pavan Sangha - Academia.edu (original) (raw)

Papers by pavan Sangha

arXiv (Cornell University), Jun 5, 2018

Line of Sight (LoS) networks were designed to model wireless communication in settings which may ... more Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer d, and positive integer ω, a graph G = (V, E) is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a cube of side size n of the ddimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any ω on narrow LoS networks (i.e. networks which can be embedded in a n × k × k. .. × k region, for some fixed k independent of n). In the unrestricted case it has been shown that the MIS problem is NP-hard when ω > 2 (the hardness proof goes through for any ω = O(n 1−δ), for fixed 0 < δ < 1). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks.

Information and Computation, 2019

Line of Sight (LoS) networks were designed to model wireless communication in settings which may ... more Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer d, and positive integer ω, a graph G = (V, E) is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a cube of side size n of the ddimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any ω on narrow LoS networks (i.e. networks which can be embedded in a n × k × k. .. × k region, for some fixed k independent of n). In the unrestricted case it has been shown that the MIS problem is NP-hard when ω > 2 (the hardness proof goes through for any ω = O(n 1−δ), for fixed 0 < δ < 1). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks.

Discrete Applied Mathematics, 2019

An embedding of two graphs G and G respectively with vertices embedded at the same locations in Z... more An embedding of two graphs G and G respectively with vertices embedded at the same locations in Z 2 9 however G has range paramter ω = 3 and G has range parameter ω = 5. The black edges are the edges of the integer grid and the blue edges are the edges of the LoS network.. .. .. .. .. . 3.4 A UD and a line of sight representation with parameter ω = 2 on the same grid. The edges of the LoS network are represented in blue.. .. .. .. .. 3.5 Curved lines represent paths and straight lines represent edges between partitions. The cycle v * P v v P v * has even length.. .. .. .. .. .. .. .. 3.6 The first set of yellow vertices in the graph represent a maximal independent set, the second set of yellow vertices represent a maximum independent set. 3.7 The top image is a schedule with intervals representing the feasible solution in blue. The bottom image represents the corresponding independent set in the LoS network constructed from the original schedule.. .. .. .. .. . 4.1 An example of a 4-planar (a planar graph with maximum degree at most 4) graph G = (V, E) on the left, and an orthogonal embedding of G on the right. 4.2 Example of a path rotation embedding in 5-dimensions, with the first vertex

Algorithms and Discrete Applied Mathematics, 2017

Line of Sight (LoS) networks provide a model of wireless communication which incorporates visibil... more Line of Sight (LoS) networks provide a model of wireless communication which incorporates visibility constraints. Vertices of such networks can be embedded in finite d-dimensional grids of size n, and two vertices are adjacent if they share a line of sight and are at distance less than ω. In this paper we study large independent sets in LoS networks. We prove that the computational problem of finding a largest independent set can be solved optimally in polynomial time for one dimensional LoS networks. However, for d ≥ 2, the (decision version of) the problem becomes NP-hard for any fixed ω ≥ 3 and even if ω is chosen to be a function of n that is O(n 1−) for any fixed > 0. In addition we show that the problem is also NP-hard when ω = n for d ≥ 3. This result extends earlier work which showed that the problem is solvable in polynomial time for gridline graphs when d = 2. Finally we describe simple algorithms that achieve constant factor approximations and present a polynomial time approximation scheme for the case where ω is constant.

Lecture Notes in Computer Science, 2013

Motivated by fundamental optimization problems in video delivery over wireless networks, we consi... more Motivated by fundamental optimization problems in video delivery over wireless networks, we consider the following problem of packing resizable items (PRI). Given is a bin of capacity B > 0, and a set I of items. Each item j ∈ I is of size sj > 0. A packed item must stay in the bin for a fixed time interval. To accommodate more items in the bin, each item j can be compressed to a size pj ∈ [0, sj) for at most a fraction qj ∈ [0, 1) of the packing interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI is strongly NP-hard already for highly restricted instances. Our main result is an approximation algorithm that packs, for any instance I of PRI, at least 2 3 OPT (I)−3 items, where OPT (I) is the number of items packed in an optimal solution. Our algorithm yields better ratio for instances in which the maximum compression time of an item is qmax ∈ (0, 12 ). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OPT (I)− 2 items. Finally, we show that a non-trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS).

This thesis provides an introduction to the fundamentals of random graph theory. The study starts... more This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory-these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number R(k, k). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties.

arXiv (Cornell University), Jun 5, 2018

Line of Sight (LoS) networks were designed to model wireless communication in settings which may ... more Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer d, and positive integer ω, a graph G = (V, E) is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a cube of side size n of the ddimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any ω on narrow LoS networks (i.e. networks which can be embedded in a n × k × k. .. × k region, for some fixed k independent of n). In the unrestricted case it has been shown that the MIS problem is NP-hard when ω > 2 (the hardness proof goes through for any ω = O(n 1−δ), for fixed 0 < δ < 1). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks.

Information and Computation, 2019

Line of Sight (LoS) networks were designed to model wireless communication in settings which may ... more Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer d, and positive integer ω, a graph G = (V, E) is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a cube of side size n of the ddimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any ω on narrow LoS networks (i.e. networks which can be embedded in a n × k × k. .. × k region, for some fixed k independent of n). In the unrestricted case it has been shown that the MIS problem is NP-hard when ω > 2 (the hardness proof goes through for any ω = O(n 1−δ), for fixed 0 < δ < 1). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks.

Discrete Applied Mathematics, 2019

An embedding of two graphs G and G respectively with vertices embedded at the same locations in Z... more An embedding of two graphs G and G respectively with vertices embedded at the same locations in Z 2 9 however G has range paramter ω = 3 and G has range parameter ω = 5. The black edges are the edges of the integer grid and the blue edges are the edges of the LoS network.. .. .. .. .. . 3.4 A UD and a line of sight representation with parameter ω = 2 on the same grid. The edges of the LoS network are represented in blue.. .. .. .. .. 3.5 Curved lines represent paths and straight lines represent edges between partitions. The cycle v * P v v P v * has even length.. .. .. .. .. .. .. .. 3.6 The first set of yellow vertices in the graph represent a maximal independent set, the second set of yellow vertices represent a maximum independent set. 3.7 The top image is a schedule with intervals representing the feasible solution in blue. The bottom image represents the corresponding independent set in the LoS network constructed from the original schedule.. .. .. .. .. . 4.1 An example of a 4-planar (a planar graph with maximum degree at most 4) graph G = (V, E) on the left, and an orthogonal embedding of G on the right. 4.2 Example of a path rotation embedding in 5-dimensions, with the first vertex

Algorithms and Discrete Applied Mathematics, 2017

Line of Sight (LoS) networks provide a model of wireless communication which incorporates visibil... more Line of Sight (LoS) networks provide a model of wireless communication which incorporates visibility constraints. Vertices of such networks can be embedded in finite d-dimensional grids of size n, and two vertices are adjacent if they share a line of sight and are at distance less than ω. In this paper we study large independent sets in LoS networks. We prove that the computational problem of finding a largest independent set can be solved optimally in polynomial time for one dimensional LoS networks. However, for d ≥ 2, the (decision version of) the problem becomes NP-hard for any fixed ω ≥ 3 and even if ω is chosen to be a function of n that is O(n 1−) for any fixed > 0. In addition we show that the problem is also NP-hard when ω = n for d ≥ 3. This result extends earlier work which showed that the problem is solvable in polynomial time for gridline graphs when d = 2. Finally we describe simple algorithms that achieve constant factor approximations and present a polynomial time approximation scheme for the case where ω is constant.

Lecture Notes in Computer Science, 2013

Motivated by fundamental optimization problems in video delivery over wireless networks, we consi... more Motivated by fundamental optimization problems in video delivery over wireless networks, we consider the following problem of packing resizable items (PRI). Given is a bin of capacity B > 0, and a set I of items. Each item j ∈ I is of size sj > 0. A packed item must stay in the bin for a fixed time interval. To accommodate more items in the bin, each item j can be compressed to a size pj ∈ [0, sj) for at most a fraction qj ∈ [0, 1) of the packing interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI is strongly NP-hard already for highly restricted instances. Our main result is an approximation algorithm that packs, for any instance I of PRI, at least 2 3 OPT (I)−3 items, where OPT (I) is the number of items packed in an optimal solution. Our algorithm yields better ratio for instances in which the maximum compression time of an item is qmax ∈ (0, 12 ). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OPT (I)− 2 items. Finally, we show that a non-trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS).

This thesis provides an introduction to the fundamentals of random graph theory. The study starts... more This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory-these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number R(k, k). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties.