2 research outputs found
Biconnectivity, -numbering and other applications of DFS using bits
We consider space efficient implementations of some classical applications of
DFS including the problem of testing biconnectivity and -edge connectivity,
finding cut vertices and cut edges, computing chain decomposition and
-numbering of a given undirected graph on vertices and edges.
Classical algorithms for them typically use DFS and some
bits\footnote{We use to denote logarithm to the base .} of information
at each vertex. Building on a recent -bits implementation of DFS due to
Elmasry et al. (STACS 2015) we provide -bit implementations for all these
applications of DFS. Our algorithms take time for some
small constant (where ). Central to our implementation is a
succinct representation of the DFS tree and a space efficient partitioning of
the DFS tree into connected subtrees, which maybe of independent interest for
designing other space efficient graph algorithms.Comment: 18 pages, 4 figures, Preliminary version of this article appeared in
the proceedings of 27th ISAAC 2016, Journal version is accepted to JCSS and
will soon appea
Frameworks for Designing In-place Graph Algorithms
Read-only memory model is a classical model of computation to study
time-space tradeoffs of algorithms. One of the classical results on the ROM
model is that any sorting algorithm that uses O(s) words of extra space
requires comparisons for and the
bound has also been recently matched by an algorithm. However, if we relax the
model (from ROM), we do have sorting algorithms (say Heapsort) that can sort
using comparisons using bits of extra space, even
keeping a permutation of the given input sequence at any point of time during
the algorithm.
We address similar questions for graph algorithms. We show that a simple
natural relaxation of ROM model allows us to implement fundamental graph search
methods like BFS and DFS more space efficiently than in ROM. By simply allowing
elements in the adjacency list of a vertex to be permuted, we show that, on an
undirected or directed connected graph having vertices and edges,
the vertices of can be output in a DFS or BFS order using bits
of extra space and time. Thus we obtain similar bounds for
reachability and shortest path distance (both for undirected and directed
graphs). With a little more (but still polynomial) time, we can also output
vertices in the lex-DFS order. As reachability in directed graphs and shortest
path distance are NL-complete, and lex-DFS is P-complete, our results show that
our model is more powerful than ROM if L P. En route, we also introduce
and develop algorithms for another relaxation of ROM where the adjacency lists
of the vertices are circular lists and we can modify only the heads of the
lists. All our algorithms are simple but quite subtle, and we believe that
these models are practical enough to spur interest for other graph problems in
these models