7 research outputs found
Data Structures for Node Connectivity Queries
Let denote the maximum number of internally disjoint paths in
an undirected graph . We consider designing a data structure that includes a
list of cuts, and answers the following query: given , determine
whether , and if so, return a pointer to an -cut of
size (or to a minimum -cut) in the list. A trivial data structure
that includes a list of cuts and requires space can
answer each query in time. We obtain the following results. In the case
when is -connected, we show that cuts suffice, and that these cuts
can be partitioned into laminar families. Thus using space we
can answers each min-cut query in time, slightly improving and
substantially simplifying a recent result of Pettie and Yin. We then extend
this data structure to subset -connectivity. In the general case we show
that cuts suffice to return an -cut of size ,and a list
of size contains a minimum -cut for every . Combining
our subset -connectivity data structure with the data structure of Hsu and
Lu for checking -connectivity, we give an space data structure
that returns an -cut of size in time, while
space enables to return a minimum -cut
\~Optimal Fault-Tolerant Reachability Labeling in Planar Graphs
We show how to assign labels of size to the vertices of a
directed planar graph , such that from the labels of any three vertices
we can deduce in time whether is reachable from
in the graph . Previously it was only known how to achieve
queries using a centralized size oracle [SODA'21]
The Structure of Minimum Vertex Cuts
In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts.
As a consequence of these investigations, we exhibit a simple O(? n)-space data structure that can quickly answer pairwise (?+1)-connectivity queries in a ?-connected graph. We also show how to compute the "closest" ?-cut to every vertex in near linear O?(m+poly(?)n) time
Connectivity Labeling for Multiple Vertex Failures
We present an efficient labeling scheme for answering connectivity queries in
graphs subject to a specified number of vertex failures. Our first result is a
randomized construction of a labeling function that assigns vertices
-bit labels, such that given the labels of
where , we can correctly report, with probability
, whether and are connected in . However, it
is possible that over all distinct queries, some are answered
incorrectly. Our second result is a deterministic labeling function that
produces -bit labels such that all connectivity queries are
answered correctly. Both upper bounds are polynomially off from an
-bit lower bound.
Our labeling schemes are based on a new low degree decomposition that
improves the Duan-Pettie decomposition, and facilitates its distributed
representation. We make heavy use of randomization to construct hitting sets,
fault-tolerant graph sparsifiers, and in constructing linear sketches. Our
derandomized labeling scheme combines a variety of techniques: the method of
conditional expectations, hit-miss hash families, and -nets for
axis-aligned rectangles.
The prior labeling scheme of Parter and Petruschka shows that and
vertex faults can be handled with - and -bit labels,
respectively, and for vertex faults, -bit
labels suffice
An Optimal Ancestry Labeling Scheme with Applications to XML Trees and Universal Posets
International audienceIn this paper we solve the ancestry-labeling scheme problem which aims at assigning the shortest possible labels (bit strings) to nodes of rooted trees, so that ancestry queries between any two nodes can be answered by inspecting their assigned labels only. This problem was introduced more than twenty years ago by Kannan et al. [STOC '88], and is among the most well-studied problems in the field of informative labeling schemes. We construct an ancestry-labeling scheme for n-node trees with label size log 2 n + O(log log n) bits, thus matching the log 2 n + Ω(log log n) bits lower bound given by Alstrup et al. [SODA '03]. Our scheme is based on a simplified ancestry scheme that operates extremely well on a restricted set of trees. In particular, for the set of n-node trees with depth at most d, the simplified ancestry scheme enjoys label size of log 2 n + 2 log 2 d + O(1) bits. Since the depth of most XML trees is at most some small constant, such an ancestry scheme may be of practical use. In addition, we also obtain an adjacency-labeling scheme that labels n-node trees of depth d with labels of size log 2 n + 3 log 2 d + O(1) bits. All our schemes assign the labels in linear time, and guarantee that any query can be answered in constant time. Finally, our ancestry scheme finds applications to the construction of small universal partially ordered sets (posets). Specifically, for any fixed integer k, it enables the construction of a universal poset of size˜Osize˜ size˜O(n k) for the family of n-element posets with tree-dimension at most k. Up to lower order terms, this bound is tight thanks to a lower bound of n k−o(1) due to Alon and Scheinerman [Order '88]
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum