I teach Computer Science and do research in Computer Science and Mathematics. Research interests: Theoretical Computer Science, Algorithms, Game Theory, Computational and Experimental Mathematics, Monte Carlo Simulations, Software Development
A nonempty set π of residues modulo π is said to be balanced if for each
π₯ β π, there is a π wi... more A nonempty set π of residues modulo π is said to be balanced if for each
π₯ β π, there is a π with 0 < π β€ πβ2 such that π₯Β±π mod π both lie in π. We denote the minimum cardinality of a balanced set modulo π by πΌ(π). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo π, when π is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence c_π = [2^(π+3) + (β1)^π]/3. Then we prove that the numbers c_π = [2^(π+3) + (β1)^π]/3 are never prime when π is odd and π > 1. Thus, the sequence simplifies to c_m = (2^m + 1)/3 with π odd. Finally, we prove that if (2^p + 1)/3 is prime, then π must be a prime.
When we were submitting the paper, we did not know about the Wagstaff primes. So at the end of our paper, we prove unnecessary that if (2^π+1)/3 is
prime, then π must be a prime. The main contribution of the paper is making a connection between the Vector game (Nedev Z., Quas A., Balanced sets and the vector game) and the Wagstaff primes. Namely, when the Vector game is played with a Wagstaff prime, there is a polynomial algorithm for finding an optimal strategy
(a balanced set mod p) for both players in the game.
In this paper we show that the following problem, the even simple path (ESP) problem for directed... more In this paper we show that the following problem, the even simple path (ESP) problem for directed planar graphs, is solvable in polynomial time: Given: a directed planar graph G = (V, E) and two nodes s (startingnode), t (targetnode) β V ; Find: a simple path (i.e., without repeated nodes) from s to t of even length. (The length of the path is the number of edges it contains.
AWOCA 2004 (15th Australasian Workshop on Combinatorial Algorithms), 2004
Given an implemented algorithm, we consider the problem of predicting its actual running time on ... more Given an implemented algorithm, we consider the problem of predicting its actual running time on a specific machine. Our approach consists of running the implemented algorithm on a small number of specially constructed inputs. The recorded running times are used to build the formulas for predicting the running times on the specific computer. Our main ideas is that for every algorithm an auxiliary procedure should be designed that constructs a 'typical' input of a given size for this algorithm. A 'typical' input of a given size will cause the algorithm to finish at a time equal to the expected running time for an input of that size. We show how to construct typical inputs for insertion sort and quicksort.
International Journal of Pure and Applied Mathematics, 2012
We introduce the notion of uniform rank. A matrix C has an uniform rank r if its (normal) rank(C)... more We introduce the notion of uniform rank. A matrix C has an uniform rank r if its (normal) rank(C) equals r and each of its minors of size r is non-zero. Then we prove that a directed graph (digraph) with n vertices is strongly connected if and only if its laplacian has an uniform rank n β 1. Finally we show how to test if a digraph is strongly connected with a single Gaussian Elimination on the digraph's laplacian.
For a prime p, we call a non-empty subset S of the group F p balanced if every element of S is th... more For a prime p, we call a non-empty subset S of the group F p balanced if every element of S is the midterm of a three-term arithmetic progression, contained in S. A result of Browkin, DiviΕ‘ and Schinzel implies that the size of a balanced subset of F p is at least log 2 p+1. In this paper we present an efficient algorithm which yields a balanced set of size (1 + o(1)) log 2 p as p grows.
We describe the design and implementation of a computer application that can search for a small s... more We describe the design and implementation of a computer application that can search for a small size balanced set modulo p, with p prime. Additionally, the application can "analyze" and "visualize" any given balanced set to show graphically the dependancies between the elements of the set. A nonempty set S of residues modulo N is balanced if for each x β S, there is a d with 0 < d β€ N/2 such that x Β± d mod N both lie in S; we use Ξ±(N) to denote the minimum cardinality of a balanced set modulo N. Small size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets in [5]
Parallel and Distributed Computing and Networks, 2004
In this paper we investigate the use of ordinary workstations and/or personnel computers for para... more In this paper we investigate the use of ordinary workstations and/or personnel computers for parallel computations. We use Java as a programming language and its Socket mechanisms for communications. With our software, any machine on the internet which is not utilized by its owner can make itself available and may eventually participate as a member of a parallel team. When its owner returns, the machine goes back to serve him, usually by running ordinary sequential software. As an example, we use sorting as a problem and merge sort as an algorithm. Statistical results were gathered using machines on the departmental network. Several optimization problems can arise when deciding which subteam will minimize the total running time of parallel merge sort. A solution is offered for two of the problems.
We analyze further the Magnus-Derek game, a two-player game played on a round table with n positi... more We analyze further the Magnus-Derek game, a two-player game played on a round table with n positions. The players jointly control the movement of a token. One player, Magnus, aims to maximize the number of positions visited while minimizing the number of rounds. The other player, Derek, attempts to minimize the number of visited positions. We present a new strategy for Magnus that succeeds in visiting the maximal number of positions in 3(n β 1) rounds, which is the optimal number of rounds up to a constant factor.
We analyze the Magnus-Derek game, a two-player game played on a round table having n positions. T... more We analyze the Magnus-Derek game, a two-player game played on a round table having n positions. The players jointly control the movement of a token. One player aims to minimize the number of positions visited, and the other aims to maximize this quantity. We give an f * (n) + log 2 (m β 1) f * (m)-round strategy, where p is the smallest odd prime factor of n, m = n p , and f * is a function yielding the number of positions visited when both players play optimally. This strategy requires significantly less computation time than the previously published strategies.
We deal with small size balanced subsets of Zp when p is prime. In a balanced set S, each element... more We deal with small size balanced subsets of Zp when p is prime. In a balanced set S, each element x β S is a midpoint between two other elementsfromS,i.e. 2x=y+z (modp),y,zβS\{x}. Wedenote the minimum cardinality of a balanced set modulo p by Ξ±(p). We prove a new lower bound for Ξ±(p). Thus we demonstrate that for every prime p > 2, Ξ±(p) β₯ log2 p + 1.41.
INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2008
We consider two complementary classes of subsets in Z n. A non-empty subset U of Z n is universal... more We consider two complementary classes of subsets in Z n. A non-empty subset U of Z n is universal if, for all x β U and all 0 < β€ n 2 , at least one of x Β± (mod n) is in U. A proper subset M of Z n is middle-inclusive if, for all a, b β M , all solutions x of 2x β‘ a + b (mod n) are in M. We derive a formula for Ξ²(n), the minimum cardinality of a universal set modulo n, and examine how universal sets arise in the context of the Vector Game, studied in an earlier paper by Nedev and Muthukrishnan.
ABSTRACT We introduce a new combinatorial game between two players: Magnus and Derek. Initially, ... more ABSTRACT We introduce a new combinatorial game between two players: Magnus and Derek. Initially, a token is placed at position 0 on a round table with n positions. In each round of the game Magnus chooses the number of positions for the token to move, and Derek decides in which direction, + (clockwise) or - (counterclockwise), the token will be moved. Magnus aims to maximize the total number of positions visited during the course of the game, while Derek aims to minimize this quantity. We define f * (n) to be the eventual size of the set of visited positions when both players play optimally. We prove a closed form expression for f * (n) in terms of the prime factorization of n, and provide algorithmic strategies for Magnus and Derek to meet this bound. We note the relevance of the game for a mobile agent exploring a ring network with faulty sense of direction, and we pose variants of the game for future study.
A nonempty set π of residues modulo π is said to be balanced if for each
π₯ β π, there is a π wi... more A nonempty set π of residues modulo π is said to be balanced if for each
π₯ β π, there is a π with 0 < π β€ πβ2 such that π₯Β±π mod π both lie in π. We denote the minimum cardinality of a balanced set modulo π by πΌ(π). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo π, when π is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence c_π = [2^(π+3) + (β1)^π]/3. Then we prove that the numbers c_π = [2^(π+3) + (β1)^π]/3 are never prime when π is odd and π > 1. Thus, the sequence simplifies to c_m = (2^m + 1)/3 with π odd. Finally, we prove that if (2^p + 1)/3 is prime, then π must be a prime.
When we were submitting the paper, we did not know about the Wagstaff primes. So at the end of our paper, we prove unnecessary that if (2^π+1)/3 is
prime, then π must be a prime. The main contribution of the paper is making a connection between the Vector game (Nedev Z., Quas A., Balanced sets and the vector game) and the Wagstaff primes. Namely, when the Vector game is played with a Wagstaff prime, there is a polynomial algorithm for finding an optimal strategy
(a balanced set mod p) for both players in the game.
In this paper we show that the following problem, the even simple path (ESP) problem for directed... more In this paper we show that the following problem, the even simple path (ESP) problem for directed planar graphs, is solvable in polynomial time: Given: a directed planar graph G = (V, E) and two nodes s (startingnode), t (targetnode) β V ; Find: a simple path (i.e., without repeated nodes) from s to t of even length. (The length of the path is the number of edges it contains.
AWOCA 2004 (15th Australasian Workshop on Combinatorial Algorithms), 2004
Given an implemented algorithm, we consider the problem of predicting its actual running time on ... more Given an implemented algorithm, we consider the problem of predicting its actual running time on a specific machine. Our approach consists of running the implemented algorithm on a small number of specially constructed inputs. The recorded running times are used to build the formulas for predicting the running times on the specific computer. Our main ideas is that for every algorithm an auxiliary procedure should be designed that constructs a 'typical' input of a given size for this algorithm. A 'typical' input of a given size will cause the algorithm to finish at a time equal to the expected running time for an input of that size. We show how to construct typical inputs for insertion sort and quicksort.
International Journal of Pure and Applied Mathematics, 2012
We introduce the notion of uniform rank. A matrix C has an uniform rank r if its (normal) rank(C)... more We introduce the notion of uniform rank. A matrix C has an uniform rank r if its (normal) rank(C) equals r and each of its minors of size r is non-zero. Then we prove that a directed graph (digraph) with n vertices is strongly connected if and only if its laplacian has an uniform rank n β 1. Finally we show how to test if a digraph is strongly connected with a single Gaussian Elimination on the digraph's laplacian.
For a prime p, we call a non-empty subset S of the group F p balanced if every element of S is th... more For a prime p, we call a non-empty subset S of the group F p balanced if every element of S is the midterm of a three-term arithmetic progression, contained in S. A result of Browkin, DiviΕ‘ and Schinzel implies that the size of a balanced subset of F p is at least log 2 p+1. In this paper we present an efficient algorithm which yields a balanced set of size (1 + o(1)) log 2 p as p grows.
We describe the design and implementation of a computer application that can search for a small s... more We describe the design and implementation of a computer application that can search for a small size balanced set modulo p, with p prime. Additionally, the application can "analyze" and "visualize" any given balanced set to show graphically the dependancies between the elements of the set. A nonempty set S of residues modulo N is balanced if for each x β S, there is a d with 0 < d β€ N/2 such that x Β± d mod N both lie in S; we use Ξ±(N) to denote the minimum cardinality of a balanced set modulo N. Small size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets in [5]
Parallel and Distributed Computing and Networks, 2004
In this paper we investigate the use of ordinary workstations and/or personnel computers for para... more In this paper we investigate the use of ordinary workstations and/or personnel computers for parallel computations. We use Java as a programming language and its Socket mechanisms for communications. With our software, any machine on the internet which is not utilized by its owner can make itself available and may eventually participate as a member of a parallel team. When its owner returns, the machine goes back to serve him, usually by running ordinary sequential software. As an example, we use sorting as a problem and merge sort as an algorithm. Statistical results were gathered using machines on the departmental network. Several optimization problems can arise when deciding which subteam will minimize the total running time of parallel merge sort. A solution is offered for two of the problems.
We analyze further the Magnus-Derek game, a two-player game played on a round table with n positi... more We analyze further the Magnus-Derek game, a two-player game played on a round table with n positions. The players jointly control the movement of a token. One player, Magnus, aims to maximize the number of positions visited while minimizing the number of rounds. The other player, Derek, attempts to minimize the number of visited positions. We present a new strategy for Magnus that succeeds in visiting the maximal number of positions in 3(n β 1) rounds, which is the optimal number of rounds up to a constant factor.
We analyze the Magnus-Derek game, a two-player game played on a round table having n positions. T... more We analyze the Magnus-Derek game, a two-player game played on a round table having n positions. The players jointly control the movement of a token. One player aims to minimize the number of positions visited, and the other aims to maximize this quantity. We give an f * (n) + log 2 (m β 1) f * (m)-round strategy, where p is the smallest odd prime factor of n, m = n p , and f * is a function yielding the number of positions visited when both players play optimally. This strategy requires significantly less computation time than the previously published strategies.
We deal with small size balanced subsets of Zp when p is prime. In a balanced set S, each element... more We deal with small size balanced subsets of Zp when p is prime. In a balanced set S, each element x β S is a midpoint between two other elementsfromS,i.e. 2x=y+z (modp),y,zβS\{x}. Wedenote the minimum cardinality of a balanced set modulo p by Ξ±(p). We prove a new lower bound for Ξ±(p). Thus we demonstrate that for every prime p > 2, Ξ±(p) β₯ log2 p + 1.41.
INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2008
We consider two complementary classes of subsets in Z n. A non-empty subset U of Z n is universal... more We consider two complementary classes of subsets in Z n. A non-empty subset U of Z n is universal if, for all x β U and all 0 < β€ n 2 , at least one of x Β± (mod n) is in U. A proper subset M of Z n is middle-inclusive if, for all a, b β M , all solutions x of 2x β‘ a + b (mod n) are in M. We derive a formula for Ξ²(n), the minimum cardinality of a universal set modulo n, and examine how universal sets arise in the context of the Vector Game, studied in an earlier paper by Nedev and Muthukrishnan.
ABSTRACT We introduce a new combinatorial game between two players: Magnus and Derek. Initially, ... more ABSTRACT We introduce a new combinatorial game between two players: Magnus and Derek. Initially, a token is placed at position 0 on a round table with n positions. In each round of the game Magnus chooses the number of positions for the token to move, and Derek decides in which direction, + (clockwise) or - (counterclockwise), the token will be moved. Magnus aims to maximize the total number of positions visited during the course of the game, while Derek aims to minimize this quantity. We define f * (n) to be the eventual size of the set of visited positions when both players play optimally. We prove a closed form expression for f * (n) in terms of the prime factorization of n, and provide algorithmic strategies for Magnus and Derek to meet this bound. We note the relevance of the game for a mobile agent exploring a ring network with faulty sense of direction, and we pose variants of the game for future study.
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π₯ β π, there is a π with 0 < π β€ πβ2 such that π₯Β±π mod π both lie in π. We denote the minimum cardinality of a balanced set modulo π by πΌ(π). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo π, when π is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence c_π = [2^(π+3) + (β1)^π]/3. Then we prove that the numbers c_π = [2^(π+3) + (β1)^π]/3 are never prime when π is odd and π > 1. Thus, the sequence simplifies to c_m = (2^m + 1)/3 with π odd. Finally, we prove that if (2^p + 1)/3 is prime, then π must be a prime.
When we were submitting the paper, we did not know about the Wagstaff primes. So at the end of our paper, we prove unnecessary that if (2^π+1)/3 is
prime, then π must be a prime. The main contribution of the paper is making a connection between the Vector game (Nedev Z., Quas A., Balanced sets and the vector game) and the Wagstaff primes. Namely, when the Vector game is played with a Wagstaff prime, there is a polynomial algorithm for finding an optimal strategy
(a balanced set mod p) for both players in the game.
We prove a new lower bound for Ξ±(p). Thus we demonstrate that for every prime p > 2, Ξ±(p) β₯ log2 p + 1.41.
π₯ β π, there is a π with 0 < π β€ πβ2 such that π₯Β±π mod π both lie in π. We denote the minimum cardinality of a balanced set modulo π by πΌ(π). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo π, when π is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence c_π = [2^(π+3) + (β1)^π]/3. Then we prove that the numbers c_π = [2^(π+3) + (β1)^π]/3 are never prime when π is odd and π > 1. Thus, the sequence simplifies to c_m = (2^m + 1)/3 with π odd. Finally, we prove that if (2^p + 1)/3 is prime, then π must be a prime.
When we were submitting the paper, we did not know about the Wagstaff primes. So at the end of our paper, we prove unnecessary that if (2^π+1)/3 is
prime, then π must be a prime. The main contribution of the paper is making a connection between the Vector game (Nedev Z., Quas A., Balanced sets and the vector game) and the Wagstaff primes. Namely, when the Vector game is played with a Wagstaff prime, there is a polynomial algorithm for finding an optimal strategy
(a balanced set mod p) for both players in the game.
We prove a new lower bound for Ξ±(p). Thus we demonstrate that for every prime p > 2, Ξ±(p) β₯ log2 p + 1.41.