Approximate Distance Oracles for Geometric Spanner Networks

1
Approximate Distance Oracles for Geometric
Spanner Networks

• Joachim Gudmundsson TUE, Netherlands
• Christos Levcopoulos Lund U., Sweden
• Giri Narasimhan Florida Intl U., Miami, USA
• Michiel Smid Carleton U., Ottawa, Canada

2
Problem

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

3
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.

4
Main Results

1. 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.

5
Main Results

• 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.

6
Previous Work

• General Weighted Graphs
• Cohen Zwick 97, Zwick98, 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)

7
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
• H1 ? H2 ? ? Hi ? ? Hp
• Each Hi 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 Hi.

O(mnlogn)
O(mnlogn)
Search
O(1)
8
(No Transcript)
9
Applications
10
Approximate Stretch Factors

• PATH NETWORKS
• O(nlogn)
• CYCLE NETWORKS
• O(nlogn)
• TREE NETWORK
• O(nlog2n) O(nlogn)
• PLANAR NETWORKS
• O(n3/2logn) O(nlogn)
• ARBITRARY NETWORKS
• O(mn1/?log2) 2? - approx O(m nlogn)
  (1e)-approx

11
Approximate Closest Pairs

• 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 AB.

12
SP in Polygonal Domain with Polygonal Obstacles

• Require that domain be t -rounded.
• Preprocessing O(nlogn)
• Space O(nlogn)
• Query on vertices O(1)
• Query on arbitrary points O(nlogn)

13
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.

14
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?

15
Thanks!
16
What are Cluster Graphs?

• Cluster graph Hi closely approximates distances
• in N for vertices (p?q) at distance at least
  ?D?i-1.
• Hi 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))

17
Constructing Cluster Graphs
18
(No Transcript)
19
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
• H1 ? H2 ? ? Hi ? ? Hp
• Each Hi 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 Hi.

O(mnlogn)
O(mnlogn)
Search
O(1)