arXiv subjects

Marek Chrobak

Publications and source records attributed to Marek Chrobak.

At least 19 recordsLinked to original sources

arXiv2026

A Gossiping Protocol for Sparse Ad-Hoc Radio Networks

We study the problem of gossiping (all-to-all information exchange) in ad-hoc radio networks. Such a network is represented by a strongly-connected directed graph with $n$ vertices, whose topology is initially unknown to the protocol. In 2004, Gasieniec, Radzik, and Xin gave a $\tilde O(n^{4/3})$-time deterministic protocol for this problem, and closing the gap between their upper bound and the $\tilde\Omega(n)$ lower bound on the time complexity of gossiping remains a central open problem. We develop a deterministic protocol for gossiping in ad-hoc radio networks that achieves running time $\tilde O((mn)^{3/5})$ for directed graphs with at most $m$ edges. Our protocol improves on the $\tilde O(n^{4/3})$ bound when $m = O(n^c)$, for $c < 11/9$. We also present a $\tilde O(\Delta^{1/2} n)$-time gossiping protocol for $\Delta$-regular graphs.

cs.DS↗

arXiv2026

Two Complexity Results on Spanning-Tree Congestion Problems

In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard, even for some restricted graph classes, several key questions regarding its computational complexity remain open, and we address some of these in our paper. (i) For graphs of degree at most $\Delta$, it is known that $\mathsf{STC}$ is $\mathbb{NP}$-hard when $\Delta\ge 8$. We provide a complete resolution of this variant, by showing that $\mathsf{STC}$ remains $\mathbb{NP}$-hard for each degree bound $\Delta\ge 3$. (ii) In the decision version of $\mathsf{STC}$, given an integer $K$, the goal is to determine whether the congestion of $G$ is at most $K$. We prove that this variant is polynomial-time solvable for $K$-edge-connected graphs.

cs.DS↗

arXiv2025

A Refutation of Elmasry's $\tilde{O}(m \sqrt{n})$-Time Algorithm for Single-Source Shortest Paths

In this note we examine the recent paper "Breaking the Bellman-Ford Shortest-Path Bound" by Amr Elmasry, where he presents an algorithm for the single-source shortest path problem and claims that its running time complexity is $\tilde{O}(m\sqrt{n})$, where $n$ is the number of vertices and $m$ is the number of edges. We show that his analysis is incorrect, by providing an example of a weighted graph on which the running time of his algorithm is $\Omega(mn)$.

cs.DS↗

arXiv2025

A 3.3904-Competitive Online Algorithm for List Update with Uniform Costs

We consider the List Update problem where the cost of each swap is assumed to be 1. This is in contrast to the ``standard'' model, in which an algorithm is allowed to swap the requested item with previous items for free. We construct an online algorithm Full-Or-Partial-Move (FPM), whose competitive ratio is at most $3.3904$, improving over the previous best known bound of $4$.

cs.DS↗

arXiv2024

Lower Bounds for Adaptive Relaxation-Based Algorithms for Single-Source Shortest Paths

We consider the classical single-source shortest path problem in directed weighted graphs. D.~Eppstein proved recently an $\Omega(n^3)$ lower bound for oblivious algorithms that use relaxation operations to update the tentative distances from the source vertex. We generalize this result by extending this $\Omega(n^3)$ lower bound to \emph{adaptive} algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances. Our model captures as a special case the operations on tentative distances used by Dijkstra's algorithm.

cs.DS↗

arXiv2024

On HTLC-Based Protocols for Multi-Party Cross-Chain Swaps

In his 2018 paper, Herlihy introduced an atomic protocol for multi-party asset swaps across different blockchains. His model represents an asset swap by a directed graph whose nodes are the participating parties and edges represent asset transfers, and rational behavior of the participants is captured by a preference relation between a protocol's outcomes. Asset transfers between parties are achieved using smart contracts. These smart contracts are quite involved and they require storage and processing of a large number of paths in the swap digraph, limiting practical significance of his protocol. His paper also describes a different protocol that uses only standard hash time-lock contracts (HTLC's), but this simpler protocol applies only to some special types of digraphs. He left open the question whether there is a simple and efficient protocol for cross-chain asset swaps in arbitrary digraphs. Motivated by this open problem, we conducted a comprehensive study of \emph{HTLC-based protocols}, in which all asset transfers are implemented with HTLCs. Our main contribution is a full characterization of swap digraphs that have such protocols.

cs.DS↗

arXiv2024

On Permutation Selectors and their Applications in Ad-Hoc Radio Networks Protocols

Selective families of sets, or selectors, are combinatorial tools used to "isolate" individual members of sets from some set family. Given a set $X$ and an element $x\in X$, to isolate $x$ from $X$, at least one of the sets in the selector must intersect $X$ on exactly $x$. We study (k,N)-permutation selectors which have the property that they can isolate each element of each $k$-element subset of $\{0,1,...,N-1\}$ in each possible order. These selectors can be used in protocols for ad-hoc radio networks to more efficiently disseminate information along multiple hops. In 2004, Gasieniec, Radzik and Xin gave a construction of a (k,N)-permutation selector of size $O(k^2\log^3 N)$. This paper improves this by providing a probabilistic construction of a (k,N)-permutation selector of size $O(k^2\log N)$. Remarkably, this matches the asymptotic bound for standard strong (k,N)-selectors, that isolate each element of each set of size $k$, but with no restriction on the order. We then show that the use of our (k,N)-permutation selector improves the best running time for gossiping in ad-hoc radio networks by a poly-logarithmic factor.

cs.DS↗

arXiv2023

A Note on Local Convergence of Iterative Processes for Pipe Network Analysis

Analysis of pipe networks involves computing flow rates and pressure differences on pipe segments in the network, given the external inflow/outflow values. This analysis can be conducted using iterative methods, among which the algorithms of Hardy Cross and Newton-Raphson have historically been applied in practice. In this note, we address the mathematical analysis of the local convergence of these algorithms. The loop-based Newton-Raphson algorithm converges quadratically fast, and we provide estimates for its convergence radius to correct some estimates in the previous literature. In contrast, we show that the convergence of the Hardy Cross algorithm is only linear. This provides theoretical confirmation of experimental observations reported earlier in the literature.

cs.CE↗

arXiv2023

A Tight Threshold Bound for Search Trees with 2-way Comparisons

We study search trees with 2-way comparisons (2WCST's), which involve separate less-than and equal-to tests in their nodes, each test having two possible outcomes, yes and no. These trees have a much subtler structure than standard search trees with 3-way comparisons (3WCST's) and are still not well understood, hampering progress towards designing an efficient algorithm for computing minimum-cost trees. One question that attracted attention in the past is whether there is an easy way to determine which type of comparison should be applied at any step of the search. Anderson, Kannan, Karloff and Ladner studied this in terms of the ratio between the maximum and total key weight, and defined two threshold values: $\lambda^-$ is the largest ratio that forces the less-than test, and $\lambda^+$ is the smallest ratio that forces the equal-to test. They determined that $\lambda^- = 1/4$, but for the higher threshold they only showed that $\lambda^+\in [3/7,4/9]$. We give the tight bound for the higher threshold, by proving that in fact $\lambda^+ = 3/7$.

cs.DS↗

arXiv2023

Structural Properties of Search Trees with 2-way Comparisons

Optimal 3-way comparison search trees (3WCST's) can be computed using standard dynamic programming in time O(n^3), and this can be further improved to O(n^2) by taking advantage of the Monge property. In contrast, the fastest algorithm in the literature for computing optimal 2-way comparison search trees (2WCST's) runs in time O(n^4). To shed light on this discrepancy, we study structure properties of 2WCST's. On one hand, we show some new threshold bounds involving key weights that can be helpful in deciding which type of comparison should be at the root of the optimal tree. On the other hand, we also show that the standard techniques for speeding up dynamic programming (the Monge property / quadrangle inequality) do not apply to 2WCST's.

cs.DS↗

arXiv2023

A $ϕ$-Competitive Algorithm for Scheduling Packets with Deadlines

In the online packet scheduling problem with deadlines (PacketSchD, for short), the goal is to schedule transmissions of packets that arrive over time in a network switch and need to be sent across a link. Each packet has a deadline, representing its urgency, and a non-negative weight, that represents its priority. Only one packet can be transmitted in any time slot, so if the system is overloaded, some packets will inevitably miss their deadlines and be dropped. In this scenario, the natural objective is to compute a transmission schedule that maximizes the total weight of packets that are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning PacketSchD, that has been a subject of intensive study since 2001, is to determine the optimal competitive ratio of online algorithms, namely the worst-case ratio between the optimum total weight of a schedule (computed by an offline algorithm) and the weight of a schedule computed by a (deterministic) online algorithm. We solve this open problem by presenting a $ϕ$-competitive online algorithm for PacketSchD (where $ϕ\approx 1.618$ is the golden ratio), matching the previously established lower bound.

cs.DS↗

arXiv2023

Better Hardness Results for the Minimum Spanning Tree Congestion Problem

In the spanning tree congestion problem, given a connected graph $G$, the objective is to compute a spanning tree $T$ in $G$ that minimizes its maximum edge congestion, where the congestion of an edge $e$ of $T$ is the number of edges in $G$ for which the unique path in $T$ between their endpoints traverses $e$. The problem is known to be $\mathbb{NP}$-hard, but its approximability is still poorly understood. In the decision version of this problem, denoted $K-\textsf{STC}$, we need to determine if $G$ has a spanning tree with congestion at most $K$. It is known that $K-\textsf{STC}$ is $\mathbb{NP}$-complete for $K\ge 8$. On the other hand, $3-\textsf{STC}$ can be solved in polynomial time, with the complexity status of this problem for $K\in \{4,5,6,7\}$ remaining an open problem. We substantially improve the earlier hardness results by proving that $K-\textsf{STC}$ is $\mathbb{NP}$-complete for $K\ge 5$. This leaves only the case $K=4$ open, and improves the lower bound on the approximation ratio to $1.2$. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we consider $K-\textsf{STC}$ restricted to graphs of radius $2$, and we prove that this variant is $\mathbb{NP}$-complete for all $K\ge 6$. Exploring further in this direction, we also examine the variant, denoted $K-\textsf{STC}D$, where the objective is to determine if the graph has a depth-$D$ spanning three of congestion at most $K$. We prove that $6-\textsf{STC}2$ is $\mathbb{NP}$-complete even for bipartite graphs. For bipartite graphs we establish a tight bound, by also proving that $5-\textsf{STC}2$ is polynomial-time solvable. Additionally, we complement this result with polynomial-time algorithms for two special cases that involve bipartite graphs and restrictions on vertex degrees.

cs.CC↗

arXiv2023

Cross-chain Swaps with Preferences

Extreme valuation and volatility of cryptocurrencies require investors to diversify often which demands secure exchange protocols. A cross-chain swap protocol allows distrusting parties to securely exchange their assets. However, the current models and protocols assume predefined user preferences for acceptable outcomes. This paper presents a generalized model of swaps that allows each party to specify its preferences on the subsets of its incoming and outgoing assets. It shows that the existing swap protocols are not necessarily a strong Nash equilibrium in this model. It characterizes the class of swap graphs that have protocols that are safe, live and a strong Nash equilibrium, and presents such a protocol for this class. Further, it shows that deciding whether a swap is in this class is NP-hard through a reduction from 3SAT, and further is $Σ_2^{\mathsf{P}}$-complete through a reduction from $\exists\forall\mathsf{DNF}$.

cs.DC↗

arXiv2023

Classification via Two-Way Comparisons

Given a weighted, ordered query set $Q$ and a partition of $Q$ into classes, we study the problem of computing a minimum-cost decision tree that, given any query $q$ in $Q$, uses equality tests and less-than comparisons to determine the class to which $q$ belongs. Such a tree can be much smaller than a lookup table, and much faster and smaller than a conventional search tree. We give the first polynomial-time algorithm for the problem. The algorithm extends naturally to the setting where each query has multiple allowed classes.

cs.DS↗

arXiv2022

Online Paging with Heterogeneous Cache Slots

It is natural to generalize the online $k$-Server problem by allowing each request to specify not only a point $p$, but also a subset $S$ of servers that may serve it. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page $p$, but also a subset $S$ of cache slots, and is satisfied by having a copy of $p$ in some slot in $S$. We call this problem Slot-Heterogenous Paging. We parameterize the problem by specifying a family $\mathcal S \subseteq 2^{[k]}$ of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size $k$ and family $\mathcal S$: - If all request sets are allowed ($\mathcal S=2^{[k]}\setminus\{\emptyset\}$), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard \Paging ($\mathcal S=\{[k]\}$). - As a function of $|\mathcal S|$ and $k$, the optimal deterministic ratio is polynomial: at most $O(k^2|\mathcal S|)$ and at least $\Omega(\sqrt{|\mathcal S|})$. - For any laminar family $\mathcal S$ of height $h$, the optimal ratios are $O(hk)$ (deterministic) and $O(h^2\log k)$ (randomized). - The special case of laminar $\mathcal S$ that we call All-or-One Paging extends standard Paging by allowing each request to specify a specific slot to put the requested page in. The optimal deterministic ratio for weighted All-or-One Paging is $\Theta(k)$. Offline All-or-One Paging is NP-hard. Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set $\mathcal P of pages, and is satisfied by fetching any page from $\mathcal P into the cache. The optimal ratios for the latter problem (with laminar family of height $h$) are at most $hk$ (deterministic) and $h\,H_k$ (randomized).

cs.DS↗

arXiv2022

On Huang and Wong's Algorithm for Generalized Binary Split Trees

Huang and Wong [1984] proposed a polynomial-time dynamic-programming algorithm for computing optimal generalized binary split trees. We show that their algorithm is incorrect. Thus, it remains open whether such trees can be computed in polynomial time. Spuler [1994] proposed modifying Huang and Wong's algorithm to obtain an algorithm for a different problem: computing optimal two-way-comparison search trees. We show that the dynamic program underlying Spuler's algorithm is not valid, in that it does not satisfy the necessary optimal-substructure property and its proposed recurrence relation is incorrect. It remains unknown whether the algorithm is guaranteed to compute a correct overall solution.

cs.DS↗

arXiv2021

A Simple Algorithm for Optimal Search Trees with Two-Way Comparisons

We present a simple $O(n^4)$-time algorithm for computing optimal search trees with two-way comparisons. The only previous solution to this problem, by Anderson et al., has the same running time, but is significantly more complicated and is restricted to the variant where only successful queries are allowed. Our algorithm extends directly to solve the standard full variant of the problem, which also allows unsuccessful queries and for which no polynomial-time algorithm was previously known. The correctness proof of our algorithm relies on a new structural theorem for two-way-comparison search trees.

cs.DS↗

arXiv2021

A Note on Scheduling Equal-Length Jobs to Maximize Throughput

We study the problem of scheduling equal-length jobs with release times and deadlines, where the objective is to maximize the number of completed jobs. Preemptions are not allowed. In Graham's notation, the problem is described as 1|r_j;p_j=p|\sum U_j. We give the following results: (1) We show that the often cited algorithm by Carlier from 1981 is not correct. (2) We give an algorithm for this problem with running time O(n^5).

cs.DS↗