HAL (Le Centre pour la Communication Scientifique Directe), Oct 14, 2019
This announces the first successful attempt at using model-checking techniques to verify the corr... more This announces the first successful attempt at using model-checking techniques to verify the correctness of self-stabilizing distributed algorithms for robots evolving in a continuous environment. The study focuses on the problem of rendezvous of two robots with lights and presents a generic verification model for the SPIN model checker. It will be presented in full at an upcoming venue.
We consider distributed systems of identical autonomous computational entities, called robots, mo... more We consider distributed systems of identical autonomous computational entities, called robots, moving and operating in the plane in synchronous Look -Compute-Move (LCM ) cycles. The algorithmic capabilities of these systems have been extensively investigated in the literature under four distinct models (OBLOT , FSTA, FCOM, LUMI), each identifying different levels of memory persistence and communication capabilities of the robots. In spite of their differences, they all always assume that the robots have unlimited amounts of energy. In this paper we remove this assumption, and start the study of the computational capabilities of robots whose energy is limited, albeit renewable. More precisely, we consider systems where an activated entity uses all its energy to execute an LCM cycle, and the energy can be restored through a period of inactivity. We first study the impact that memory persistence and communication capabilities have on the computational power of such energyconstrained systems of robots; we do so by analyzing the computational relationship between the four models under this energy constraint. We provide a complete characterization of this relationship. Among the many results of this cross-model analysis we prove that for energyconstrained robots, FCOM is more powerful than FSTA (that is, it is better to communicate than to remember). Integral part of the proof is
The paper details the first successful attempt at using model-checking techniques to verify the c... more The paper details the first successful attempt at using model-checking techniques to verify the correctness of distributed algorithms for robots evolving in a continuous environment. The study focuses on the problem of rendezvous of two robots with lights. There exist many different rendezvous algorithms that aim at finding the minimal number of colors needed to solve rendezvous in various synchrony models (e.g., FSYNC, SSYNC, ASYNC). While these rendezvous algorithms are typically very simple, their analysis and proof of correctness tend to be extremely complex, tedious, and error-prone as impossibility results are based on subtle interactions between robots activation schedules. The paper presents a generic verification model written for the SPIN model-checker. In particular, we explain the subtle design decisions that allow to keep the search space finite and tractable, as well as prove several important theorems that support them. As a sanity check, we use the model to verify several known rendezvous algorithms in six different models of synchrony. In each case, we find that the results obtained from the model-checker are consistent with the results known in the literature. The model-checker outputs a counter-example execution in every case that is known to fail. In the course of developing and proving the validity of the model, we identified several fundamental theorems, including the ability for a well chosen algorithm and ASYNC scheduler to produce an emerging property of memory in a system of oblivious mobile robots, and why it is not a problem for luminous rendezvous algorithms.
We consider a Gathering problem for n autonomous mobile robots with persistent memory called ligh... more We consider a Gathering problem for n autonomous mobile robots with persistent memory called light in an asynchronous scheduler (ASYNC). It is well known that Gathering is impossible when robots have no lights in basic common models, if the system is semi-synchronous (SSYNC) or even centralized (only one robot is active in each time). It is known that Gathering can be solved by robots with 10 colors of lights in ASYNC. This result is obtained by combining the following results. (1) The simulation of SSYNC robots with k colors by ASYNC robots with 5k colors , and (2) Gathering is solved by SSYNC robots with 2 colors . In this paper, we improve the result by reducing the number of colors and show that Gathering can be solved by ASYNC robots with 3 colors of lights. We also show that we can construct a simulation algorithm of any unfair SSYNC algorithm using k colors by ASYNC robots with 3k colors, where unfairness does not guarantee that every robot is activated infinitely often. Combining this simulation and the Gathering algorithm by SSYNC robots with 2 colors [27], we obtain a Gathering algorithm by ASYNC robots with 6 colors. Our main result can be obtained by reducing the number of colors from 6 to 3.
This paper considers a situation where each process has a set of operations Op and executes each ... more This paper considers a situation where each process has a set of operations Op and executes each operation in Op infinitely often in distributed systems. Then, let Oe ⊂ Op be a subset of operations, which cannot be executed by a process while its closed neighborhood execute operations in Op∖ Oe. For such a situation, this new concept of Neighborhood Mutual Remainder (NMR) is defined. A distributed algorithm that satisfies the NMR requirement should satisfy the following two properties:(1) Liveness is satisfied if a process executes each operation in Op infinitely often, and (2) safety is satisfied if, when each process executes operations in Oe, no process in its closed neighborhood executes operations in Op∖ Oe. We formalize the concept of NMR and give a simple self-stabilizing algorithm to demonstrate the design paradigm to achieve NMR. We present an application of NMR to a LOOK-COMPUTE-MOVE (LCM) robot system for implementing a move-atomic property, where robots possess an indepe...
We study the Rendezvous problem for 2 autonomous mobile robots in asynchronous settings with pers... more We study the Rendezvous problem for 2 autonomous mobile robots in asynchronous settings with persistent memory called light. It is well known that Rendezvous is impossible in a basic model when robots have no lights, even if the system is semi-synchronous. On the other hand, Rendezvous is possible if robots have lights of various types with a constant number of colors . If robots can observe not only their own lights but also other robots' lights, their lights are called full-light. If robots can only observe the state of other robots' lights, the lights are called external-light. In this paper, we focus on robots with external-lights in asynchronous settings and a particular class of algorithms (called L-algorithms), where an L-algorithm computes a destination based only on the current colors of observable lights. When considering L-algorithms, Rendezvous can be solved by robots with full-lights and 3 colors in general asynchronous settings (called ASYNC) and the number of colors is optimal under these assumptions. In contrast, there exists no L-algorithms in ASYNC with external-lights regardless of the number of colors . In this paper, we consider a fairly large subclass of ASYNC in which Rendezvous can be solved by L-algorithms using external-lights with a finite number of colors, and we show that the algorithms are optimal in the number of colors they use.
The MapReduce framework has firmly established itself as one of the most widely used parallel com... more The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be notoriously difficult, however. In continuation of Goodrich et al.'s early efforts, explicitly espousing the goal of putting the MapReduce framework on footing equal to that of long-established models such as the PRAM, we investigate the obvious complexity question of how the computational power of MapReduce algorithms compares to that of combinational Boolean circuits commonly used for parallel computations. Relying on the standard MapReduce model introduced by Karloff et al. a decade ago, we develop an intricate simulation technique to show that any problem in NC (i.e., a problem solved by a logspace-uniform family of Boolean circuits of polynomial size and a depth polylogarithmic in the input size) can be solved by a MapReduce computation in O(T(n)/ ...
We consider the gathering problem for asynchronous and oblivious robots that cannot communicate e... more We consider the gathering problem for asynchronous and oblivious robots that cannot communicate explicitly with each other, but are endowed with visibility sensors that allow them to see the positions of the other robots. Most of the investigations on the gathering problem on the discrete universe are done on ring shaped networks due to the number of symmetric configurations. We extend in this paper the study of the gathering problem on torus shaped networks assuming robots endowed with local weak multiplicity detection. That is, robots cannot make the difference between nodes occupied by only one robot from those occupied by more than one robots unless it is their current node. As a consequence, solutions based on creating a single multiplicity node as a landmark for the gathering cannot be used. We present in this paper a deterministic algorithm that solves the gathering problem starting from any rigid configuration on an asymmetric unoriented torus shaped network.
In this paper, we define a new concept neighborhood mutual remainder (NMR). An NMR distributed al... more In this paper, we define a new concept neighborhood mutual remainder (NMR). An NMR distributed algorithms should satisfy global fairness, l-exclusion and repeated local rendezvous requirements. We give a simple self-stabilizing algorithm to demonstrate the design paradigm to achieve NMR, and also present applications of NMR to a Look-Compute-Move robot system. 2012 ACM Subject Classification Theory of computation → Distributed algorithms
We consider a problem on a passively-mobile sensor network with a base station; the base station ... more We consider a problem on a passively-mobile sensor network with a base station; the base station counts the number of sensors in the network. In [6], these passively-mobile sensor networks are modeled by extending the model of population protocols and self-stabilizing protocols to count the number of existing sensors, where self-stabilizing counting means from any initial states of sensors and some initialization of the base station (unless the base station is initialized, this problem can not be solved in general), the base station eventually counts the exact number of sensors in the system. In this setting, Beauquier et al. show several protocols to solve the self-stabilizing counting (See Table ). In this paper, we focus on space complexity of the self-stabilizing counting protocols (that is, the number of states sensors can possess, denoted by «(P), where P is an upper bound of the number of states) and improve it by showing selfstabilizing counting protocols using «(P) 2P and «(P) 3P 2, respectively. Since previous best known protocol needs «(P) 4P and a lower bound of «(P) is P, we can shrink the gap lying that feasibility.
Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to... more Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex. This problem has been shown to be LOGAPX-complete for the most general setting, and APX-hard and 2-approximable in polynomial time for dense request sets, where R forms a clique. In this paper, we investigate the complexity of MCD with sparse (tree) structures. We first show that MCD is APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree ∆ of the tree: MCD for tree request set with constant ∆ is solvable in polynomial time, while that with ∆ = Ω(n) is 2.56-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that the problem can be solved in polynomial time if G is a tree.
Proceedings of the twelfth annual symposium on Computational geometry - SCG '96, 1996
Given a set S of n points in the plane, an e-strongly convex 6-hull of S is defined as a convex p... more Given a set S of n points in the plane, an e-strongly convex 6-hull of S is defined as a convex polygon P with the vertices taken from S such that no point of S lies farther than d outside P and such that even if the vertices of .F' arc perturbed by as much as e, P remains convex (e.g., the convex hull of S is a O-strongly convex O-hull of S). This paper presents the first parallel robust method for the generalized convex hull problem. Wc show that an c-strongly convex O (e + ~)-hull of S can be * This work is partly supported by The Hori Information Science Promotion Foundation(1995). t Department of Electrica.land Computer Engineering,
This paper presents the following algorithms to compute the sum of n d-bit integers on reconfigur... more This paper presents the following algorithms to compute the sum of n d-bit integers on reconfigurable parallel computation models: (1) a constant-time algorithm on a reconfigurable mesh of the bit model of size &log('(')) n x dfi, (2) an O(log*n)-time algorithm on a reconfigurable mesh of the bit model of size dm x ddm, (3) an O(logd + log*n)-time algorithm on a reconfigurable mesh of the word model of size n/(log d + log* n) x n/(log d + log* n), and (4) an O(log*n)-time algorithm on a VLSI reconfigurable circuit of area O(dn/log* n).
Given a set S of n points in the plane, an -strongly convex -hull of S is deÿned as a convex poly... more Given a set S of n points in the plane, an -strongly convex -hull of S is deÿned as a convex polygon P with the vertices taken from S such that no point of S lies farther than outside P and such that even if the vertices of P are perturbed by as much as , P remains convex. This paper presents the ÿrst parallel robust method for this generalized convex hull problem (note that the convex hull of S is the 0-strongly convex 0-hull of S). We show that an -strongly convex O( + ÿ)-hull of S can be constructed in O(log 3 n) time using n processors with imprecise computations, where ÿ is the error unit of primitive operations. This result also implies an improved sequential algorithm. Our algorithm consists of two parts: (1) computing a convex O( + ÿ) -hull of n points, in O(log 3 n) time using n processors, and (2) constructing an -strongly convex O( + ÿ)-hull of a convex polygon with n vertices, in O(log 2 n) time with n processors. We also ÿnd an approximate bridge of two sets with n points each, in O(log 2 n) time using n processors, which we use as a subroutine. All these algorithms are fundamental and have their own applications. The parallel computational model in this paper is the EREW PRAM.
IEICE Transactions on Information and Systems, 2005
This paper considers a neighborhood broadcasting protocol in undirected de Bruijn and Kautz netwo... more This paper considers a neighborhood broadcasting protocol in undirected de Bruijn and Kautz networks. The neighborhood broadcasting problem(NBP) is the problem of disseminating a message from an originator vertex to only its neighbors. Our protocol works under the single-port and half-duplex model and solves NBP in 5 log 2 (n + 1) + O(1) time units on the undirected de Bruijn graph UB (n, d) with n d vertices and the undirected Kautz graph UK (n, d) with n d +n d-1 vertices, where 2n is the maximum degree of these graphs. This completion time is asymptotically optimal in this model.
Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to... more Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex. This problem has been shown to be LOGAPXcomplete for the most general setting, and APX-hard and 2-approximable in polynomial time for dense request sets, where R forms a clique. In this paper, we investigate the complexity of MCD with sparse (tree) structures. We first show that MCD is APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree ∆ of the tree: MCD for tree request set with constant ∆ is solvable in polynomial time, while that with ∆ = Ω(n) is 2.56-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that the problem can be solved in polynomial time if G is a tree.
We consider a Gathering problem for n autonomous mobile robots with persistent memory called ligh... more We consider a Gathering problem for n autonomous mobile robots with persistent memory called light in an asynchronous scheduler (ASYNC). It is well known that Gathering is impossible when robots have no lights in basic common models, if the system is semi-synchronous (SSYNC) or even centralized (only one robot is active in each time). It is known that Gathering can be solved by robots with 10 colors of lights in ASYNC. This result is obtained by combining the following results. (1) The simulation of SSYNC robots with k colors by ASYNC robots with 5k colors [6], and (2) Gathering is solved by SSYNC robots with 2 colors . In this paper, we improve the result by reducing the number of colors and show that Gathering can be solved by ASYNC robots with 3 colors of lights. We also show that we can construct a simulation algorithm of any unfair SSYNC algorithm using k colors by ASYNC robots with 3k colors, where unfairness does not guarantee that every robot is activated infinitely often. Combining this simulation and the Gathering algorithm by SSYNC robots with 2 colors [24], we obtain a Gathering algorithm by ASYNC robots with 6 colors. Our main result can be obtained by reducing the number of colors from 6 to 3.
We study the Gathering problem for n autonomous mobile robots in semi-synchronous settings with p... more We study the Gathering problem for n autonomous mobile robots in semi-synchronous settings with persistent memory called light. It is well known that Gathering is impossible in a basic model when robots have no lights, if the system is semi-synchronous or even centralized (only one robot is active in each time) [5, 21]. On the other hand, Rendezvous (Gathering when n = 2) is possible if robots have lights of various types with a constant number of colors [10, 22]. If robots can observe not only their own lights but also other robots' lights, their lights are called full-light. If robots can only observe the state of other robots' lights, their lights are called external-light. If robots can only observe their own lights, their lights are called internal-light. In this paper, we extend the model of robots with lights so that Gathering algorithms can be discussed properly. Then we show Gathering algorithms with three types of lights in the semi-synchronous settings and reveal relationship between the power of lights and other additional assumptions. The most algorithms shown here are optimal in the number of colors they use.
HAL (Le Centre pour la Communication Scientifique Directe), Oct 14, 2019
This announces the first successful attempt at using model-checking techniques to verify the corr... more This announces the first successful attempt at using model-checking techniques to verify the correctness of self-stabilizing distributed algorithms for robots evolving in a continuous environment. The study focuses on the problem of rendezvous of two robots with lights and presents a generic verification model for the SPIN model checker. It will be presented in full at an upcoming venue.
We consider distributed systems of identical autonomous computational entities, called robots, mo... more We consider distributed systems of identical autonomous computational entities, called robots, moving and operating in the plane in synchronous Look -Compute-Move (LCM ) cycles. The algorithmic capabilities of these systems have been extensively investigated in the literature under four distinct models (OBLOT , FSTA, FCOM, LUMI), each identifying different levels of memory persistence and communication capabilities of the robots. In spite of their differences, they all always assume that the robots have unlimited amounts of energy. In this paper we remove this assumption, and start the study of the computational capabilities of robots whose energy is limited, albeit renewable. More precisely, we consider systems where an activated entity uses all its energy to execute an LCM cycle, and the energy can be restored through a period of inactivity. We first study the impact that memory persistence and communication capabilities have on the computational power of such energyconstrained systems of robots; we do so by analyzing the computational relationship between the four models under this energy constraint. We provide a complete characterization of this relationship. Among the many results of this cross-model analysis we prove that for energyconstrained robots, FCOM is more powerful than FSTA (that is, it is better to communicate than to remember). Integral part of the proof is
The paper details the first successful attempt at using model-checking techniques to verify the c... more The paper details the first successful attempt at using model-checking techniques to verify the correctness of distributed algorithms for robots evolving in a continuous environment. The study focuses on the problem of rendezvous of two robots with lights. There exist many different rendezvous algorithms that aim at finding the minimal number of colors needed to solve rendezvous in various synchrony models (e.g., FSYNC, SSYNC, ASYNC). While these rendezvous algorithms are typically very simple, their analysis and proof of correctness tend to be extremely complex, tedious, and error-prone as impossibility results are based on subtle interactions between robots activation schedules. The paper presents a generic verification model written for the SPIN model-checker. In particular, we explain the subtle design decisions that allow to keep the search space finite and tractable, as well as prove several important theorems that support them. As a sanity check, we use the model to verify several known rendezvous algorithms in six different models of synchrony. In each case, we find that the results obtained from the model-checker are consistent with the results known in the literature. The model-checker outputs a counter-example execution in every case that is known to fail. In the course of developing and proving the validity of the model, we identified several fundamental theorems, including the ability for a well chosen algorithm and ASYNC scheduler to produce an emerging property of memory in a system of oblivious mobile robots, and why it is not a problem for luminous rendezvous algorithms.
We consider a Gathering problem for n autonomous mobile robots with persistent memory called ligh... more We consider a Gathering problem for n autonomous mobile robots with persistent memory called light in an asynchronous scheduler (ASYNC). It is well known that Gathering is impossible when robots have no lights in basic common models, if the system is semi-synchronous (SSYNC) or even centralized (only one robot is active in each time). It is known that Gathering can be solved by robots with 10 colors of lights in ASYNC. This result is obtained by combining the following results. (1) The simulation of SSYNC robots with k colors by ASYNC robots with 5k colors , and (2) Gathering is solved by SSYNC robots with 2 colors . In this paper, we improve the result by reducing the number of colors and show that Gathering can be solved by ASYNC robots with 3 colors of lights. We also show that we can construct a simulation algorithm of any unfair SSYNC algorithm using k colors by ASYNC robots with 3k colors, where unfairness does not guarantee that every robot is activated infinitely often. Combining this simulation and the Gathering algorithm by SSYNC robots with 2 colors [27], we obtain a Gathering algorithm by ASYNC robots with 6 colors. Our main result can be obtained by reducing the number of colors from 6 to 3.
This paper considers a situation where each process has a set of operations Op and executes each ... more This paper considers a situation where each process has a set of operations Op and executes each operation in Op infinitely often in distributed systems. Then, let Oe ⊂ Op be a subset of operations, which cannot be executed by a process while its closed neighborhood execute operations in Op∖ Oe. For such a situation, this new concept of Neighborhood Mutual Remainder (NMR) is defined. A distributed algorithm that satisfies the NMR requirement should satisfy the following two properties:(1) Liveness is satisfied if a process executes each operation in Op infinitely often, and (2) safety is satisfied if, when each process executes operations in Oe, no process in its closed neighborhood executes operations in Op∖ Oe. We formalize the concept of NMR and give a simple self-stabilizing algorithm to demonstrate the design paradigm to achieve NMR. We present an application of NMR to a LOOK-COMPUTE-MOVE (LCM) robot system for implementing a move-atomic property, where robots possess an indepe...
We study the Rendezvous problem for 2 autonomous mobile robots in asynchronous settings with pers... more We study the Rendezvous problem for 2 autonomous mobile robots in asynchronous settings with persistent memory called light. It is well known that Rendezvous is impossible in a basic model when robots have no lights, even if the system is semi-synchronous. On the other hand, Rendezvous is possible if robots have lights of various types with a constant number of colors . If robots can observe not only their own lights but also other robots' lights, their lights are called full-light. If robots can only observe the state of other robots' lights, the lights are called external-light. In this paper, we focus on robots with external-lights in asynchronous settings and a particular class of algorithms (called L-algorithms), where an L-algorithm computes a destination based only on the current colors of observable lights. When considering L-algorithms, Rendezvous can be solved by robots with full-lights and 3 colors in general asynchronous settings (called ASYNC) and the number of colors is optimal under these assumptions. In contrast, there exists no L-algorithms in ASYNC with external-lights regardless of the number of colors . In this paper, we consider a fairly large subclass of ASYNC in which Rendezvous can be solved by L-algorithms using external-lights with a finite number of colors, and we show that the algorithms are optimal in the number of colors they use.
The MapReduce framework has firmly established itself as one of the most widely used parallel com... more The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be notoriously difficult, however. In continuation of Goodrich et al.'s early efforts, explicitly espousing the goal of putting the MapReduce framework on footing equal to that of long-established models such as the PRAM, we investigate the obvious complexity question of how the computational power of MapReduce algorithms compares to that of combinational Boolean circuits commonly used for parallel computations. Relying on the standard MapReduce model introduced by Karloff et al. a decade ago, we develop an intricate simulation technique to show that any problem in NC (i.e., a problem solved by a logspace-uniform family of Boolean circuits of polynomial size and a depth polylogarithmic in the input size) can be solved by a MapReduce computation in O(T(n)/ ...
We consider the gathering problem for asynchronous and oblivious robots that cannot communicate e... more We consider the gathering problem for asynchronous and oblivious robots that cannot communicate explicitly with each other, but are endowed with visibility sensors that allow them to see the positions of the other robots. Most of the investigations on the gathering problem on the discrete universe are done on ring shaped networks due to the number of symmetric configurations. We extend in this paper the study of the gathering problem on torus shaped networks assuming robots endowed with local weak multiplicity detection. That is, robots cannot make the difference between nodes occupied by only one robot from those occupied by more than one robots unless it is their current node. As a consequence, solutions based on creating a single multiplicity node as a landmark for the gathering cannot be used. We present in this paper a deterministic algorithm that solves the gathering problem starting from any rigid configuration on an asymmetric unoriented torus shaped network.
In this paper, we define a new concept neighborhood mutual remainder (NMR). An NMR distributed al... more In this paper, we define a new concept neighborhood mutual remainder (NMR). An NMR distributed algorithms should satisfy global fairness, l-exclusion and repeated local rendezvous requirements. We give a simple self-stabilizing algorithm to demonstrate the design paradigm to achieve NMR, and also present applications of NMR to a Look-Compute-Move robot system. 2012 ACM Subject Classification Theory of computation → Distributed algorithms
We consider a problem on a passively-mobile sensor network with a base station; the base station ... more We consider a problem on a passively-mobile sensor network with a base station; the base station counts the number of sensors in the network. In [6], these passively-mobile sensor networks are modeled by extending the model of population protocols and self-stabilizing protocols to count the number of existing sensors, where self-stabilizing counting means from any initial states of sensors and some initialization of the base station (unless the base station is initialized, this problem can not be solved in general), the base station eventually counts the exact number of sensors in the system. In this setting, Beauquier et al. show several protocols to solve the self-stabilizing counting (See Table ). In this paper, we focus on space complexity of the self-stabilizing counting protocols (that is, the number of states sensors can possess, denoted by «(P), where P is an upper bound of the number of states) and improve it by showing selfstabilizing counting protocols using «(P) 2P and «(P) 3P 2, respectively. Since previous best known protocol needs «(P) 4P and a lower bound of «(P) is P, we can shrink the gap lying that feasibility.
Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to... more Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex. This problem has been shown to be LOGAPX-complete for the most general setting, and APX-hard and 2-approximable in polynomial time for dense request sets, where R forms a clique. In this paper, we investigate the complexity of MCD with sparse (tree) structures. We first show that MCD is APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree ∆ of the tree: MCD for tree request set with constant ∆ is solvable in polynomial time, while that with ∆ = Ω(n) is 2.56-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that the problem can be solved in polynomial time if G is a tree.
Proceedings of the twelfth annual symposium on Computational geometry - SCG '96, 1996
Given a set S of n points in the plane, an e-strongly convex 6-hull of S is defined as a convex p... more Given a set S of n points in the plane, an e-strongly convex 6-hull of S is defined as a convex polygon P with the vertices taken from S such that no point of S lies farther than d outside P and such that even if the vertices of .F' arc perturbed by as much as e, P remains convex (e.g., the convex hull of S is a O-strongly convex O-hull of S). This paper presents the first parallel robust method for the generalized convex hull problem. Wc show that an c-strongly convex O (e + ~)-hull of S can be * This work is partly supported by The Hori Information Science Promotion Foundation(1995). t Department of Electrica.land Computer Engineering,
This paper presents the following algorithms to compute the sum of n d-bit integers on reconfigur... more This paper presents the following algorithms to compute the sum of n d-bit integers on reconfigurable parallel computation models: (1) a constant-time algorithm on a reconfigurable mesh of the bit model of size &log('(')) n x dfi, (2) an O(log*n)-time algorithm on a reconfigurable mesh of the bit model of size dm x ddm, (3) an O(logd + log*n)-time algorithm on a reconfigurable mesh of the word model of size n/(log d + log* n) x n/(log d + log* n), and (4) an O(log*n)-time algorithm on a VLSI reconfigurable circuit of area O(dn/log* n).
Given a set S of n points in the plane, an -strongly convex -hull of S is deÿned as a convex poly... more Given a set S of n points in the plane, an -strongly convex -hull of S is deÿned as a convex polygon P with the vertices taken from S such that no point of S lies farther than outside P and such that even if the vertices of P are perturbed by as much as , P remains convex. This paper presents the ÿrst parallel robust method for this generalized convex hull problem (note that the convex hull of S is the 0-strongly convex 0-hull of S). We show that an -strongly convex O( + ÿ)-hull of S can be constructed in O(log 3 n) time using n processors with imprecise computations, where ÿ is the error unit of primitive operations. This result also implies an improved sequential algorithm. Our algorithm consists of two parts: (1) computing a convex O( + ÿ) -hull of n points, in O(log 3 n) time using n processors, and (2) constructing an -strongly convex O( + ÿ)-hull of a convex polygon with n vertices, in O(log 2 n) time with n processors. We also ÿnd an approximate bridge of two sets with n points each, in O(log 2 n) time using n processors, which we use as a subroutine. All these algorithms are fundamental and have their own applications. The parallel computational model in this paper is the EREW PRAM.
IEICE Transactions on Information and Systems, 2005
This paper considers a neighborhood broadcasting protocol in undirected de Bruijn and Kautz netwo... more This paper considers a neighborhood broadcasting protocol in undirected de Bruijn and Kautz networks. The neighborhood broadcasting problem(NBP) is the problem of disseminating a message from an originator vertex to only its neighbors. Our protocol works under the single-port and half-duplex model and solves NBP in 5 log 2 (n + 1) + O(1) time units on the undirected de Bruijn graph UB (n, d) with n d vertices and the undirected Kautz graph UK (n, d) with n d +n d-1 vertices, where 2n is the maximum degree of these graphs. This completion time is asymptotically optimal in this model.
Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to... more Given an n-vertex graph G = (V, E) and a set R ⊆ {{x, y} | x, y ∈ V } of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex. This problem has been shown to be LOGAPXcomplete for the most general setting, and APX-hard and 2-approximable in polynomial time for dense request sets, where R forms a clique. In this paper, we investigate the complexity of MCD with sparse (tree) structures. We first show that MCD is APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree ∆ of the tree: MCD for tree request set with constant ∆ is solvable in polynomial time, while that with ∆ = Ω(n) is 2.56-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that the problem can be solved in polynomial time if G is a tree.
We consider a Gathering problem for n autonomous mobile robots with persistent memory called ligh... more We consider a Gathering problem for n autonomous mobile robots with persistent memory called light in an asynchronous scheduler (ASYNC). It is well known that Gathering is impossible when robots have no lights in basic common models, if the system is semi-synchronous (SSYNC) or even centralized (only one robot is active in each time). It is known that Gathering can be solved by robots with 10 colors of lights in ASYNC. This result is obtained by combining the following results. (1) The simulation of SSYNC robots with k colors by ASYNC robots with 5k colors [6], and (2) Gathering is solved by SSYNC robots with 2 colors . In this paper, we improve the result by reducing the number of colors and show that Gathering can be solved by ASYNC robots with 3 colors of lights. We also show that we can construct a simulation algorithm of any unfair SSYNC algorithm using k colors by ASYNC robots with 3k colors, where unfairness does not guarantee that every robot is activated infinitely often. Combining this simulation and the Gathering algorithm by SSYNC robots with 2 colors [24], we obtain a Gathering algorithm by ASYNC robots with 6 colors. Our main result can be obtained by reducing the number of colors from 6 to 3.
We study the Gathering problem for n autonomous mobile robots in semi-synchronous settings with p... more We study the Gathering problem for n autonomous mobile robots in semi-synchronous settings with persistent memory called light. It is well known that Gathering is impossible in a basic model when robots have no lights, if the system is semi-synchronous or even centralized (only one robot is active in each time) [5, 21]. On the other hand, Rendezvous (Gathering when n = 2) is possible if robots have lights of various types with a constant number of colors [10, 22]. If robots can observe not only their own lights but also other robots' lights, their lights are called full-light. If robots can only observe the state of other robots' lights, their lights are called external-light. If robots can only observe their own lights, their lights are called internal-light. In this paper, we extend the model of robots with lights so that Gathering algorithms can be discussed properly. Then we show Gathering algorithms with three types of lights in the semi-synchronous settings and reveal relationship between the power of lights and other additional assumptions. The most algorithms shown here are optimal in the number of colors they use.
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Papers by Koichi Wada