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We present a deterministic O(n log log n) time algorithm for finding shortest cycles and minimum cuts in planar graphs. The algorithm improves the previously known fastest algorithm by Italiano et al. in STOC'11 by a factor of log n. This speedup is obtained through the use of dense distance graphs combined with a divide-and-conquer approach.

Let G = (V, E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t ∈ V \ {s}. We show how to solve this problem in near-linear O(n log 3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n−1 times, giving a total time bound of… (More)

- Jakub Lacki
- SODA
- 2011

This paper presents a new deterministic algorithm for decremental maintenance of the transitive closure in a directed graph. The algorithm processes any sequence of edge deletions in <i>O</i>(<i>mn</i>) time and answers queries in constant time. Until now such time bound has only been achieved by a randomized Las Vegas algorithm.
In addition to that, a few… (More)

It was experimentally observed that the majority of real-world networks are scale-free and follow power law degree distribution. The aim of this paper is to study the algorithmic complexity of such " typical " networks. The contribution of this work is twofold. First, we define a deterministic condition for checking whether a graph has a power law degree… (More)

This article presents a new deterministic algorithm for decremental maintenance of the transitive closure in a directed graph. The algorithm processes any sequence of edge deletions in <i>O</i>(<i>mn</i>) time and answers queries in constant time. Previously, such time bound has only been achieved by a randomized Las Vegas algorithm. In addition to that, a… (More)

We consider two core algorithmic problems for probabilistic verification: the maximal end-component decomposition and the almost-sure reachability set computation for Markov decision processes (MDPs). For MDPs with treewidth k, we present two improved static algorithms for both the problems that run in time O(n · k 2.38 · 2 k) and O(m · log n · k),… (More)

In this paper we consider the problem of maintaining information about graphs with history -- so called <i>graph timeline</i>. A graph timeline is a sequence of graphs G<sub>1</sub>,..., G<sub>t</sub>, in which consecutive graphs are obtained from previous ones by small modifications, e.g., by adding or removing a single edge. We aim to devise algorithms… (More)

In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an n-vertex graph G=(V,E,w) with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set S ⊆ V, which changes over time. The changes applied… (More)

In this paper we study the problem of answering connectivity queries about a graph timeline. A graph timeline is a sequence of undirected graphs G 1 ,. .. , G t on a common set of vertices of size n such that each graph is obtained from the previous one by an addition or a deletion of a single edge. We present data structures, which preprocess the timeline… (More)