The challenges posed by big data in the 21st Century are complex: Under the previous common sense, we considered that polynomial-time algorithms are practical; however, when we handle big data, even a linear-time algorithm may be too slow. Thus, sublinear- and constant-time algorithms are required. The academic research project, “Foundations of Innovative Algorithms for Big Data,” which was started in 2014 and will finish in September 2021, aimed at developing various techniques and frameworks to design algorithms for big data. In this project, we introduce a “Sublinear Computation Paradigm.” Toward this purpose, we first provide a survey of constant-time algorithms, which are the most investigated framework of this area, and then present our recent results on sublinear progressive algorithms. A sublinear progressive algorithm first outputs a temporary approximate solution in constant time, and then suggests better solutions gradually in sublinear-time, finally finds the exact solution. We present Sublinear Progressive Algorithm Theory (SPA Theory, for short), which enables to make a sublinear progressive algorithm for any property if it has a constant-time algorithm and an exact algorithm (an exponential-time one is allowed) without losing any computation time in the big-O sense.

In the rendezvous problem, two computing entities (called agents) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous neighborhood rendezvous problem, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in O(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in o(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of O($O(sqrt{n})$) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $ ilde{O}(sqrt{nDelta/delta} + n/delta)$ rounds for graphs of the minimum degree larger than $sqrt{n}$, where n is the number of vertices in the graph, and δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is O(n), and achieves $ ilde{O}left( rac{n}{sqrt{delta}} ight)$-round time complexity without using whiteboards. These algorithms attain o(Δ)-round complexity in the case of $delta = {omega}(sqrt{n} log n)$ and δ=ω(n2/3log4/3n) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(n)-round lower bound if either one of them is removed.

In this study, Bayesian image denoising, in which the prior distribution is assumed to be a Gaussian Markov random field (GMRF), is considered. Recently, an effective algorithm for Bayesian image denoising with a standard GMRF prior has been proposed, which can help implement the overall procedure and optimize its parameters in O(n)-time, where n is the size of the image. A new GMRF-type prior, referred to as a hierarchical GMRF (HGMRF) prior, is proposed, which is obtained by applying a hierarchical Bayesian approach to the standard GMRF prior; in addition, an effective denoising algorithm based on the HGMRF prior is proposed. The proposed HGMRF method can help implement the overall procedure and optimize its parameters in O(n)-time, as well as the previous GMRF method. The restoration quality of the proposed method is found to be significantly higher than that of the previous GMRF method as well as that of a non-local means filter in several cases. Furthermore, numerical evidence implies that the proposed HGMRF prior is more suitable for the image prior than the standard GMRF prior.

In this paper, we consider Bayesian image denoising based on a Gaussian Markov random field (GMRF) model, for which we propose an new algorithm. Our method can solve Bayesian image denoising problems, including hyperparameter estimation, in O(n)-time, where n is the number of pixels in a given image. From the perspective of the order of the computational time, this is a state-of-the-art algorithm for the present problem setting. Moreover, the results of our numerical experiments we show our method is in fact effective in practice.

The subject of the paper is to propose an O(|V|+|E|) algorithm for the 3-edge-connectivity augmentation problem (UW-3-ECA) defined by "Given an undirected graph G0=(V,E), find an edge set E of minimum cardinality such that the graph (V,EE ) (denoted as G0+E ) is 3-edge-connected, where each edge of E connects distinct vertices of V." Such a set E is called a solution to the problem. Let UW-3-ECA(S) (UW-3-ECA(M), respectively) denote UW-3-ECA in which G0+E is required to be simple (G0+E may have multiple edges). Note that we can assume that G0 is simple in UW-3-ECA(S). UW-3-ECA(M) is divided into two subproblems (1) and (2) as follows: (1) finding all k-edge-connected components of a given graph for every k3, and (2) determining a minimum set of edges whose addition to G0 result in a 3-edge-connected graph. Concerning the subproblem (1), we use an O(|V|+|E|) algorithm that has already been existing. The paper proposes an O(|V|+|E|) algorithm for the subproblem (2). Combining these algorithms makes an O(|V|+|E|) algorithm for finding a solution to UW-3-ECA(M). Furthermore, it is shown that a solution E to UW-3-ECA(M) is also a solution to UW-3-ECA(S) if |V|4, partly solving an open problem UW-k-ECA(S) that is a generalization of UW-3-ECA(S).

The subject of the paper is to propose a simple O(|V|+|E|) algorithm for finding all 3-edge-components of a given undirected multigraph G=(V, E). An 3-edge-connected component of G is defined as a maximal set of vertices such that G has at least three edge-disjoint paths between every pair of vertices in the set. The algorithm is based on the depth-first search (DFS) technique. For any fixed DFS-tree T of G, cutpairs of G are partitioned into two types: a type 1 pair consists of an edge of T and a back edge; a type 2 pair consists of two edges of T. All type 1 pairs can easily be determined in O(|V|+|E|) time. The point is that an edge set KE(T) in which any type 2 pair is included can be found in O(|V|+|E|) time. All 3-edge-components of G appear as connected components if we delete from G all edges contained in type 1 pairs or in the edge set KE(T).