98 research outputs found
Join-Reachability Problems in Directed Graphs
For a given collection G of directed graphs we define the join-reachability
graph of G, denoted by J(G), as the directed graph that, for any pair of
vertices a and b, contains a path from a to b if and only if such a path exists
in all graphs of G. Our goal is to compute an efficient representation of J(G).
In particular, we consider two versions of this problem. In the explicit
version we wish to construct the smallest join-reachability graph for G. In the
implicit version we wish to build an efficient data structure (in terms of
space and query time) such that we can report fast the set of vertices that
reach a query vertex in all graphs of G. This problem is related to the
well-studied reachability problem and is motivated by emerging applications of
graph-structured databases and graph algorithms. We consider the construction
of join-reachability structures for two graphs and develop techniques that can
be applied to both the explicit and the implicit problem. First we present
optimal and near-optimal structures for paths and trees. Then, based on these
results, we provide efficient structures for planar graphs and general directed
graphs
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
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Dynamic Data Structures for Series Parallel Digraphs
We consider the problem of dynamically maintaining general series parallel directed acyclic graphs (GSP dags), two-terminal series parallel directed acyclic graphs (TTSP dags) and looped series parallel directed graphs (looped SP digraphs). We present data structures for updating (by both inserting and deleting either a group of edges or vertices) GSP dags, TTSP clags and looped SP digraphs of m edges and n vertices in O( log n) worst-case time. The time required to check whether there is a path between two given vertices is O(log n), while a path of length k can be traced out in O(k + log n) time. For GSP and TTSP dags, our data structures are able to report a regular expression describing all the paths between two vertices x and y in O(h + log n), where h †n is the total number of vertices which are contained in paths from x to y. Although GSP dags can have as many as O(n2) edges, we use an implicit representation which requires only O(n) space. Motivations for studying dynamic graphs arise in several areas, such as communication networks, Incremental compilation environments and the design of very high level languages, while the dynamic maintenance of series parallel graphs is also relevant in reducible flow diagrams
Planar Reachability in Linear Space and Constant Time
We show how to represent a planar digraph in linear space so that distance
queries can be answered in constant time. The data structure can be constructed
in linear time. This representation of reachability is thus optimal in both
time and space, and has optimal construction time. The previous best solution
used space for constant query time [Thorup FOCS'01].Comment: 20 pages, 5 figures, submitted to FoC
Hardness Results for Dynamic Problems by Extensions of Fredman and Saksâ Chronogram Method
We introduce new models for dynamic computation based on the cell probe model of Fredman and Yao. We give these models access to nondeterministic queries or the right answer +-1 as an oracle. We prove that for the dynamic partial sum problem, these new powers do not help, the problem retains its lower bound of Omega(log n/log log n). From these results we easily derive a large number of lower bounds of order Omega(log n/log log n) for conventional dynamic models like the random access machine. We prove lower bounds for dynamic algorithms for reachability in directed graphs, planarity testing, planar point location, incremental parsing, fundamental data structure problems like maintaining the majority of the prefixes of a string of bits and range queries. We characterise the complexity of maintaining the value of any symmetric function on the prefixes of a bit string
Max -Flow Oracles and Negative Cycle Detection in Planar Digraphs
We study the maximum -flow oracle problem on planar directed graphs
where the goal is to design a data structure answering max -flow value (or
equivalently, min -cut value) queries for arbitrary source-target pairs
. For the case of polynomially bounded integer edge capacities, we
describe an exact max -flow oracle with truly subquadratic space and
preprocessing, and sublinear query time. Moreover, if
-approximate answers are acceptable, we obtain a static oracle
with near-linear preprocessing and query time and a
dynamic oracle supporting edge capacity updates and queries in
worst-case time.
To the best of our knowledge, for directed planar graphs, no (approximate)
max -flow oracles have been described even in the unweighted case, and
only trivial tradeoffs involving either no preprocessing or precomputing all
the possible answers have been known.
One key technical tool we develop on the way is a sublinear (in the number of
edges) algorithm for finding a negative cycle in so-called dense distance
graphs. By plugging it in earlier frameworks, we obtain improved bounds for
other fundamental problems on planar digraphs. In particular, we show: (1) a
deterministic time algorithm for negatively-weighted SSSP in
planar digraphs with integer edge weights at least . This improves upon the
previously known bounds in the important case of weights polynomial in , and
(2) an improved bound on finding a perfect matching in a
bipartite planar graph.Comment: Extended abstract to appear in SODA 202
On Fully Dynamic Strongly Connected Components
We consider maintaining strongly connected components (SCCs) of a directed graph subject to edge insertions and deletions. For this problem, we show a randomized algebraic data structure with conditionally tight O(n^1.529) worst-case update time. The only previously described subquadratic update bound for this problem [Karczmarz, Mukherjee, and Sankowski, STOC\u2722] holds exclusively in the amortized sense.
For the less general dynamic strong connectivity problem, where one is only interested in maintaining whether the graph is strongly connected, we give an efficient deterministic black-box reduction to (arbitrary-pair) dynamic reachability. Consequently, for dynamic strong connectivity we match the best-known O(n^1.407) worst-case upper bound for dynamic reachability [van den Brand, Nanongkai, and Saranurak FOCS\u2719]. This is also conditionally optimal and improves upon the previous O(n^1.529) bound. Our reduction also yields the first fully dynamic algorithms for maintaining the minimum strong connectivity augmentation of a digraph
Min-Cost Flow in Unit-Capacity Planar Graphs
In this paper we give an O~((nm)^(2/3) log C) time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by C. For planar multigraphs, this improves upon the best known algorithms for general graphs: the O~(m^(10/7) log C) time algorithm of Cohen et al. [SODA 2017], the O(m^(3/2) log(nC)) time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the O~(sqrt(n) m log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Omega(m^(3/2)) time barrier for min-cost flow problem in planar graphs.
To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in O~(m^(3/2) log C) time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: r-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs
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