On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs

On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs

Reordering method and hierarchies for quantum and classical ordered binary decision diagrams

© Springer International Publishing AG 2017.We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between determin-istic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is 2Ω(n/logn) and present quantum OBDD of width O(n2). It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs. Additionally, we prove the width hierarchy for complexity classes of Boolean functions for bounded error probabilistic OBDDs. And using “reordering” method we extend a hierarchy for k-OBDD of polynomial size, for k = o(n/log3n). Moreover, we proved a similar hierarchy for bounded error probabilistic k-OBDD. And for deterministic and proba-bilistic k-OBDDs of superpolynomial and subexponential size

Extension of the hierarchy for k-OBDDs of small width

In this paper we explore the well-known k-OBDD model of branching programs. We develop a method of representation of the k-OBDD computation process as an "automata-communication protocol" computation process. Our method allows us to extend the hierarchy proved by Bolling-Sauerhoff-Sieling-Wegener in 1996 for k-OBDDs. Moreover, using the PJM function (a modification of well-known PJ and ISA functions), we prove a new hierarchy. © 2013 Allerton Press, Inc

On analysis of input data for jobs shop scheduling problem with respect to workers productivity

© 2016,International Journal of Pharmacy and Technology. All rights reserved.We explore one kind of Job shop scheduling problem,where productivity of workers depends on previous behavior. We consider two parts in “working day”: working period and time-off period. Worker increase their efficient on working period and time-off period. Playing with schedule give us different productivity for workers. We consider big input data,which size does not allow use brute-force search or back-tracking algorithm. It means we should use a heuristic solutions. We suggest linear-time solution which works for specific input data structure. We analyze the input and describe the conditions which allow to apply our solution. Considered conditions was following: length of periods,number of jobs,number of “small” and “big” jobs,speed of productivity increasing and decreasing,number of workers,minimal and maximal productivity and others. We compare our solutions with greedy algorithmic solution,which just assign free worker to first available job,and we have benefit about 15 percents with respect to that greedy algorithmic solution

Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluating. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.Comment: UCNC2019 Conference pape

New Size Hierarchies for Two Way Automata

© 2018, Pleiades Publishing, Ltd. We introduce a new type of nonuniform two-way automaton that can use a different transition function for each tape square. We also enhance this model by allowing to shuffle the given input at the beginning of the computation. Then we present some hierarchy and incomparability results on the number of states for the types of deterministic, nondeterministic, and bounded-error probabilistic models. For this purpose, we provide some lower bounds for all three models based on the numbers of subfunctions and we define two witness functions

Collaborative filtering approach in adaptive learning

© 2016,International Journal of Pharmacy and Technology. All rights reserved.Nowadays an adaptive approach in education is gaining in popularity. But what does this adaptive approach mean? Adaptive learning (also known as Adaptive education) means that the education system has a personal approach for each student or for groups of students that fits to the students’ abilities. Teacher must pick up the most relevant topic for explanation,exercises and tests for such an education process. Also the teacher should adapt the order of learning topics for the current student. It is a very big and hard job,but machine learning algorithms can solve some of these tasks instead of teacher. What do we have? We have a set of lessons and students. In each step of the education process we select one lesson that fits best for a current student. This problem can be solved by recommender systems of algorithms. Recommender systems predict rating or “preference’” that a user would give to the item,and by similar way an adaptive education system also predict lessons “ratings” for user. In the paper we will define “rating” of lessons and what does it mean “fits the best”. Also we give some explanations of a chosen machine learning algorithm

Very narrow quantum OBDDs and width hierarchies for classical OBDDs

In the paper we investigate a model for computing of Boolean functions - Ordered Binary Decision Diagrams (OBDDs), which is a restricted version of Branching Programs. We present several results on the comparative complexity for several variants of OBDD models. - We present some results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2, but any classical OBDD (deterministic or stable bounded-error probabilistic) needs width 2 k+1. - We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical nondeterminism. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width, but any classical nondeterministic OBDD for this function needs non-constant width. - We also present new hierarchies on widths of deterministic and nondeterministic OBDDs. We focus both on small and large widths. © 2014 Springer International Publishing