A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (... more A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (1, −1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0, 0) and finish at (2n, 0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p 1 and p 2 , for the number of Dyck paths starting at p 1 , ending at p 2 , and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.
We present a direct and fairly simple proof of the following incidence bound: Let P be a set of m... more We present a direct and fairly simple proof of the following incidence bound: Let P be a set of m points and L a set of n lines in R d , for d ≥ 3, which lie in a common algebraic two-dimensional surface of degree D that does not contain any 2-flat, so that no 2-flat contains more than s ≤ D lines of L. Then the number of incidences between P and L is I(P, L) = O m 1/2 n 1/2 D 1/2 + m 2/3 min{n, D 2 } 1/3 s 1/3 + m + n .
The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood [29], is a natural generalizat... more The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood [29], is a natural generalization of the classic Tower of Hanoi (TH) problem. There, a generalized placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a given parameter k ≥ 1. The objective is to compute a shortest move-sequence transferring a legal (under the above rule) configuration of n disks on three pegs to another legal configuration. In SOFSEM'07, Dinitz and the second author [7] established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all values of n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all values of n > 3k and n ≤ k, and conjectured that the same phenomenon also occurs in the complementary range k < n ≤ 3k. In this paper we settle the conjecture of Dinitz and the second author in the affirmative, and show that the average-case complexity of the BTH problem is asymptotically the same as the worst-case complexity, for all values of n and k. We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift , and the pancake problem . ⋆
We show that the number of incidences between m distinct points and n distinct lines in R 4 is O ... more We show that the number of incidences between m distinct points and n distinct lines in R 4 is O 2 c √ log m (m 2/5 n 4/5 + m) + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n , for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor 2 c √ log m when m ≤ n 6/7 or m ≥ n 5/3 . Except for this factor, the bound is tight in the worst case.
We give a fairly elementary and simple proof that shows that the number of incidences between m p... more We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R 3 , so that no plane contains more than s lines, is O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n). This bound, originally obtained by Guth and Katz [9] as a major step in their solution of Erdős's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth [7]. The present paper presents a different and simpler derivation, with better bounds than those in , and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.
Annual Symposium on Computational Geometry - SOCG'14, 2014
We show that the number of incidences between m distinct points and n distinct lines in R 4 is O ... more We show that the number of incidences between m distinct points and n distinct lines in R 4 is O 2 c √ log m (m 2/5 n 4/5 + m) + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n , for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor 2 c √ log m when m ≤ n 6/7 or m ≥ n 5/3 . Except for this factor, the bound is tight in the worst case.
The cohomology groups of line bundles over complex tori (or abelian varieties) are classically st... more The cohomology groups of line bundles over complex tori (or abelian varieties) are classically studied invariants of these spaces. In this article, we compute the cohomology groups of line bundles over various holomorphic, non-commutative deformations of complex tori. Our analysis interpolates between two extreme cases. The first case is a calculation of the space of (cohomological) theta functions for line bundles over constant, commutative deformations. The second case is a calculation of the cohomologies of non-commutative deformations of degree-zero line bundles.
The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed gra... more The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed graph G with two distinguished vertices, S and D, and allowing moves only along arcs of this graph. An optimal solution for such a graph G is an algorithm that completes the task of moving a tower of any given number of disks from S to D in a minimal number of disk moves.
The concept of distributed communication bit complexity was introduced by Dinitz, Rajsbaum, and M... more The concept of distributed communication bit complexity was introduced by Dinitz, Rajsbaum, and Moran. They studied bit complexity of Consensus and Leader Election, arriving at more or less exact bounds. This paper answers two questions on Leader Election, which remained there open. The first is to close the gap between the known upper and lower bounds, for electing a leader by two linked processors. The second is whether the suggested algorithm, sending 1.5n bits while electing a leader in a chain of even length n, is optimal, in the case when n is known to the processors. For both problems, absolutely exact bounds are found. Moreover, the presented lower bound proofs show that there is no optimal algorithm other than the suggested ones.
A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (... more A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (1, −1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0, 0) and finish at (2n, 0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p 1 and p 2 , for the number of Dyck paths starting at p 1 , ending at p 2 , and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.
We present a direct and fairly simple proof of the following incidence bound: Let P be a set of m... more We present a direct and fairly simple proof of the following incidence bound: Let P be a set of m points and L a set of n lines in R d , for d ≥ 3, which lie in a common algebraic two-dimensional surface of degree D that does not contain any 2-flat, so that no 2-flat contains more than s ≤ D lines of L. Then the number of incidences between P and L is I(P, L) = O m 1/2 n 1/2 D 1/2 + m 2/3 min{n, D 2 } 1/3 s 1/3 + m + n .
The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood [29], is a natural generalizat... more The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood [29], is a natural generalization of the classic Tower of Hanoi (TH) problem. There, a generalized placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a given parameter k ≥ 1. The objective is to compute a shortest move-sequence transferring a legal (under the above rule) configuration of n disks on three pegs to another legal configuration. In SOFSEM'07, Dinitz and the second author [7] established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all values of n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all values of n > 3k and n ≤ k, and conjectured that the same phenomenon also occurs in the complementary range k < n ≤ 3k. In this paper we settle the conjecture of Dinitz and the second author in the affirmative, and show that the average-case complexity of the BTH problem is asymptotically the same as the worst-case complexity, for all values of n and k. We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift , and the pancake problem . ⋆
We show that the number of incidences between m distinct points and n distinct lines in R 4 is O ... more We show that the number of incidences between m distinct points and n distinct lines in R 4 is O 2 c √ log m (m 2/5 n 4/5 + m) + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n , for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor 2 c √ log m when m ≤ n 6/7 or m ≥ n 5/3 . Except for this factor, the bound is tight in the worst case.
We give a fairly elementary and simple proof that shows that the number of incidences between m p... more We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R 3 , so that no plane contains more than s lines, is O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n). This bound, originally obtained by Guth and Katz [9] as a major step in their solution of Erdős's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth [7]. The present paper presents a different and simpler derivation, with better bounds than those in , and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.
Annual Symposium on Computational Geometry - SOCG'14, 2014
We show that the number of incidences between m distinct points and n distinct lines in R 4 is O ... more We show that the number of incidences between m distinct points and n distinct lines in R 4 is O 2 c √ log m (m 2/5 n 4/5 + m) + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n , for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor 2 c √ log m when m ≤ n 6/7 or m ≥ n 5/3 . Except for this factor, the bound is tight in the worst case.
The cohomology groups of line bundles over complex tori (or abelian varieties) are classically st... more The cohomology groups of line bundles over complex tori (or abelian varieties) are classically studied invariants of these spaces. In this article, we compute the cohomology groups of line bundles over various holomorphic, non-commutative deformations of complex tori. Our analysis interpolates between two extreme cases. The first case is a calculation of the space of (cohomological) theta functions for line bundles over constant, commutative deformations. The second case is a calculation of the cohomologies of non-commutative deformations of degree-zero line bundles.
The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed gra... more The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed graph G with two distinguished vertices, S and D, and allowing moves only along arcs of this graph. An optimal solution for such a graph G is an algorithm that completes the task of moving a tower of any given number of disks from S to D in a minimal number of disk moves.
The concept of distributed communication bit complexity was introduced by Dinitz, Rajsbaum, and M... more The concept of distributed communication bit complexity was introduced by Dinitz, Rajsbaum, and Moran. They studied bit complexity of Consensus and Leader Election, arriving at more or less exact bounds. This paper answers two questions on Leader Election, which remained there open. The first is to close the gap between the known upper and lower bounds, for electing a leader by two linked processors. The second is whether the suggested algorithm, sending 1.5n bits while electing a leader in a chain of even length n, is optimal, in the case when n is known to the processors. For both problems, absolutely exact bounds are found. Moreover, the presented lower bound proofs show that there is no optimal algorithm other than the suggested ones.
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Papers by Noam Solomon