Distributed Exact Shortest Paths in Sublinear Time

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in $O(n)$ time, where $n$ is the number of vertices in the input graph $G$. Peleg and Rubinovich (FOCS'99) showed a lower bound of $\tilde{\Omega}(D + \sqrt{n})$ for this problem, where $D$ is the hop-diameter of $G$. Whether or not this problem can be solved in $o(n)$ time when $D$ is relatively small is a major notorious open question. Despite intensive research \cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires $O((n \log n)^{5/6})$ time, for $D = O(\sqrt{n \log n})$, and $O(D^{1/3} \cdot (n \log n)^{2/3})$ time, for larger $D$. This running time is sublinear in $n$ in almost the entire range of parameters, specifically, for $D = o(n/\log^2 n)$. For the all-pairs shortest paths problem, our algorithm requires $O(n^{5/3} \log^{2/3} n)$ time, regardless of the value of $D$. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset $G"$ of a skeleton graph $G'$ of $G$ without first computing $G'$ itself. We then conduct a Bellman-Ford exploration in $G' \cup G"$, while computing the required edges of $G'$ on the fly. As a result, our algorithm computes exactly those edges of $G'$ that it really needs, rather than computing approximately the entire $G'$

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

Connectivity Lower Bounds in Broadcast Congested Clique

We prove three new lower bounds for graph connectivity in the 1-bit broadcast congested clique model, BCC(1). First, in the KT-0 version of BCC(1), in which nodes are aware of neighbors only through port numbers, we show an ?(log n) round lower bound for Connectivity even for constant-error randomized Monte Carlo algorithms. The deterministic version of this result can be obtained via the well-known "edge-crossing" argument, but, the randomized version of this result requires establishing new combinatorial results regarding the indistinguishability graph induced by inputs. In our second result, we show that the ?(log n) lower bound result extends to the KT-1 version of the BCC(1) model, in which nodes are aware of IDs of all neighbors, though our proof works only for deterministic algorithms. This result substantially improves upon the existing ?(log^* n) deterministic lower bound (Jurdzi?ski et el., SIROCCO 2018) for this problem. Since nodes know IDs of their neighbors in the KT-1 model, it is no longer possible to play "edge-crossing" tricks; instead we present a reduction from the 2-party communication complexity problem Partition in which Alice and Bob are given two set partitions on [n] and are required to determine if the join of these two set partitions equals the trivial one-part set partition. While our KT-1 Connectivity lower bound holds only for deterministic algorithms, in our third result we extend this ?(log n) KT-1 lower bound to constant-error Monte Carlo algorithms for the closely related ConnectedComponents problem. We use information-theoretic techniques to obtain this result. All our results hold for the seemingly easy special case of Connectivity in which an algorithm has to distinguish an instance with one cycle from an instance with multiple cycles. Our results showcase three rather different lower bound techniques and lay the groundwork for further improvements in lower bounds for Connectivity in the BCC(1) model

Learning the facts in medical school is not enough: which factors predict successful application of procedural knowledge in a laboratory setting?

BACKGROUND: Medical knowledge encompasses both conceptual (facts or “what” information) and procedural knowledge (“how” and “why” information). Conceptual knowledge is known to be an essential prerequisite for clinical problem solving. Primarily, medical students learn from textbooks and often struggle with the process of applying their conceptual knowledge to clinical problems. Recent studies address the question of how to foster the acquisition of procedural knowledge and its application in medical education. However, little is known about the factors which predict performance in procedural knowledge tasks. Which additional factors of the learner predict performance in procedural knowledge? METHODS: Domain specific conceptual knowledge (facts) in clinical nephrology was provided to 80 medical students (3(rd) to 5(th) year) using electronic flashcards in a laboratory setting. Learner characteristics were obtained by questionnaires. Procedural knowledge in clinical nephrology was assessed by key feature problems (KFP) and problem solving tasks (PST) reflecting strategic and conditional knowledge, respectively. RESULTS: Results in procedural knowledge tests (KFP and PST) correlated significantly with each other. In univariate analysis, performance in procedural knowledge (sum of KFP+PST) was significantly correlated with the results in (1) the conceptual knowledge test (CKT), (2) the intended future career as hospital based doctor, (3) the duration of clinical clerkships, and (4) the results in the written German National Medical Examination Part I on preclinical subjects (NME-I). After multiple regression analysis only clinical clerkship experience and NME-I performance remained independent influencing factors. CONCLUSIONS: Performance in procedural knowledge tests seems independent from the degree of domain specific conceptual knowledge above a certain level. Procedural knowledge may be fostered by clinical experience. More attention should be paid to the interplay of individual clinical clerkship experiences and structured teaching of procedural knowledge and its assessment in medical education curricula

Optimisation of the adhesion of polypropylene-based materials during extrusion-based additive manufacturing

Polypropylene (PP) parts produced by means of extrusion-based additive manufacturing, also known as fused filament fabrication, are prone to detaching from the build platform due to their strong tendency to shrink and warp. Apart from incorporating high volume fractions of fillers, one approach to mitigate this issue is to improve the adhesion between the first deposited layer and the build platform. However, a major challenge for PP is the lack of adhesion on standard platform materials, as well as a high risk of welding on PP-based platform materials. This study reports the material selection of build platform alternatives based on contact angle measurements. The adhesion forces, investigated by shear-off measurements, between PP-based filaments and the most promising platform material, an ultra-high-molecular-weight polyethylene (UHMW-PE), were optimised by a thorough parametric study. Higher adhesion forces were measured by increasing the platform and extrusion temperatures, increasing the flow rate and decreasing the thickness of the first layer. Apart from changes in printer settings, an increased surface roughness of the UHMW-PE platform led to a sufficient, weld-free adhesion for large-area parts of PP-based filaments, due to improved wetting, mechanical interlockings, and an increased surface area between the two materials in contact

Leader election in SINR model with arbitrary power control

Post-print (lokagerð höfundar)We consider the Leader Election Problem in the Signal-to-Interference-plusNoise-Ratio (SINR) model where nodes can adjust their transmission power. We show that in this setting it is possible to elect a leader in two communication rounds, with high probability. Previously, it was known that Θ(log n) rounds were sufficient and necessary when using uniform power, where n is the number of nodes in the network. We then examine how much power control is needed to achieve fast leader election. We show that every 2-round leader election algorithm in the SINR model running correctly w.h.p. requires a power range 2Ω(n) , even when n is known. We complement this with an algorithm that uses power range 2O˜(n)1, when n is known, and 2O˜(n 1.5 ) , when n is not known. We also explore tradeoffs between time and power used, and show that to elect a leader in t rounds, a range of possible power levels of size exp(n 1/Θ(t) ) is sufficient and necessaryThis work was funded by Icelandic Research Fund grants 152679-05, 174484-05, AFOSR FA9550-13-1-0042, NSF CCF-1461559 and NSF CCF-0939370."Peer Reviewed