L-convex functions are nonlinear discrete functions on integer points that are computationally tractable in optimization. In this paper, a discrete Hessian matrix and a local quadratic expansion are defined for L-convex functions. We characterize L-convex functions in terms of the discrete Hessian matrix and the local quadratic expansion.

Self-stabilization is a theoretical framework of non-masking fault-tolerant distributed algorithms. In this paper, we investigate a self-stabilizing distributed approximation for the minimum k-dominating set (KDS) problem in general networks. The minimum KDS problem is a generalization of the well-known dominating set problem in graph theory. For a graph G = (V,E), a set D_k V is a KDS of G if and only if each vertex not in D_k is adjacent to at least k vertices in D_k. The approximation ratio of our algorithm is , where Δ is the maximum degree of G, in the networks of which the minimum degree is more than or equal to k.

The aim of this work is to investigate the possibility of designing zero-knowledge identification schemes based on hard-on-average problems. It includes a new two-party identification protocol whose security relies on a discrete mathematics problem classified as DistNP-Complete under the average-case analysis, the so-called Distributional Matrix Representability Problem. Thanks to the use of the search version of the mentioned problem, the zero-knowledge property is formally proved by black-box simulation, and consequently the security of the proposed scheme is actually guaranteed. Furthermore, with the proposal of a new zero-knowledge proof based on a problem never used before for this purpose, the set of tools for designing cryptographic applications is enlarged.

In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.

In this paper we give an algorithm to generates all series-parallel graphs with at most m edges. This algorithm generate each series-parallel graph in constant time on average.

A topological book embedding of a graph is an embedding in a book that carries the vertices in the spine of the book and the edges in the pages; edges are allowed to cross the spine. Enomoto showed that for any graph G having n vertices, there exists a three-page book embedding of G in which each edge crosses the spine log n times. This paper generalizes the result and shows that for any graph G having n vertices, there exists a d + 1-page book embedding of G in which each edge crosses the spine log_d n times.

Side channel attacks (SCA) are serious attacks on mobile devices. In SCA, the attacker can observe the side channel information while the device performs the cryptographic operations, and he/she can detect the secret stored in the device using such side channel information. Ha-Moon proposed a novel countermeasure against side channel attacks in elliptic curve cryptosystems (ECC). The countermeasure is based on the signed scalar multiplication with randomized concept, and does not pay the penalty of speed. Ha-Moon proved that the countermeasure is secure against side channel attack theoretically, and confirmed its immunity experimentally. Thus Ha-Moon's countermeasure seems to be very attractive. In this paper we propose a novel attack against Ha-Moon's countermeasure, and show that the countermeasure is vulnerable to the proposed attack. The proposed attack utilizes a Markov chain for detecting the secret. The attacker determines the transitions in the Markov chain using side channel information, then detects the relation between consecutive two bits of the secret key, instead of bits of the secret key as they are. The use of such relations drastically reduces the search space for the secret key, and the attacker can easily reveal the secret. In fact, around twenty observations of execution of the countermeasure are sufficient to detect the secret in the case of the standard sizes of ECC. Therefore, the single use of Ha-Moon's countermeasure is not recommended for cryptographic use.

We show that the necessary and sufficient condition for the existence of a balanced C₄-bowtie decomposition of the complete multi-graph λK_n is λ(n - 1) 0 (mod 16) and n 7. Decomposition algorithms are also given.

The recent burst growth of the Internet use overloads networking systems and degrades the quality of communications, e.g., bandwidth loss, packet drops, delay of responses, etc. To overcome such degradation of the communication quality, the notion of Quality of Service (QoS) has received attention in practice. In general, QoS switches have several queues and each queue has several slots to store arriving packets. Since network traffic changes frequently, QoS switches need to control arriving packets to maximize the total priorities of transmitted packets, where the priorities are given by nonnegative values and correspond to the quality of service required for each packet. In this paper, we derive lower bounds for the competitive ratio of deterministic multi-queue nonpreemptive QoS problem of priorities 1 and α 1: 1 + /α ln if α α^*; 1/(1 - e^-τ0) if 1 α < α^*, where α^* 1.657 and τ₀ is a root of the equality that e^-τ(1/α + τ)=1 - e^-τ. As an immediate result, this shows a lower bound 1.466 for the competitive ratio of deterministic multi-queue nonpreemptive QoS problem of single priority, which slightly improves the best known lower bound 1.366.

SOBER-128 is a stream cipher designed by Rose and Hawkes in 2003. It can be also used for generating Message Authentication Codes (MACs) and an authenticated encryption. The developers claimed that it is difficult to forge MACs generated by both functions of SOBER-128, though, the security assumption in the proposal paper is not realistic in some instances. In this paper, we examine the security of these message authentication mechanisms of SOBER-128 under security channel model. As a result, we show that both a MAC generation and an authenticated encryption are vulnerable against differential cryptanalysis. The success probabilities of the MAC forgery attack are estimated at 2^-6 and 2^-27 respectively. In addition, we show that some secret bits are revealed if a key is used many times.

This paper investigates a relationship between inkdot and one-pebble for two-dimensional finite automata (2-fa's). Especially we show that (1) alternating inkdot 2-fa's are more powerful than nondeterministic one-pebble 2-fa's, and (2) there is a set accepted by an alternating inkdot 2-fa, but not accepted by any alternating one-pebble 2-fa with only universal states.

The supertask approach was proposed by Moir and Ramamthy as a means of supporting non-migratory tasks in Pfair-scheduled systems. In this approach, tasks bound to the same processor are combined into a single server task, called a supertask, which is scheduled as an ordinary Pfair task. When a supertask is scheduled, one of its component tasks is selected for execution. In previous work, Holman et al. showed that component-task deadlines can be guaranteed by inflating each supertask's utilization. In addition, their experimental results showed that the required inflation factors should be small in practice. Consequently, the average inflation produced by their rules is much greater than that actually required by the supertasks. In this paper, we first propose a notion of Transient Behavior Prediction for supertasks, which predicts the latest possible finish time of subtasks that belong to supertasks. On the basis of the notion, we present an efficient schedulability algorithm for Pfair supertasks in which the deadlines of all component tasks can be guaranteed. In addition, we propose a task merging process which combines the unschedulable supertasks with some Pfair tasks; hence, a newly supertask can be scheduled in the system. Finally, we propose the new reweighting functions that can be used when the previous two methods fail. Our reweighting functions produce smaller inflation factor than the previous work does. To demonstrate the efficacy of the supertasking approach, we present the experimental evaluations of our algorithm, which decreases substantially a number of reweights and the size of inflation when there are many supertasks in the Pfair-scheduled systems.

Just-in-time scheduling problem is the problem of finding an optimal schedule such that each job finishes exactly at its due date. We study the problem under a realistic assumption called periodic time slots. In this paper, we prove that this problem cannot be approximated, assuming P≠NP. Next, we present a heuristic algorithm, assuming that the number of machines is one. The key idea is a reduction of the problem to a network flow problem. The heuristic algorithm is fast because its main part consists of computation of the minimum cost flow that dominates the total time. Our algorithm is O(n³) in the worst case, where n is the number of jobs. Next, we show some simulation results. Finally, we show cases in which our algorithm returns an optimal schedule and is a factor 1.5 approximation algorithm, respectively, and also give an approximation ratio depending on the upper bound of set-up times.

This paper proposes an extension field named TypeII AOPF. This extension field adopts TypeII optimal normal basis, cyclic vector multiplication algorithm, and Itoh-Tsujii inversion algorithm. The calculation costs for a multiplication and inversion in this field is clearly given with the extension degree. For example, the arithmetic operations in TypeII AOPF F_p⁵ is about 20% faster than those in OEF F_p⁵. Then, since CVMA is suitable for parallel processing, we show that TypeII AOPF is superior to AOPF as to parallel processing and then show that a multiplication in TypeII AOPF becomes about twice faster by parallelizing the CVMA computation in TypeII AOPF.

The abstract ray-tracing problem asks, for a given scene consisting of a light source, a light receiver and finitely many obstacles in a space, and a given positive integer ε > 0, whether a ray going out from the light source could reach the light receiver with intensity at least ε. The problem is known to be PSPACE-hard, and it is very unlikely that there exists an efficient algorithm to solve the problem without adding any restriction. In this paper, we show that the problem can be solved in polynomial time under some weak practical restrictions.

The XTR public key cryptosystem was introduced in 2000. XTR is suitable for a variety of environments including low-end smart cards, and is regarded as an excellent alternative to RSA and ECC. Moreover, it is remarked that XTR single exponentiation (XTR-SE) is less susceptible than usual exponentiation routines to environmental attacks such as the timing attack and the differential power analysis (DPA). This paper investigates the security of side channel attack (SCA) on XTR. In this paper, we show the immunity of XTR-SE against the simple power analysis if the order of the computation of XTR-SE is carefully considered. In addition, we show that XTR-SE is vulnerable to the data-bit DPA, the address-bit DPA, the doubling attack, the modified refined power analysis, and the modified zero-value attack. Moreover, we propose some countermeasures against these attacks. We also show experimental results of the efficiency of the countermeasures. From our implementation results, if we compare XTR with ECC with countermeasures against "SCAs," we think XTR is as suitable to smart cards as ECC.

This paper proposes a group signature scheme with efficient membership revocation. Though group signature schemes with efficient membership revocation based on a dynamic accumulator were proposed, the previous schemes force a member to change his secret key whenever he makes a signature. Furthermore, for the modification, the member has to obtain a public membership information of O(_nN) bits, where _n is the length of the RSA modulus and N is the total number of joining members and removed members. In our scheme, the signer needs no modification of his secret, and the public membership information has only K bits, where K is the maximal number of members. Then, for middle-scale groups with the size that is comparable to the RSA modulus size (e.g., up to about 1000 members for 1024 bit RSA modulus), the public membership information is a single small value only, while the signing/verification also remains efficient.

The wavelength-division multiplexing (WDM) technology has been popular in communication societies for providing very large communication bands by multiple lightpaths with different wavelengths on a single optical fiber. Particularly, a double-ring optical network architecture based on the packet-over-WDM technology such as the HORNET architecture, has been extensively studied as a next generation platform for metropolitan area networks (MANs). Each node in this architecture is equipped with a wavelength-fixed optical-drop and a fast tunable transmitter so that a lightpath can be established between any pair of nodes without wavelength conversions. In this paper, we formulate the optical-drop wavelength assignment problem (ODWAP) for efficient wavelength reuse under heterogeneous traffic in this network, and prove the NP-completeness of its decision problem. Then, we propose a simple heuristic algorithm for the basic case of ODWAP. Through extensive simulations, we demonstrate the effectiveness of our approach in reducing waiting times for packet transmissions when a small number of wavelengths are available to retain the network cost for MANs.

In this paper, we propose a proof scheme of shuffle, which is an honest verifier zero-knowledge proof of knowledge such as the protocols by Groth and Furukawa. Unlike the previous schemes proposed by Furukawa-Sako, Groth, and Furukawa, our scheme can be used as the shuffle of the elements encrypted by Paillier's encryption scheme, which has an additive homomorphic property in the message part. The ElGamal encryption scheme used in the previous schemes does not have this property.

Generalized Tsume-Shogi (GTS) is Tsume-Shogi on the board of size n n for arbitrary n. The problem to decide the existence of a winning sequence of moves (where the attacker must always check) on an instance of GTS was proved to be exptime-complete by Yokota et al. (2000). This paper considers the complexity of yozume problem of GTS, which is, roughly speaking, the problem whether a given position of GTS has a winning sequence other than given sequences (though the actual rule of yozume is more complicated). The detection of yozume is an important issue in designing Tsume-Shogi problems, since the modern designing rule strongly prohibits it. We define a function problem of GTS appropriately to formulate yozume problem as its Another Solution Problem (ASP; the problem to decide the existence of solutions other than given ones). Moreover, we extend the existing framework for investigating ASPs so that it can be applied to exptime-complete problems. In particular, since the decision of correctness of given winning sequences is not easy, we establish a framework to treat ASP of function problems with promises. On the basis of these results, we prove that the decision version of yozume problem of GTS is exptime-complete as a promise problem using the existing reduction which was constructed by Yokota et al. to prove the exptime-completeness of GTS.

In this paper, we focus on the problem that in an ad hoc network, how to send a message securely between two users using the certificate dispersal system. In this system, special data called certificate is issued between two users and these issued certificates are stored among the network. Our final purpose on this certificate dispersal problem is to construct certificate graphs with lower dispersability cost which indicates the average number of certificates stored in each node in an ad hoc network. As our first step, when a certificate graph is given, we construct two efficient certificate dispersal algorithms for strongly connected graphs and directed graphs in this paper. We can show that for a strongly connected graph G =(V, E) and a directed graph H =(V ′, E ′), new upper bounds on dispersability cost on the average number of certificates stored in one node are O(D_G +|E|/|V|) and O(p_G d_max +|E ′|/|V ′|) respectively, where D_G is the diameter of G, d_max is the maximum diameter of strongly connected components of H and p_G is the number of strongly connected components of H. Furthermore, we give some new lower bounds for the problem and we also show that our algorithms are optimal for several graph classes.

"Can one walk to infinity on Gaussian primes taking steps of bounded length?" We adopted computational techniques to probe into this open problem. We propose an efficient method to search for the farthest point reachable from the origin, which can be parallelized easily, and have confirmed the existence of a moat of width k =, whereas the best previous result was k = due to Gethner et al. The amount of computation needed for k = is about 5000 times larger than that for k =. A refinement of Vardi's estimate for the farthest distance reachable from the origin is proposed. The proposed estimate incorporates discreteness into Vardi's that is based on percolation theory.

Up to present, proposed are many multi-signature schemes in which signers use respective moduli in the signature generation process. The FDH-based schemes are proposed by Mitomi et al. and Lysyanskaya et al.. The PSS-based schemes are proposed by Kawauchi et al. and Komano et al.. The FDH-based schemes have the advantage that the signature size is independent of the number of the signers. However, since the signature generation algorithm is deterministic, it has a bad reduction rate as a defect. Consequently, the signers must unfortunately use the keys large enough to keep the security. On the other hand, in the PSS-based schemes, good reduction rates can be obtained since the signature generation algorithms are probabilistic. However, the size of the random component shall overflow the security parameter, and thereby the signature size shall grow by the total size of the random components used the signers. That means, if the size of the random component is smaller, the growth of the signature size can be kept smaller. In this paper, we propose new probabilistic multi-signature scheme, which can be proven secure despite that smaller random components are used. We compare the proposed scheme and two existing schemes. Finally, we conclude that the proposed scheme is so-called optimal due to.

In this paper, we propose a global routing method for 2-layer BGA packages. In our routing model, the global routing for each net is uniquely determined by a via assignment of each net. Our global routing method starts from an initial monotonic via assignment and incrementally improves the via assignment to optimize the total wire length and the wire congestion. Experimental results show that our proposed method generates a better global routing efficiently.

The (k + δ)-edge-connectivity augmentation problem for a specified set of vertices ((k + δ)ECA-SV) is defined as follows: "Given an undirected graph G =(V,E), a specified set of vertices Γ V, a subgraph G ′=(V,E ′) with λ(Γ;G ′) = k of G and a cost function c: E Z⁺ (nonnegative integers), find a set E^* E - E ′of edges, each connecting distinct vertices of V, of minimum total cost such that λ(Γ;G″) k + δ for G"=(V,E ′∪E^*)," where λ(Γ;G″) is the minimum value of the maximum number of edge disjoint paths between any pair of vertices in Γ of G". The paper proposes an O(Δ+|V||E|) time 2-approximation algorithm FSAR for (k + 1)ECA-SV with a restriction λ(V;G ′) = λ(Γ;G ′), where Δ is the time complexity of constructing a structural graph of a given graph G ′.

An efficient computational Zero-Knowledge Proof of Knowledge whose security relies on the NP-completeness of the Independent Set Problem is presented here. The proposed algorithm is constructed from a bit commitment scheme based on the hardness of the Discrete Logarithm Problem, which guarantees the fulfillment of soundness, completeness and computational zero-knowledge properties, and allows avoiding the use of the Graph Isomorphism Problem, which is present in every known Zero-Knowledge Proofs for the Independent Set Problem.

A new edge-coloring algorithm for bipartite graphs is presented. This algorithm, based on the framework of the O(m log d + (m/d) log (m/d) log d) algorithm by Makino-Takabatake-Fujishige and the O(m log m) one by Alon, finds an optimal edge-coloring of a bipartite graph with m edges and maximum degree d in O(m log d + (m/d) log (m/d)) time. This algorithm does not require elaborate data structures, which the best known O(m log d) algorithm due to Cole-Ost-Schirra depends on.

In recent years, there is an increasing trend of using biometric identifiers for personal authentication. Encouraged by advances in smart card technologies, the fingerprint matching gets increasingly embedded into smart cards for an effective personal authentication method. However, current generation of low cost smart cards are usually equipped with limited hardware resources such as an 8-bit or 16-bit microcontroller. The fingerprint matching typically is a time consuming, computationally intensive and costly process. Therefore, it is still a challenge to integrate the fingerprint matching into a smart card. In this paper, we present a fast memory-efficient fingerprint matching using minutia ridge shape feature. This feature offers advantages of smaller template size, smaller memory requirement, faster matching time and robust matching against image distortion over conventional minutiae-based feature. The implementation result shows that the proposed method can be embedded in smart cards for a real-time Match-on-Card system.

In a switched-capacitor (SC) circuit, the major block is an operational transconductance amplifier (OTA) designed in order to form a feedback loop. However, the OTA is the block that consumes most of the power in SC circuits. This paper proposes the use of a class-C inverter instead of the OTA in SC circuits and a corresponding switches configuration for extremely low power applications. A detailed analysis and design trade-offs are also provided. Simulation and experimental results show that sufficient performance can be obtained even though a class-C inverter is used. The second-order biquad filter and the second-order SC sigma-delta (ΣΔ) modulator based on a class-C inverter are designed. These circuits have been fabricated with a 0.35-µm CMOS process. The measurement results of the fabricated SC biquad filter show a 59-dB signal-to-noise-plus-distortion ratio (SNDR) for a 0.2-V_p-p input signal and 0.9-V dynamic ranges. The power consumption of the biquad filter is only 0.4 µW with a 1-V power supply. The measurement results of the fabricated ΣΔ modulator show a 61-dB peak SNR for a 1.6-kHz bandwidth with a sample rate of 200 kHz. The modulator consumes 0.8 µW with a 1-V power supply.

A new interacting multiple model (IMM) algorithm using intelligent input estimation (IIE) is proposed for maneuvering target tracking. In the proposed method, the acceleration level for each sub-model is determined by IIE-the estimation of the unknown target acceleration by a fuzzy system using the relation between the residuals of the maneuvering filter and the non-maneuvering filter. The genetic algorithm (GA) is utilized to optimize a fuzzy system for a sub-model within a fixed range of target acceleration. Then, multiple models are represented as the acceleration levels estimated by these fuzzy systems, which are optimized for different ranges of target acceleration. In computer simulation for an incoming anti-ship missile, it is shown that the proposed method has better tracking performance compared with the adaptive interacting multiple model (AIMM) algorithm.

In this paper, a modified adaptive fuzzy sliding mode controller for a certain class of uncertain nonlinear systems is presented. We incorporate the fuzzy sliding mode control technique with a modified adaptive fuzzy control technique to design a modified adaptive fuzzy sliding mode controller so that the proposed controller is robust against the unmodeled dynamics and the approximation errors. Firstly, we establish a fuzzy model to describe the dynamic characteristics of the given uncertain nonlinear system. Then, based on the fuzzy model, a fuzzy sliding mode controller is designed. By considering both the information of tracking error and modeling error, the modified adaptive laws for tuning the adjustable parameters of the fuzzy model are derived based on the Lyapunov synthesis approach. Since the modified adaptive laws contain both the tracking error and the modeling error, it implies that the fuzzy model parameters would continuously converge until both the tracking error and modeling error converges to zero. An inverted pendulum control system is simulated to demonstrate the control performance by using the proposed method.

Let G = (V,E) be a connected graph such that each edge e ∈ E and each vertex v ∈ V are weighted by nonnegative reals w(e) and h(v), respectively. Let r be a vertex designated as a root, and p be a positive integer. The minmax rooted-subtree cover problem (MRSC) asks to find a partition X = {X₁,X₂,...,X_p of V and a set of p subtrees T₁,T₂,...,T_p such that each T_i contains X_i∪{r} so as to minimize the maximum cost of the subtrees, where the cost of T_i is defined to be the sum of the weights of edges in T_i and the weights of vertices in X_i. Similarly, the minmax rooted-cycle cover problem (MRCC) asks to find a partition X = {X₁,X₂,...,X_p} of V and a set of p cycles C₁,C₂,...,C_p such that C_i contains X_i∪{r} so as to minimize the maximum cost of the cycles, where the cost of C_i is defined analogously with the MRSC. In this paper, we first propose a (3-2/(p+1))-approximation algorithm to the MRSC with a general graph G, and we then give a (6-4/(p+1))-approximation algorithm to the MRCC with a metric (G,w).

We investigate the error exponent in the lossy source coding with a fidelity criterion. Marton (1974) established a formula of the reliability function for the stationary memoryless source with finite alphabet. In this paper, we consider a stationary memoryless source assuming that the alphabet space is a metric space and not necessarily finite nor discrete. Our aim is to prove that Marton's formula for the reliability function remains true even if the alphabet is general.

New decoding algorithms for binary linear codes based on the concave-convex procedure are presented. Numerical experiments show that the proposed decoding algorithms surpass Belief Propagation (BP) decoding in error performance. Average computational complexity of one of the proposed decoding algorithms is only a few times greater than that of the BP decoding.

Recent studies show that several FFT-based high-accuracy frequency estimation methods achieve very good performance. In this letter, we select three methods, which are the zero-padding, weighted multipoint interpolated DFT, and phase difference approximation respectively, and discuss the window selection for each method. Experiments show that the window selection primarily depends on the signal-to-noise ratio (SNR). As a result, the optimal window selection for each method and, reversely, the optimal selection of the estimation method for a specific window are discussed as a function of SNR. Consideration on the computational load and the resolution problem is also briefly discussed.

A new algorithm for computing precise estimates of the motion vectors of moving objects in a sequence of images is presented. The proposed method is a fusion of block-matching motion estimation and global optimization technique. To avoid some contradictions between global optimization techniques and piece-wise smooth values of sought motion vectors, a hidden segmentation model is utilized. Computer simulation and experimental results demonstrate a good performance of the method in terms of dynamic motion analysis.

This paper presents high-speed low-power small-area accumulator designs to be used in DDFS systems. To reduce the Numerically Controlled Oscillator (NCO) design complexity and size, only the most significant bits of the accumulator drive the phase to amplitude mapping block. Those bits need to be updated on every sampling clock, while the least significant bits of the accumulator are not visible to the rest of the DDFS design and can be updated less frequently, which motivated the development of new accumulator designs. Without performance degradation, the proposed designs relieve constraints in implementation, and hence they can be employed for GHz-range DDFS, reduce power consumption up to 82% compared to standard accumulator design, and minimize chip area. For further power reduction, the proposed designs place the phase modulation adder at the front of the accumulator.

Conventional delay-insensitive (DI) data encodings require 2N+1 wires for transferring N-bit. To reduce complexity and power dissipation of wires in designing a large scaled chip, a DI data transfer mechanism based on current-mode multiple valued logic (CMMVL), where N-bit data transfer can be performed with only N+1 wires, is proposed. The effectiveness of the proposed data transfer mechanism is validated by comparisons with conventional data transfer mechanisms using dual-rail and 1-of-4 encodings through simulation at the 0.25-µm CMOS technology. Simulation results with wire lengths of 4 mm or larger demonstrate that the CMMVL scheme significantly reduces delay-power product values of the dual-rail encoding with data rate of 5 MHz or more and the 1-of-4 encoding with data rate of 18 MHz or more.

We present a simple decoding algorithm that modifies soft bit-flipping algorithm for decoding LDPC codes. In our method, a new parameter is explored to distinguish the variables (symbols) belonging to the same number of unsatisfied constraints. A token is also assigned in the method to avoid repeated flipping of the same variable, rather than using a constant taboo length. Our scheme shows a similar computational load as the taboo-based algorithm, while having a similar decoding performance as the belief propagation algorithm.