Solving connectivity problems via basic Linear Algebra
We consider some classical graph connectivity problems like
reachability, shortest path and disjoint paths in different settings.
For the first two problems, we show efficient parallel algorithms in the
dynamic framework, confirming a conjecture of Patnaik and Immerman ’94
for directed reachability. Very recently we have been able to extend
this for more general updates. For the shortest disjoint paths problem,
we show efficient (and parallel) algorithms in planar graphs where all
the terminals lie on a single face. This answers an open question of
Colin de Verdiere and Schrijver ’08. Interestingly the basic idea behind
all these algorithms is to compute some linear algebraic properties of
the matrix associated with the given graph, which I shall try to convey
in this talk.
This is based on joint works with Samir Datta, Raghav Kulkarni,
Siddharth Iyer, Thomas Schwentick, and Thomas Zeume.