 90, 45, 135 degree lines)

R-trees - variations n n n A 3: L-shapes; holes (h. B-tree) A 4: TV-trees [Lin+, VLDB-Journal 1994] A 5: SR-trees [Katayama+, SIGMOD 97] (used in Informedia)

R-trees - conclusions n n n Popular method; like multi-d B-trees guaranteed utilization good search times (for low-dim. at least) R*-, Hilbert- and SR-trees: still used IBM (Informix) ships Data. Blade with R-trees

References n n n Guttman, A. (June 1984). R-Trees: A Dynamic Index Structure for Spatial Searching. Proc. ACM SIGMOD, Boston, Mass. Jagadish, H. V. (May 23 -25, 1990). Linear Clustering of Objects with Multiple Attributes. ACM SIGMOD Conf. , Atlantic City, NJ. Lin, K. -I. , H. V. Jagadish, et al. (Oct. 1994). “The TVtree - An Index Structure for High-dimensional Data. ” VLDB Journal 3: 517 -542.

References, cont’d n n n Pagel, B. , H. Six, et al. (May 1993). Towards an Analysis of Range Query Performance. Proc. of ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), Washington, D. C. Robinson, J. T. (1981). The k-D-B-Tree: A Search Structure for Large Multidimensional Dynamic Indexes. Proc. ACM SIGMOD. Roussopoulos, N. , S. Kelley, et al. (May 1995). Nearest Neighbor Queries. Proc. of ACM-SIGMOD, San Jose, CA.