Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1311.5081

Combining the Shortest Paths and the Bottleneck Paths Problems

Tong-Wook Shinn·Tadao Takaoka
View original source ↗Read source document ↗Export metadata ↓

Abstract

We combine the well known Shortest Paths (SP) problem and the Bottleneck Paths (BP) problem to introduce a new problem called the Shortest Paths for All Flows (SP-AF) problem that has relevance in real life applications. We first solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem on directed graphs with unit edge costs in $O(mn)$ worst case time bound. We then present two algorithms to solve SSSP-AF on directed graphs with integer edge costs bounded by $c$ in $O(m^2 + nc)$ and $O(m^2 + mn\log{(\frac{c}{m})})$ time bounds. Finally we extend our algorithms for the SSSP-AF problem to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem in $O(m^{2}n + nc)$ and $O(m^{2}n + mn^{2}\log{(\frac{c}{mn})})$ time bounds. All algorithms presented in this paper are practical for implementation.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tong-Wook Shinn, Tadao Takaoka. 2013-11-20. Combining the Shortest Paths and the Bottleneck Paths Problems. https://arxiv.org/abs/1311.5081

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Online Flexible Busy Time Scheduling on Heterogeneous Machines

We study the online busy time scheduling model on heterogeneous machines. In our setting, jobs with uniform processing time arrive online with a deadline that becomes known to the algorithm at the job's arrival time. An algorithm has access to machines, each with different associated capacities and costs. The goal is to schedule jobs on machines by their deadline, so that the total cost incurred by the scheduling algorithm is minimized. While busy time scheduling has been well-studied, relatively little is known when machines are heterogeneous (i.e., have different costs and capacities), despite this natural theoretical generalization being the most practical model for clients using cloud computing services. We make significant progress in understanding this model by designing a deterministic online algorithm with competitive ratio 8(2p-1)/p < 16 when all jobs have uniform processing time p. A randomized version of this algorithm is 4(2p-1)/(p \ln 2)-competitive against an oblivious adversary. For unit-processing-time jobs, we give lower bounds of 4 and e (where e is Euler's number) on the competitive ratio of deterministic and randomized online algorithms, respectively. For unit-processing-time jobs with agreeable deadlines, we provide a deterministic 2-competitive online algorithm and a matching lower bound.

cs.DS↗
arXiv2026

The Binary Tree Mechanism is Optimal for Differentially Private Continual Counting

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. For fixed privacy parameters, the standard binary tree mechanism achieves expected $\ell_\infty$ error $O(\log^{3/2} n)$ under approximate differential privacy and $O(\log^2 n)$ under pure differential privacy. Whether these dependences on the stream length are necessary has remained a central open problem. For fixed $\varepsilon\in(0,1)$, we prove a lower bound of $Ω(\log^{3/2} n)$ under approximate DP with sufficiently small fixed $δ>0$, and a lower bound of $Ω(\log^2 n)$ under pure DP. These bounds establish the optimality of the binary tree mechanism in both settings. The bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. Both proofs use the same decomposition and accumulation of residual noise along a tree. As a consequence of the approximate-DP bound, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

cs.DS↗
arXiv2026

Directed Hamiltonian-Cycle Parity in $O^*((3/2)^n)$ Deterministic Time and Polynomial Space

We give a deterministic algorithm that computes the parity of the number of Hamiltonian cycles in an $n$-vertex directed graph in $O(n^4(3/2)^n)$ time and $O(n^2)$ bits of working space, improving the $O^*(φ^n)$ bound of Björklund and Husfeldt. Their local-degree formula reduces the problem to a weighted sum over solutions of structured quadratic equations. We cover the corresponding ternary state space by binary subcubes, each inducing an affine system. The Kuang--Wang cover can be regenerated within the target bound; canonical ownership resolves its overlaps, while self-loop conditional expectations bound every affine solution visit. Rollback elimination shares the work across cover prefixes. The same cover gives a Las Vegas algorithm listing all $L$ solutions of $m$ affine product constraints in $N$ Boolean variables in expected time $\operatorname{poly}(N,m)((3/2)^m+L)$ and polynomial space. Finally, we show that complete enumeration can require $Ω((3/2)^n)$ visits even on strongly connected digraphs after an optimal self-loop choice. This is a limitation of the enumeration method, not a general lower bound for Hamiltonian-cycle parity.

cs.DS↗