approximate distance oracles for geometric spanner networks

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 Narasimhan Dagstuhl '04 2

Problem

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


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.


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.


Main Results

3. Let V be a set of points in d with interpoint distances in the range [D, Dk]. 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.


Previous Work

General Weighted GraphsCohen & Zwick ’97, Zwick’98, Dor et al. ’00, Thorup & Zwick ‘01: Preprocess , Space , ApproxKlein ’02 (Planar Networks); Query O(k)

Baswana & Sen ’04 (Unweighted Graphs)

Geometric Graphs & DomainsClarkson ‘87, Arikati et al. ’96, Chen ‘95, Chiang & Mitchell ’99, Chen et al. ’00

Preprocess , Space , Approx 3,Query O(log n)


Basic IdeaPreprocessin

g• 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

H1 H2 … Hi … Hp Each Hi has only edges of length in the range (Di-1 tDi] and degree bounded by a constant.

• For query (p,q), find i such that |pq| (Di-1 Di].• Report distance between p and q in Hi.

Search

O(m+nlogn)

O(m+nlogn)

O(1)


Applications


PATH NETWORKSO(nlogn)

CYCLE NETWORKSO(nlogn)

TREE NETWORKO(nlog2n) O(nlogn)

PLANAR NETWORKSO(n3/2logn) O(nlogn)

ARBITRARY NETWORKSO(mn1/log2) [2 - approx]

O(m + nlogn) [(1+)-approx]

Approximate Stretch Factors


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


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


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.


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!


What are Cluster Graphs?

• Cluster graph Hi closely approximates distances in N for vertices (pq) at distance at least Di-1.• Hi has degree bounded by a constant. (Size = O(n))• Shortest path queries for vertices (pq) such that |pq| (Di-1 Di] 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))


Constructing Cluster Graphs


Basic IdeaPreprocessin

g• 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

H1 H2 … Hi … Hp Each Hi has only edges of length in the range (Di-1 tDi] and degree bounded by a constant.

• For query (p,q), find i such that |pq| (Di-1 Di].• Report distance between p and q in Hi.

Search

O(m+nlogn)

O(m+nlogn)

O(1)

approximate distance oracles for geometric spanner networks

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