Hollow Heaps

We introduce the hollow heap, a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete-min take $O(1)$ time, worst case as well as amortized; delete and delete-min take $O(\log n)$ amortized time on a heap of $n$ items. Hollow heaps are by far the simplest structure to achieve this. Hollow heaps combine two novel ideas: the use of lazy deletion and re-insertion to do decrease-key operations, and the use of a dag (directed acyclic graph) instead of a tree or set of trees to represent a heap. Lazy deletion produces hollow nodes (nodes without items), giving the data structure its name.Comment: 27 pages, 7 figures, preliminary version appeared in ICALP 201

Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs

We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with n nodes and m edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time ~O(m^{3/4} n^{3/2}), which gives the first improvement over Megiddo\u27s ~O(n^3) algorithm [JACM\u2783] for sparse graphs (We use the notation ~O(.) to hide factors that are polylogarithmic in n.) We further demonstrate how to obtain both an algorithm with running time n^3/2^{Omega(sqrt(log n)} on general graphs and an algorithm with running time ~O(n) on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest

Faster Algorithms for Computing Maximal 2-Connected Subgraphs in Sparse Directed Graphs

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with $\widetilde{O}(mn)$ time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced $O(n^2)$ time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time $O(m^{3/2})$, which further improves the running times for sparse graphs. The notion of 2-connectivity naturally generalizes to k-connectivity for $k>2$. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time $O(m^{3/2} \log{n})$, improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in $O(n^2 \log n)$ time.Comment: Revised version of SODA 2017 paper including details for k-edge-connected subgraph

ARRIVAL: Next Stop in CLS

We study the computational complexity of ARRIVAL, a zero-player game on $n$-vertex switch graphs introduced by Dohrau, G\"{a}rtner, Kohler, Matou\v{s}ek, and Welzl. They showed that the problem of deciding termination of this game is contained in $\text{NP} \cap \text{coNP}$. Karthik C. S. recently introduced a search variant of ARRIVAL and showed that it is in the complexity class PLS. In this work, we significantly improve the known upper bounds for both the decision and the search variants of ARRIVAL. First, we resolve a question suggested by Dohrau et al. and show that the decision variant of ARRIVAL is in $\text{UP} \cap \text{coUP}$. Second, we prove that the search variant of ARRIVAL is contained in CLS. Third, we give a randomized $\mathcal{O}(1.4143^n)$-time algorithm to solve both variants. Our main technical contributions are (a) an efficiently verifiable characterization of the unique witness for termination of the ARRIVAL game, and (b) an efficient way of sampling from the state space of the game. We show that the problem of finding the unique witness is contained in CLS, whereas it was previously conjectured to be FPSPACE-complete. The efficient sampling procedure yields the first algorithm for the problem that has expected runtime $\mathcal{O}(c^n)$ with $c<2$.Comment: 13 pages, 6 figure