13 research outputs found
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
Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs
We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with O?(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in O?(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.
Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in O?(n?m) worst-case time and queries in O(?m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008]
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
Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form
Algebraic techniques have had an important impact on graph algorithms so far.
Porting them, e.g., the matrix inverse, into the dynamic regime improved
best-known bounds for various dynamic graph problems. In this paper, we develop
new algorithms for another cornerstone algebraic primitive, the Frobenius
normal form (FNF). We apply our developments to dynamic and fault-tolerant
exact distance oracle problems on directed graphs.
For generic matrices over a finite field accompanied by an FNF, we show
(1) an efficient data structure for querying submatrices of the first
powers of , and (2) a near-optimal algorithm updating the FNF explicitly
under rank-1 updates.
By representing an unweighted digraph using a generic matrix over a
sufficiently large field (obtained by random sampling) and leveraging the
developed FNF toolbox, we obtain: (a) a conditionally optimal distance
sensitivity oracle (DSO) in the case of single-edge or single-vertex failures,
providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a
multiple-failures DSO improving upon the state of the art (vd. Brand and
Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved
dynamic distance oracles in the case of single-edge updates, and (d) a dynamic
distance oracle supporting vertex updates, i.e., changing all edges incident to
a single vertex, in worst-case time and distance queries in
time.Comment: To appear at FOCS 202
Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs
We consider the problem of computing shortest paths in weighted unit-disk
graphs in constant dimension . Although the single-source and all-pairs
variants of this problem are well-studied in the plane case, no non-trivial
exact distance oracles for unit-disk graphs have been known to date, even for
.
The classical result of Sedgewick and Vitter [Algorithmica '86] shows that
for weighted unit-disk graphs in the plane the search has average-case
performance superior to that of a standard shortest path algorithm, e.g.,
Dijkstra's algorithm. Specifically, if the corresponding points of a
weighted unit-disk graph are picked from a unit square uniformly at random,
and the connectivity radius is , finds a shortest path in
in expected time when , even though has
edges in expectation. In other words, the work done by the
algorithm is in expectation proportional to the number of vertices and not the
number of edges.
In this paper, we break this natural barrier and show even stronger sublinear
time results. We propose a new heuristic approach to computing point-to-point
exact shortest paths in unit-disk graphs. We analyze the average-case behavior
of our heuristic using the same random graph model as used by Sedgewick and
Vitter and prove it superior to . Specifically, we show that, if we are
able to report the set of all points of from an arbitrary rectangular
region of the plane in time, then a shortest path between
arbitrary two points of such a random graph on the plane can be found in
expected time. In particular, the state-of-the-art range
reporting data structures imply a sublinear expected bound for all
and expected bound for
after only near-linear preprocessing of the point set.Comment: Full version of a SoCG'21 paper. Abstract truncated to meet arxiv
requirement
Systemy agentowe w zarządzaniu wiedzą
The functioning of modern organizations under the influence of a competitive environment
requires them to systematically improve their competence. There is no doubt that knowledge is one
of the most important resources and appropriate Knowledge Management allows organizations to
quickly and effectively respond to changes in the area of their operations. The dynamic development of Information Technology has caused that it is difficult to imagine a modern Knowledge
Management system in an organization without the support of efficient IT solutions. One of the future concepts in the design and implementation of such systems are agent systems. The purpose
of this article is to present the applicability of agent systems in Knowledge Management in
organizations, with their characteristics, classification, and a sample implementation architecture
Services in the E-Banking as an Element Facilitate the Development of Information Society
In this article author presents the some interesting products in e-banks. Author tries to evaluate
the impact of this products taking into consideration development of information society