Estimating All Pairs Shortest Paths in Restricted Graph Families:

1
Estimating All Pairs Shortest Paths in Restricted
Graph Families A Unified Approach
Feodor F. Dragan Department of Computer Science
Kent State University Ohio, USA
2
The APSP Problem

• APSP (a classical fundamental problem)
• Given a graph,
• find shortest paths between all pairs of
  vertices in the graph.
• There has been a renewed interest in it recently
  for general graphs as well as for special graph
  classes.
• We consider unweighted, undirected graphs.
• naïve approach O(nm) ( for dense
  graphs)
• via matrix multiplications O(M(n) log n)
  Seidel92

Coppersmith/Winograd 87

• not practical, large hidden constants
• best combinatorial
  Basch/Khanna/Motwani 95
• A better ( ) combinatorial algorithm ?
  similar time bound for Boolean matrix
  multiplication.

3
Two Ways To Go

• consider the APASP problem
• stretch t all pairs paths
• Awerbuch/Berger/Cowen/Peleg93,
  Cohen93
• (via t-spanners, for )
• distances with an additive one-sided error
• Aingworth/Chekuri/Indyk/Motwani 96
  2
• Dor/Halperin/Zwick 96
  2
• Computing all distances with an additive
  one-sided error of at most 1 is as hard as
  Boolean matrix multiplication.
• consider special graph classes
• design simple and efficient (optimal
  time) algorithms for special graph classes which
  are interesting from practical point of view)

error
4
Special Graph Classes

• Optimal algorithms are known for
• interval graphs Attalah/Chen/Lee93,
• 
  Mirchandani96,
• 
  Ravi/Marathe/Rangan96,
• 
  Sridhar/Joshi/Chandrasekharan93
• circular-arc graphs Attalah/Chen/Lee93,
• 
  Sridhar/Joshi,Chandrasekharan93
• permutation graphs Dahlhaus92
• strongly chordal graphs Balachandhran,
  Rangan96,
• 
  Han/Chandrasekharan/Sridhar97,
• 
  Dahlhaus92
• chordal bipartite graphs Ho/Chang99
• distance hereditary graphs Dahlhaus92
• dually chordal graphs Brandstaedt/Chepoi/Dragan
  98
• Parallel algorithms for some graph classes are
  also considered.

5
Distances in Polygons via Distances in Visibility
Graphs. (a part of motivation)

• visibility graphs of spiral polygons are
• interval graphs Everett/Corneil90
• Motwani/Ragunathan/Saran89

link-distance
N
N
Dent orientations
W
E
W
E
S
S
W
E
S
A class 3 polygon ( no N dent )
6
Distances in Chordal Graphs

• Han/Chandrasekharan/Sridhar97
• APSP can be solved in for G if
  is given
• computing for chordal graph is as hard as
  for general graphs
• Sridhar/Han/Chandrasekharan95
• after linear time (sophisticated) preprocessing
  step, for any
  a value such that
  can be
  computed in O(1) time
• ? all distances with one-sided error of at most 1
  in time
• From Brandstaedt/Chepoi/Dragan99 it also
  follows
• for any chordal graph G(V,E) there is a tree
  T(V,U) such that
• (tree T can be constructed in linear time)

7
Our Contribution

• a very simple and efficient approach for solving
  APASP problem on weakly chordal graphs and
  subclasses.
• the same approach works well also on graphs with
  small size of largest induced cycle
• it gives a unified way to solve the APSP and
  APASP problems on different graph classes
  (including chordal, AT-free, strongly chordal,
  chordal bipartite, and distance hereditary graphs)

2
Chordal Bipartite
3
Trees
House-Hole-free
1
Strongly Chordal
Chordal
Weakly Chordal
0
HHD-free
Interval
Distance-Hereditary
Weakly chordal hierarchy
8
The Method

• We assume that our graph is given with a vertex
  ordering

• Algorithm APASP

9
Results
bounds are tight
k-1
3
BFS
BFS
BFS
0, if is given
2
LBFS
BFS
LBFS
1
LBFS
BFS
lex
0
lex
lex
10
Proof Technique (LBFS and lex)
Let G be an arbitrary graph with a vertex
ordering. Lemma1. Assume there exist integers
such that
Then,

• Let G be a House-Hole-free (HH-free) graph with a
  LBFS-ordering.
• Lemma2.
• if is given then error is 0.
• we need to examine only for
  vertices x,y with d(x,y)2
• Let d(v,u)2.
• G is HH-free with LBFS ordering ?
  , i.e., s2
• G is HHD-free (or Chordal) with LBFS? s1
• G is distance-hereditary with LBFS? s0
• G is strongly chordal (or chordal bipartite) with
  lex-ordering ? s0

x
y
mn(x)
11
Concluding Remarks and Open Problems

• we presented a very simple and efficient (
  time) approach for solving APASP problem on
  weakly chordal graphs and subclasses.
• the same approach works well also on graphs with
  small size of largest induced cycle
• it gives a unified way to solve the APSP and
  APASP problems on different graph classes
  (including chordal, AT-free, strongly chordal,
  chordal bipartite, and distance hereditary
  graphs)
• with one shoot we obtained many known results on
  distances in particular graph classes
• for which other graph classes (with special
  vertex orderings) can this approach give good
  results (approximations)?
• can these ideas be used for designing good
  routing/ labeling schemes in those graph classes?