arXiv subjects

Shuji Kijima

Publications and source records attributed to Shuji Kijima.

18 recordsLinked to original sources

arXiv2026

GPU-Parallelization of Markov Chain Pool Decoding with Unbiased MCMC

Markov chain pool decoding (MCPD) devised by Knill et al. (1996) identifies likely positive clones from noisy pooled-test results. The standard MCPD estimates clone-wise posterior probabilities using Gibbs sampling, but it may allocate excessive computational effort to low-scoring clones. This paper focuses on parallelizing MCPD on GPU architectures. Whereas the standard MCPD employs systematic-scan updates, we propose a score-weighted update scheme that updates high-scoring clones more frequently. We prove that the stationary distribution of the proposed Markov chain coincides with the target posterior distribution. To enable efficient GPU parallelization, we further incorporate the unbiased MCMC framework of Jacob et al. (2020) and employ a slot-refilling technique based on the arguments by Glynn and Heidelberger (1991) about the coupling of Markov chains. Experiments involving 1,298 clones, 97 pools, and three true positives demonstrate improved recovery compared with uniform decoders, while maintaining high overlap under high-noise conditions.

stat.CO↗

arXiv2025

Sample Complexity of Identifying the Nonredundancy of Nontransitive Games in Dueling Bandits

Dueling bandit is a variant of the Multi-armed bandit to learn the binary relation by comparisons. Most work on the dueling bandit has targeted transitive relations, that is, totally/partially ordered sets, or assumed at least the existence of a champion such as Condorcet winner and Copeland winner. This work develops an analysis of dueling bandits for non-transitive relations. Jan-ken (a.k.a. rock-paper-scissors) is a typical example of a non-transitive relation. It is known that a rational player chooses one of three items uniformly at random, which is known to be Nash equilibrium in game theory. Interestingly, any variant of Jan-ken with four items (e.g., rock, paper, scissors, and well) contains at least one useless item, which is never selected by a rational player. This work investigates a dueling bandit problem to identify whether all $n$ items are indispensable in a given win-lose relation. Then, we provide upper and lower bounds of the sample complexity of the identification problem in terms of the determinant of $A$ and a solution of $\mathbf{x}^{\top} A = \mathbf{0}^{\top}$ where $A$ is an $n \times n$ pay-off matrix that every duel follows.

cs.GT↗

arXiv2024

An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes

It is a celebrated fact that a simple random walk on an infinite $k$-ary tree for $k \geq 2$ returns to the initial vertex at most finitely many times during infinitely many transitions; it is called transient. This work points out the fact that a simple random walk on an infinitely growing $k$-ary tree can return to the initial vertex infinitely many times, it is called recurrent, depending on the growing speed of the tree. Precisely, this paper is concerned with a simple specific model of a random walk on a growing graph (RWoGG), and shows a phase transition between the recurrence and transience of the random walk regarding the growing speed of the graph. To prove the phase transition, we develop a coupling argument, introducing the notion of less homesick as graph growing (LHaGG). We also show some other examples, including a random walk on $\{0,1\}^n$ with infinitely growing $n$, of the phase transition between the recurrence and transience. We remark that some graphs concerned in this paper have infinitely growing degrees.

math.PR↗

arXiv2024

A Smoothed Analysis of the Space Complexity of Computing a Chaotic Sequence

This work is motivated by a question whether it is possible to calculate a chaotic sequence efficiently, e.g., is it possible to get the $n$-th bit of a bit sequence generated by a chaotic map, such as $\beta$-expansion, tent map and logistic map in $\mathrm{o}(n)$ time/space? This paper gives an affirmative answer to the question about the space complexity of a tent map. We show that the decision problem of whether a given bit sequence is a valid tent code is solved in $\mathrm{O}(\log^{2} n)$ space in a sense of the smoothed complexity.

cs.CC↗

arXiv2023

The Space Complexity of Generating Tent Codes

This paper is motivated by a question whether it is possible to calculate a chaotic sequence efficiently, e.g., is it possible to get the $n$-th bit of a bit sequence generated by a chaotic map, such as $\beta$-expansion, tent map and logistic map in $o(n)$ time/space? This paper gives an affirmative answer to the question about the space complexity of a tent map. We prove that a tent code of $n$-bits with an initial condition uniformly at random is exactly generated in $O(\log^2 n)$ space in expectation.

cs.CC↗

arXiv2022

An analysis of load-balancing algorithms on edge-Markovian evolving graphs

Analysis of algorithms on time-varying networks (often called evolving graphs) is a modern challenge in theoretical computer science. The edge-Markovian is a relatively simple and comprehensive model of evolving graphs: every pair of vertices which is not a current edge independently becomes an edge with probability $p$ at each time-step, as well as every edge disappears with probability $q$. Clearly, the edge-Markovian graph changes its shape depending on the current shape, and the dependency refuses some useful techniques for an independent sequence of random graphs which often behaves similarly to a static random graph. It motivates this paper to develop a new technique for analysis of algorithms on edge-Markovian evolving graphs. Specifically speaking, this paper is concerned with load-balancing, which is a popular subject in distributed computing, and we analyze the so-called random matching algorithms, which is a standard scheme for load-balancing. We prove that major random matching algorithms achieve nearly optimal load balance in $O(r \log (\Delta n))$ steps on edge-Markovian evolving graphs, where $r = \max\{p/(1-q), (1-q)/p\}$, $n$ is the number of vertices (i.e., processors) and $\Delta$ denotes the initial gap of loads unbalance. We remark that the independent sequences of random graphs correspond to $r=1$. To avoid the difficulty of an analysis caused by a complex correlation with the history of an execution, we develop a simple proof technique based on history-independent bounds. As far as we know, this is the first theoretical analysis of load-balancing on randomly evolving graphs, not only for the edge-Markovian but also for the independent sequences of random graphs.

cs.DM↗

arXiv2020

How Many Vertices Does a Random Walk Miss in a Network with Moderately Increasing the Number of Vertices?

Real networks are often dynamic. In response to it, analyses of algorithms on {\em dynamic networks} attract more and more attentions in network science and engineering. Random walks on dynamic graphs also have been investigated actively in more than a decade, where in most cases the edge set changes but the vertex set is static. The vertex sets are also dynamic in many real networks. Motivated by a new technology of the analysis of random walks on dynamic graphs, this paper introduces a simple model of graphs with increasing the number of vertices, and presents an analysis of random walks associated with the cover time on such graphs. In particular, we reveal that a random walk asymptotically covers the vertices all but a constant number if the vertex set grows {\em moderately}.

math.PR↗

arXiv2019

Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an ${\rm M}^{\natural}$-Concave Set Function

Submodularity is an important concept in combinatorial optimization, and it is often regarded as a discrete analog of convexity. It is a fundamental fact that the set of minimizers of any submodular function forms a distributive lattice. Conversely, it is also known that any finite distributive lattice is isomorphic to the minimizer set of a submodular function, through the celebrated Birkhoff's representation theorem. ${\rm M}^{\natural}$-concavity is a key concept in discrete convex analysis. It is known for set functions that the class of ${\rm M}^{\natural}$-concavity is a proper subclass of submodularity. Thus, the minimizer set of an ${\rm M}^{\natural}$-concave function forms a distributive lattice. It is natural to ask if any finite distributive lattice appears as the minimizer set of an ${\rm M}^{\natural}$-concave function. This paper affirmatively answers the question.

cs.DM↗

arXiv2019

Exploration of Finite 2D Square Grid by a Metamorphic Robotic System

We consider exploration of finite 2D square grid by a metamorphic robotic system consisting of anonymous oblivious modules. The number of possible shapes of a metamorphic robotic system grows as the number of modules increases. The shape of the system serves as its memory and shows its functionality. We consider the effect of global compass on the minimum number of modules necessary to explore a finite 2D square grid. We show that if the modules agree on the directions (north, south, east, and west), three modules are necessary and sufficient for exploration from an arbitrary initial configuration, otherwise five modules are necessary and sufficient for restricted initial configurations.

cs.DC↗

arXiv2019

Finding Submodularity Hidden in Symmetric Difference

A set function $f$ on a finite set $V$ is submodular if $f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y)$ for any pair $X, Y \subseteq V$. The symmetric difference transformation (SD-transformation) of $f$ by a canonical set $S \subseteq V$ is a set function $g$ given by $g(X) = f(X \vartriangle S)$ for $X \subseteq V$,where $X \vartriangle S = (X \setminus S) \cup (S \setminus X)$ denotes the symmetric difference between $X$ and $S$. Submodularity and SD-transformations are regarded as the counterparts of convexity and affine transformations in a discrete space, respectively. However, submodularity is not preserved under SD-transformations, in contrast to the fact that convexity is invariant under affine transformations. This paper presents a characterization of SD-stransformations preserving submodularity. Then, we are concerned with the problem of discovering a canonical set $S$, given the SD-transformation $g$ of a submodular function $f$ by $S$, provided that $g(X)$ is given by a function value oracle. A submodular function $f$ on $V$ is said to be strict if $f(X) + f(Y) > f(X \cup Y) + f(X \cap Y)$ holds whenever both $X \setminus Y$ and $Y \setminus X$ are nonempty. We show that the problem is solved by using ${\rm O}(|V|)$ oracle calls when $f$ is strictly submodular, although it requires exponentially many oracle calls in general.

cs.DM↗

arXiv2019

Can Walker Localize The Middle Point of A Line-segment?

This paper poses a question about a simple localization problem. The question is if an {\em oblivious} walker on a line-segment can localize the middle point of the line-segment in {\em finite} steps observing the direction (i.e., Left or Right) and the distance to the nearest end point. This problem is arisen from {\em self-stabilizing} location problems by {\em autonomous mobile robots} with {\em limited visibility}, that is a widely interested abstract model in distributed computing. Contrary to appearances, it is far from trivial if this simple problem is solvable or not, and unsettled yet. This paper is concerned with three variants of the problem with a minimal relaxation, and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.

cs.DC↗

arXiv2017

Plane Formation by Synchronous Mobile Robots without Chirality

We consider a distributed system consisting of autonomous mobile computing entities, called robots, moving in a specified space. The robots are anonymous, oblivious, and have neither any access to the global coordinate system nor any explicit communication medium. Each robot observes the positions of other robots and moves in terms of its local coordinate system. To investigate the self-organization power of robot systems, formation problems in the two dimensional space (2D-space) have been extensively studied. Yamauchi et al. (DISC 2015) introduced robot systems in the three dimensional space (3D-space). While existing results for 3D-space assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. One of the most fundamental agreement problems in 3D-space is the plane formation problem that requires the robots to land on a common plane, that is not predefined. It has been shown that the solvability of the plane formation problem by robots with chirality is determined by the rotation symmetry of their initial local coordinate systems because the robots cannot break it. We show that when the robots lack chirality, the combination of rotation symmetry and reflection symmetry determines the solvability of the plane formation problem because a set of symmetric local coordinate systems without chirality is obtained by rotations and reflections. This richer symmetry results in the increase of unsolvable instances compared with robots with chirality and a flaw of existing plane formation algorithm. In this paper, we give a characterization of initial configurations from which the robots without chirality can form a plane and a new plane formation algorithm for solvable instances.

cs.DC↗

arXiv2016

An FPTAS for the Volume of a ${\cal V}$-polytope ---It is Hard to Compute The Volume of The Intersection of Two Cross-polytopes

Given an $n$-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio $(n/\log n)^n$. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a deterministic approximation of the volume of a ${\cal V}$-polytope, that is a polytope with few vertices and (possibly) exponentially many facets, this paper investigates the volume of a "knapsack dual polytope," which is known to be #P-hard due to Khachiyan (1989). We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes, and give FPTASs for those volume computations. Interestingly, the volume of the intersection of two cross-polytopes (i.e., $L_1$-balls) is #P-hard, unlike the cases of $L_{\infty}$-balls or $L_2$-balls.

cs.CC↗

arXiv2016

The Parity Hamiltonian Cycle Problem

Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that the walk may use an edge more than once. First, we give a complete characterization of the graphs which have PHCs, and give a linear time algorithm to find a PHC, in which every edge appears at most four times, in fact. In contrast, we show that finding a PHC is NP-hard if a closed walk is allowed to use each edge at most z times for each z=1,2,3 (PHCz for short), even when a given graph is two-edge connected. We then further investigate the PHC3 problem, and show that the problem is in P when an input graph is four-edge connected. Finally, we are concerned with three (or two)-edge connected graphs, and show that the PHC3 is in P for any C_>=5-free or P6-free graphs. Note that the Hamiltonian cycle problem is known to be NP-hard for those graph classes.

cs.CC↗

arXiv2016

Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space

Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three dimensional space. This paper presents a necessary and sufficient condition for fully-synchronous robots to solve the plane formation problem that does not depend on obliviousness i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedrdon.

cs.DC↗

arXiv2015

Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper investigates deterministic random walks, which is a deterministic process analogous to a random walk. While there are several progresses on the analysis of the vertex-wise discrepancy (i.e., $L_\infty$ discrepancy), little is known about the {\em total variation discrepancy} (i.e., $L_1$ discrepancy), which plays a significant role in the analysis of an FPRAS based on MCMC. This paper investigates upper bounds of the $L_1$ discrepancy between the expected number of tokens in a Markov chain and the number of tokens in its corresponding deterministic random walk. First, we give a simple but nontrivial upper bound ${\rm O}(mt^*)$ of the $L_1$ discrepancy for any ergodic Markov chains, where $m$ is the number of edges of the transition diagram and $t^*$ is the mixing time of the Markov chain. Then, we give a better upper bound ${\rm O}(m\sqrt{t^*\log t^*})$ for non-oblivious deterministic random walks, if the corresponding Markov chain is ergodic and lazy. We also present some lower bounds.

cs.DM↗

arXiv2015

Deterministic Random Walks for Rapidly Mixing Chains

The rotor-router model is a deterministic process analogous to a simple random walk on a graph. This paper is concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational transition probabilities. We investigate the discrepancy of the number of tokens at a single vertex between the functional-router model and its corresponding Markov chain, and give an upper bound in terms of the mixing time of the Markov chain.

cs.DM↗

arXiv2014

Extended Formulations for Sparsity Matroids

We show the existence of a polynomial-size extended formulation for the base polytope of a $(k,\ell)$-sparsity matroid. For an undirected graph $G=(V,E)$, the size of the formulation is $O(|V||E|)$ when $k \geq \ell$ and $O(|V|^2 |E|)$ when $k \leq \ell$. To this end, we employ the technique developed by Faenza et al. recently that uses a randomized communication protocol.

math.CO↗