Approximate Distance Oracles for Geometric Spanner Networks Joachim Gudmundsson TUE, Netherlands Christos Levcopoulos Lund U., Sweden Giri Narasimhan Florida Int’l U., Miami, USA Michiel Smid Carleton U., Ottawa, Canada

Giri NarasimhanDagstuhl '042 Problem Preprocess a geometric spanner network so that approximate shortest path lengths between two query vertices can be reported efficiently (using subquadratic space).

Giri NarasimhanDagstuhl '043 Main Results  1. Let N be a geometric t-spanner for a set S of n points in  d with m edges. N can be preprocessed so that (1+  )-approximate shortest path lengths between two query points from S can be reported efficiently. Preprocessing O(m + nlogn) Space O(m + nlogn) Query O(1) Floor function not used. Only indirection. No restrictions on interpoint distances. 

Giri NarasimhanDagstuhl '044 Main Results  2. Let N’ be a geometric t-spanner network of a set S of n points in  d. A (1+  )-spanner N of N’ can be computed in O(m + nlogn) time such that N has only O(n) edges. Floor function not used. Only indirection. No restrictions on interpoint distances.

Giri NarasimhanDagstuhl '045 Main Results 3. Let V be a set of points in  d with interpoint distances in the range [D, D  k ]. We can preprocess V in O(n logn) time and O(n) space such that for any two points p,q  V, we can compute in O(1) time, BIndex(p,q) =  log  (|pq|/D)  without the use of the floor function.

Giri NarasimhanDagstuhl '046 Previous Work General Weighted Graphs Cohen & Zwick ’97, Zwick’98, Dor et al. ’00, Thorup & Zwick ‘01: Preprocess, Space, Approx Klein ’02 (Planar Networks); Query O(k) Baswana & Sen ’04 (Unweighted Graphs) Geometric Graphs & Domains Clarkson ‘87, Arikati et al. ’96, Chen ‘95, Chiang & Mitchell ’99, Chen et al. ’00 Preprocess, Space, Approx 3, Query O(log n)

Giri NarasimhanDagstuhl '047 Basic Idea Preprocessing Given a t-spanner network N, construct a (1+  )-spanner N’ of N with O(n) edges Build a sequence of p = O(logn) cluster graphs H 1  H 2  …  H i  …  H p Each H i has only edges of length in the range (  D  i-1  tD  i ] and degree bounded by a constant. For query (p,q), find i such that |pq|  (D  i-1  D  i ]. Report distance between p and q in H i. Search O(m+nlogn) O(1)

Giri NarasimhanDagstuhl '048

Giri NarasimhanDagstuhl '049 Applications

Giri NarasimhanDagstuhl '0410 PATH NETWORKS O(nlogn) CYCLE NETWORKS O(nlogn) TREE NETWORK O(nlog 2 n) O(nlogn) PLANAR NETWORKS O(n 3/2 logn) O(nlogn) ARBITRARY NETWORKS O(mn 1/  log 2 ) [2  - approx] O(m + nlogn) [(1+  )-approx] Approximate Stretch Factors

Giri NarasimhanDagstuhl '0411 Preprocess point set S such that for any query sets Red, Blue  S, the approx closest pair in (Red,Blue) can be reported in time O(m log m), where m = |A|+|B|. Approximate Closest Pairs

Giri NarasimhanDagstuhl '0412 Require that domain be t -rounded. Preprocessing O(nlogn) Space O(nlogn) Query on vertices O(1) Query on arbitrary points O(nlogn) SP in Polygonal Domain with Polygonal Obstacles

Giri NarasimhanDagstuhl '0413 Open Problems Output the SP in O(k) time. Reduce the space complexity of O(nlogn). Generalize to arbitrary geometric networks HARD! SP queries in dynamic spanner graphs. Add edge(s) to best improve stretch factor of a graph. Remove edge(s) to get minimum increase of stretch factor.

Giri NarasimhanDagstuhl '0414 More Open Problems Find the center of a given geometric graph. Given a graph, how to compute a subgraph with minimum stretch factor, such that the subgraph is a Spanning tree, Path, Planar graph Replace input graph by a set of points. Other applications?

Thanks!

Giri NarasimhanDagstuhl '0416 What are Cluster Graphs? Cluster graph H i closely approximates distances in N for vertices (p  q) at distance at least  D  i-1. H i has degree bounded by a constant. (Size = O(n)) Shortest path queries for vertices (p  q) such that |pq|  (D  i-1  D  i ] can be reported in constant time. All O(log n) cluster graphs of N can be constructed efficiently in O(nlogn) time. (Time and space = O(nlogn))

Giri NarasimhanDagstuhl '0417 Constructing Cluster Graphs

Giri NarasimhanDagstuhl '0418

Giri NarasimhanDagstuhl '0419 Basic Idea Preprocessing Given a t-spanner network N, construct a (1+  )-spanner N’ of N with O(n) edges Build a sequence of p = O(logn) cluster graphs H 1  H 2  …  H i  …  H p Each H i has only edges of length in the range (  D  i-1  tD  i ] and degree bounded by a constant. For query (p,q), find i such that |pq|  (D  i-1  D  i ]. Report distance between p and q in H i. Search O(m+nlogn) O(1)