1. Ch. 16: Sweep-Zones
Basic Question: Is it possible to compute nearest neighbors in expected time O(n*log(n)) ???
Basic Idea: Generalize sweep-lines to sweep-zones !!!
Def.: The sweep-zone SZ of an area is the set of regions touching the upper boundary of an area from below.
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

2. UB-Tree Insertion 18/19 1 3 6 7 2 4 8 9 10 6 5 15 11 16 12 17 18 13 14
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

3. Sweep-Zone Algorithm 1: i
{ Z-regions have been read in increasing Z-order up to region Ri-1, i.e. area(R i-1) with upper boundary B(R i-1) }
{ set of cached regions C(R i) is the set of regions in SZi-1 = SZ(area(R i-1)) plus region Ri }
1. for every point p  Ri let l(p) and h(p) be the lower and higher neighbor of p on Z-curve, compute l(p) and h(p).
2. let q = l(p) if dist(p,l(p)) < dist (p, h(p)) = h(p) otherwise
3. Let Q(p) be the query box with center p and side length 2*dist(p,q) q p
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

4. 5. Cache regions intersecting Q(p) to enforce linear I/O time
4. Retrieve Q(p) from cache or disk and compute the nearest neighbor (p)
{ Note: retrieval of Q(p) should take time O(log n), finding (p) should be nearly constant }
5. Cache regions intersecting Q(p) to enforce linear I/O time
6. If Ri was the last region in Z-order then exit
7. Release all regions from C(Ri) which are not in SZi
8. i:= i+1; read next region R i in Z-order;
9. Goto step 1
{ all nearest neighbors are known, now cluster }
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

5. Sweep-Zone Algorithm 2:
Basic Idea: run algorithm forward to compute lower (w.r. to Z-order) nearest neighbor (p) of p and backward to compute upper (w.r. to Z-order) nearest neighbor (p) of p, then (p) = closest of {(p), (p)}
i.e. modify step 4 in Sweep-Zone algorithm 1 to compute Q(p)  area(Ri)
Advantages: all pages are read in increasing or decreasing Z-order only (sequential reads) and cache requirements are smaller
Disadvantage: data must be read twice, tradeoff???
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

6. Cache Contents for Algorithm 2: 1 10 9 8 7 6 5 2 1 11 10 6 5
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

7. 2. Determine regions that can be released, i.e. SZi - SZi-1
Cache Modification
1. Determine extension of next region to be read using upper part of UB-index
2. Determine regions that can be released, i.e. SZi - SZi-1
3. Release regions from cache
4. Read next region, i.e. transfer it from disk to cache
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

8. expected cache size ~ 1.5 * sqrt (18) = 6.4
Observations:
expected cache size ~ 1.5 * sqrt (18) = 6.4
maximal occurring cache size = 6
average cache size =
Cache Organization: keep cache organized as a set of regions sorted in Z-order, e.g. AVL-tree with elementary operations append single element and delete set of elements
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

9. which algorithm is faster which algorithm requires less resources
Open Questions:
which algorithm is faster
which algorithm requires less resources
what are the tradeoffs between I/O, cache size, CPU-time, total time, etc.
analytic comparison of both algorithms?
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000

10. this is a local optimization of Algorithm 2:
if Q(p)  area (Ri) then (p) = (p)
and we can ignore the computation of (p) in the backward phase
Algorithm 4
if (p) = (p) then discard p entirely from the backward phase, i.e. reduce the amount of data and computations for the second phase, but then we have to write out the non-discarded points
Open Question: under what conditions is Algorithm 4 better than Algorithm 3?
July 20, 2000
R. Bayer, Ch. 16, DWH-SS2000