5 research outputs found
Promised streaming algorithms and finding pseudo-repetitions
As the size of data available for processing increases, new models of
computation are needed. This motivates the study of data streams, which are sequences of
information for which each element can be read only after the previous one. In
this work we study two particular types of streaming variants: promised graph streaming algorithms and
combinatorial queries on large words. We give an &omega(n) lower
bound for working memory, where n is the number of vertices of the graph, for a variety
of problems for which the graphs are promised to be forests. The crux
of the proofs is based on reductions from the field of communication complexity.
Finally, we give an upper bound for two problems related to finding
pseudo-repetitions on words via anti-/morphisms, for which we also propose streaming versions
Streaming Algorithms for Connectivity Augmentation
We study the -connectivity augmentation problem (-CAP) in the
single-pass streaming model. Given a -edge connected graph
that is stored in memory, and a stream of weighted edges with weights in
, the goal is to choose a minimum weight subset such that is -edge connected. We give a
-approximation algorithm for this problem which requires to store
words. Moreover, we show our result is tight: Any
algorithm with better than -approximation for the problem requires
bits of space even when . This establishes a gap between the
optimal approximation factor one can obtain in the streaming vs the offline
setting for -CAP.
We further consider a natural generalization to the fully streaming model
where both and arrive in the stream in an arbitrary order. We show that
this problem has a space lower bound that matches the best possible size of a
spanner of the same approximation ratio. Following this, we give improved
results for spanners on weighted graphs: We show a streaming algorithm that
finds a -approximate weighted spanner of size at most
for integer , whereas the best prior
streaming algorithm for spanner on weighted graphs had size depending on . Using our spanner result, we provide an optimal -approximation for
-CAP in the fully streaming model with words of space.
Finally we apply our results to network design problems such as Steiner tree
augmentation problem (STAP), -edge connected spanning subgraph (-ECSS),
and the general Survivable Network Design problem (SNDP). In particular, we
show a single-pass -approximation for SNDP using
words of space, where is the maximum connectivity requirement
k-connectivity in the semi-streaming model
We present the first semi-streaming algorithms to determine k-connectivity of an undirected graph with k being any constant. The semi-streaming model for graph algorithms was introduced by Muthukrishnan in 2003 and turns out to be useful when dealing with massive graphs streamed in from an external storage device. Our two semi-streaming algorithms each compute a sparse subgraph of an input graph G and can use this subgraph in a postprocessing step to decide k-connectivity of G. To this end the first algorithm reads the input stream only once and uses time O(k 2 n) to process each input edge. The second algorithm reads the input k + 1 times and needs time O(k + α(n)) per input edge. Using its constructed subgraph the second algorithm can also generate all l-separators of the input graph for all l < k