1. Distance Approximating Trees in Graphs Brandstaedt & Chepoi & Dragan, ESA’97, J. of Algorithms ’99, European J. of Combinatorics 2000
A graph G=(V,E)
Vertex set
Edges
Adjacent
Incident
Path
Connected
A tree T=(V,E’)
connected and minimum number of edges

2. Distance Approximating Trees in Graphs
Shortest path
Distance in graphs
The problem : approximate by a simpler distance (e.g., by )
applications in
Communication networks
Data analysis
Motion planning
Network design
Phylogeny reconstruction
Numerical taxonomy

3. Distance Approximating Trees in Graphs
Case t-spanners
Multiplicative Tree t-Spanners
for any
Additive Tree r-Spanners

4. Distance Approximating Trees in Graphs Tree Spanners
Problem: Given G and integer t, decide whether G has a multiplicative tree t-Spanner.
For all graphs (Cai & Corneil)
NP-complete for t>3
Linear for t=1,2
Open for t=3
For special graph classes
Multiplicative tree 3-spanners in linear time for interval and permutation graphs (Madanlal & Venkatesan & Rangan)
Additive tree 2-spanners in linear time for interval and distance hereditary graphs (Prisner)
Additive tree 4-spanner for cocomparability graphs (Prisner)
For chordal graphs
For every fixed integer t there is a chordal graph without tree t-spanners (additive as well as multiplicative) (McKee)
Question: Whether strongly chordal graphs have tree t-spanners with small t (Prisner, STACS’97)

5. Distance Approximating Trees in Graphs Our result
Every strongly chordal graph (even every dually chordal graph) has a multiplicative 4-spanner which is also an additive 3-spanner Such a tree can be constructed in linear time.
A graph G is chordal if it does not contain any chordless cycle of length at least four.
A chordal graph is strongly chordal if it does not contain any induced sun.

6. Distance Approximating Trees in Graphs
A strongly chordal graph and an additive tree 3-spanner of it, produced by our algorithm. This graph has no additive tree 2-spanner.

7. Distance Approximating Trees in Graphs
Case (it is allowed to use new edges) distance -approximating trees
A tree T=(V,E’) is a distance –approximating tree of a graph G=(V,E) if
for any

8. Distance Approximating Trees in Graphs Distance -approximating trees
Problem: Given G and integer , decide whether G has a distance -approximating tree.
Our results ( is the length of a longest chordless simple cycle in G)
Applications
Given a chordal graph G. After linear time preprocessing, for any two vertices of G, the distance with an error at most 2 can be computed in only O(1) time.
Efficient approximate solutions of several NP-complete problems related to distances.