Fast, precise and dynamic distance queries Yair BartalHebrew U. Lee-Ad GottliebWeizmann → Hebrew U. Liam RodittyBar Ilan Tsvi KopelowitzBar Ilan → Weizmann Moshe LewensteinBar Ilan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:  AAA

Fast, precise and dynamic distance queries 2 Distance oracles A distance oracle for a point set S with distance function d() preprocesses S so that given any two points x,y in S, d(x,y) (or an approximation thereof) can be retrieved quickly. Interesting cases  Expensive to store all ~ n 2 point pairs Sublinear space  Expensive to query distance function d() for example, when d() is graph-induced

Fast, precise and dynamic distance queries 3 Distance oracles Introduced by [TZ-05]  Setting: weighted graph  Approximation ratio: 2k-1 (k>1)  Query time: O(k)  Space: n 1+1/k Other possible parameters:  Setting Planar, Euclidean, graph, metric  Approximation to d(x,y) O(k), O(logn), 1+   Query time O(k), O(logn), O(1)  Space O(n), n 1+1/k  Dynamic updates addition of removal or points or graph edges

Fast, precise and dynamic distance queries 4 Efficient classification for metric data 4 Preliminaries: Doubling dimension Definition: Ball B(x,r) = all points within distance r from x. The doubling constant (of a metric M) is the minimum value such that every ball can be covered by balls of half the radius  First used by [Assoud ‘83], algorithmically by [Clarkson ‘97].  The doubling dimension is ddim(M)=log 2 (M)  Euclidean: ddim(R d ) = O(d) Packing property of doubling spaces  A set with diameter diam and minimum inter-point distance a, contains at most (diam/a) O(ddim) points Here ≥7.

Fast, precise and dynamic distance queries 5 Survey of oracle results ReferenceSettingDistortionQuery timespace TZ-05weighted graph2k-1 k>1O(k)n 1+1/k MN-06MetricO(k)O(1)n 1+1/k Kle-02, Tho-04 Planar graph 1+  O(  -1 )O(n log n/  ) HM-06Doubling metric 1+  O(ddim)  -O(ddim) n BGKRL-11Doubling metric, dynamic 1+  O(1)  -O(ddim) n + 2 O(ddim log ddim) n Caveat: word RAM model, and assuming a word is sufficient to store any single interpoint distance. Related model: Distance labeling [Tal-04, Sli-05]

Fast, precise and dynamic distance queries 6 Overview of techniques Some tools we’ll need (both static and dynamic versions): Point hierarchies for doubling spaces  By now a standard construction… Metric embeddings  Into trees  Into Euclidean space Tree search structures  Level ancestor queries in O(1) time  Least common ancestor (LCA) queries in O(1) time

Fast, precise and dynamic distance queries 7 Preliminaries: Spanners Oracle central idea: Motivated by an observation originally made in the context of low-stretch spanners.  [GGN-04, GR-08a, GR-08b] A spanner of G is a subgraph H  H contains all vertices of G  H contains a subset of the edges of G Interesting properties of H:  Stretch, degree, hop diameter G H

Fast, precise and dynamic distance queries 8 Point hierarchies To explain the observation motivating the oracle, we need to introduce point hierarchies  Hierarchies are the starting point for problems in doubling spaces  NNS, spanners, routing, embeddings… A point hierarchy is composed of levels of r-nets An r-net for a point set S is a set of balls of radius r centered at points of S  Packing: The centers are separated from each other by some minimum distance r  Covering: The balls Cover all the points of S.

Fast, precise and dynamic distance queries 9 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 10 Radius = 1 Covering: all points are covered Packing Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 11 Covering: all 1-net points are covered Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 12 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 13 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 14 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 15 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 16 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 17 Point hierarchies 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 18 Another perspective DAG 1-net 2-net 4-net 8-net Number of levels: log(aspect ratio)

Fast, precise and dynamic distance queries 19 Another perspective Make arbitrary parent-child assignments 1-net 2-net 4-net 8-net Number of levels: log(aspect ratio) DAG → Spanning tree

Fast, precise and dynamic distance queries 20 Another perspective Spanning tree 1-net 2-net 4-net 8-net Number of levels: log(aspect ratio)

Fast, precise and dynamic distance queries 21 Towards an oracle Oracle stores all tree parent-child tree links  O(n) space Define c-neighbors: r-net point pairs within distance c = 3r/   Store all distances between c-neighbors, and between their children   -O(ddim) n space Note that the c-neighbor property is hereditary  If nodes a,b are c-neighbors in tree level r  Then the ancestor a’,b’ of a,b in any tree level r+i are c-neighbors as well (or are the same node)  Proof: d(a’,b’) ≤ d(a’,a) + d(a,b) + d(b,b’) ≤ 2(r+i) + cr + 2(r+i) < c(r+i)

Fast, precise and dynamic distance queries 22 c-neighbors 1-net 2-net 4-net 8-net

Fast, precise and dynamic distance queries 23 Spanner observation Let x,y denote two points in S, and by extension their corresponding tree leaf nodes. Let x’,y’ be the highest tree ancestors of x,y that are not c- neighbors.  Note that d(x’,y’) is stored by the oracle, since the parents of x’,y’ are c- neighbors. Spanner Theorem:  d(x,y) = (1±  ) d(x’,y’)  Proof by illustration…

Fast, precise and dynamic distance queries 24 Spanner observation 1-net 2-net 4-net 8-net x y x’ y’

Fast, precise and dynamic distance queries 25 Spanner observation ≤ 6 > 12/  Distortion: (12/  +12)/(12/  ) ≤ 1+  1-net 2-net 4-net 8-net x y x’ y’

Fast, precise and dynamic distance queries 26 Oracle query  For x,y in S, find d(x,y) Oracle does this instead:  For x,y in S, find x’,y’ (the highest ancestors that are not c-neighbors)  Return stored d(x’,y’) Left with the following question:  Ancestral non-neighbors query: Find the highest tree ancestors that are not c-neighbors  We could view this as an abstract problem on trees and ignore the metric…

Fast, precise and dynamic distance queries 27 Ancestral non-neighbors query Some ideas (static case): Recall that neighborliness is hereditary  Brute force → try all ancestors: O(log aspect ratio)  Binary search → using level ancestor queries: O(log log aspect ratio)  Balanced tree + brute force: O(log n)  Balanced tree + binary search: O(log log n) But we can do better:  Make use of the tree structure  Get some help from the metric structure

Fast, precise and dynamic distance queries 28 Ancestral neighbors query Lemma: d(x,y) is closely related to the tree level r of ancestors x’,y’  r = log d(x,y) – log c ± O(1) Corollary  A b-approximation to d(x,y) pinpoints the level of x’,y’ to log b + O(1) possible tree levels

Fast, precise and dynamic distance queries 29 Oracle query Oracle Step 1: Run the oracle of MS-09 (similar in flavor to TZ- 05, MN-06) on x,y with parameter k = O(log n)  Approximation ratio: O(k) = O(log n)  Query time: O(1)  Space: n (1+1/k) = O(n) By the Corollary, an approximation ratio of O(log n) to d(x,y) limits the tree level of x’,y’ to O(log log n) possible levels.

Fast, precise and dynamic distance queries 30 Oracle query O(loglog n) levels

Fast, precise and dynamic distance queries 31 Oracle query Snowflake embedding of [Ass-04] and [GKL-03]  Given a set S in metric space  Embed S into O(ddim log ddim) Euclidean space  Distortion O(ddim) into the snowflake d ½ Oracle Step 2:  Recall that the level of x’,y’ has been narrowed down to O(loglogn) candidate levels.  Embed neighborhoods of O(loglogn) levels into Euclidean space

Fast, precise and dynamic distance queries 32 Oracle query What’s going on?  We’ve narrowed down the level of x’,y’ to O(loglogn) levels  These neighborhoods are small  Build a snowflake for each neighborhood O(ddim) = O(log 1/3 n) dimensions O(log ddim + loglog n) bits per dimension  So the Euclidean representation of each point fits into o(log ½ n) bits (into a word) Lemma: The embedded (snowflake) distance between two points can be returned in O(1) time  Proof outline: The distance between two vectors w,z is w·w - 2w·z + z·z.  A dot product can be computed in O(1) time by manipulating the multiplication operator

Fast, precise and dynamic distance queries 33 Oracle query Dot product via multiplication, proof by example: w = (1,2,3,4) z = (5,6,7,8) w’ = z’ = w’z’= =

Fast, precise and dynamic distance queries 34 Oracle query Result of Step 2:  O(ddim) approximation to the snowflake distance x,y (or rather, their ancestors in the appropriate neighborhood)  By the corollary, restricts the candidate levels of x’,y’ to O(log ddim) levels Oracle Step 3:  Preprocessing: In neighborhoods of O(log dim) levels, store a pointer from each pair to highest ancestors which are not c-neighbors  Space 2 O(ddim log ddim) per neighborhood or point  O(1) query time

Fast, precise and dynamic distance queries 35 Dynamic oracle Steps that needed to be made dynamic:  HierarchyAlready done [CG-06]  MS-09 oracleProblem! Answer: Tree embedding[Bar96]  Level ancestor queryProblem! Answer: Jump trees  Snowflake embedding Problem! Extension of above techniques… Conclusion:  There exists a dynamic 1+  approximate distortion oracle for doubling spaces with O(1) query time, which uses  -O(ddim) n +2 O(ddim log ddim) n space and can be updated in time 2 -O(ddim) log n + 2 O(ddim log ddim)