Given a set P of non-intersecting polygonal obstacles in R^2 defined with n vertices, we compute ... more Given a set P of non-intersecting polygonal obstacles in R^2 defined with n vertices, we compute a sketch Ω of P whose size is independent of n. We utilize Ω to devise an algorithm to compute a (1+ϵ)-approximate Euclidean shortest path between two points given with the description of P. When P comprises of convex polygonal obstacles, we devise a (2+ϵ)-approximation algorithm to efficiently answer two-point Euclidean distance queries.
In this paper we study the problem of market equilibrium. Firstly we consider the model (Fischer ... more In this paper we study the problem of market equilibrium. Firstly we consider the model (Fischer model) where there is a supply of money associated with each buyer and a quantity associated with each item. The market equilibrium problem is to compute a price vector which ensures market clearing, i.e. the demand of a good equals its supply, and subject to his endowment, each buyer maximizes his utility. Each item has a utility for a given buyer. We assume that the utility function is an increasing, differentiable, concave function. We show that under the assumptions of gross substitutibility, an auction algorithm can determine approximate market clearing. Not only does this algorithm extends the class of utility functions for which market equilibrium can be determined polynomially, the auction algorithm is efficient and in this model O(n) faster than the auction algorithm for the case of linear utilities in the Arrow-Debreu model. We outline an extension of our method to the Arrow-De...
Abstract—In this paper we address the issue of designing multipath routing algorithms. Multi-path... more Abstract—In this paper we address the issue of designing multipath routing algorithms. Multi-path routing has the potential of improving the throughput but requires buffers at the destination. Our model assumes a network with capacitated edges and a delay function associated with the network links (edges). We consider the problem of establishing a specified throughput from multiple source to destination pairs in the network, given bounds on the buffer sizes available at the network intermediate nodes and a bound on the maximum delay that the paths are allowed to have. We formulate the problem using a linear programming model. We also present practical heuristics and present the experimental results on an existing network topology. The results are promising and demonstrate the effectiveness of simultaneous multipaths in the multi-commodity scenario. I.
Many interactive 3D graphics applications like manufacturing, design, networked-gaming, need to a... more Many interactive 3D graphics applications like manufacturing, design, networked-gaming, need to access 3D data remotely. Transfer of large data sets consumes lot of bandwidth and space. In this paper we describe a lossless scheme for compressing 3D-surface data, represented by triangular meshes. We present a new algorithm which exploits geometry (vertex) information for encoding connectivity. The experimental results obtained are better than Edgebreaker, the existing best method.
We devise a polynomial-time approximation scheme for the classical geometric problem of finding a... more We devise a polynomial-time approximation scheme for the classical geometric problem of finding an approximate short path amid weighted regions. In this problem, a triangulated region P comprising of n vertices, a positive weight associated with each triangle, and two points s and t that belong to P are given as the input. The objective is to find a path whose cost is at most (1+epsilon)OPT where OPT is the cost of an optimal path between s and t. Our algorithm initiates a discretized-Dijkstra wavefront from source s and progresses the wavefront till it strikes t. This result is about a cubic factor (in n) improvement over the Mitchell and Papadimitriou '91 result, which is the only known polynomial time algorithm for this problem to date. Further, with polynomial time preprocessing of P, a set of data structures are computed which allow answering approximate weighted shortest path queries in polynomial time.
In this paper we investigate Object Oriented Binary Space Partitioning. The Binary Space Partitio... more In this paper we investigate Object Oriented Binary Space Partitioning. The Binary Space Partition Tree (BSP-Trees) is a widely used and e ective data structure for solid modeling and hidden surface removal. We present algorithms for eÆciently constructing Object BSP Trees(OBSP) in 2-Dimensions. The term object used arises since the construction of our tree utilizes the property of hierarchical ellipsoid covers. Besides being object-oriented, the major advantage of the OBSP trees is the reduction in the fragmentation of the scene. This paper formulates algorithms and data-structures for representing polyhedral objects and operations such as composition, deletion and motion of objects based on Object Hierarchy using the modied version of the traditional BSP tree. We improve on the design presented in [11]. Implementation details are presented which show increased eÆciency resulting through our approach.
1. ABSTRACT Torus Networks are an important and widely used architecture in modern supercomputers... more 1. ABSTRACT Torus Networks are an important and widely used architecture in modern supercomputers. However, these supercomputers frequently employ static routing protocols, which have poor worst case performance under heavy network traffic. In this paper, we first characterize bottlenecking behavior under the current standard protocol, then demonstrate that even simplistic dynamic routing protocols offer lower, more scalable latencies over a 3D torus network. As supercomputing moves towards exascale, such scalability will be necessary to cope with unprecedented workloads. These results were generated using the CODES highly parallel simulator, built on top of the ROSS parallel discrete event simulator.
The crisis induced by the Coronavirus pandemic has severely impacted educational institutes. Even... more The crisis induced by the Coronavirus pandemic has severely impacted educational institutes. Even with vaccination efforts underway, it is not clear that sufficient confidence will be achieved for schools to reopen soon. This paper considers the impact of vaccination rates and testing rates to reduce infections and hospitalizations and evaluates strategies that will allow educational institute in urban settings to reopen. These strategies are also applicable to businesses and would help plan reopening in order to help the economy. Our analysis is based on a graph model where nodes represent population groups and edges represent population exchanges due to commuting populations. The commuting population is associated with edges and is associated with one of the end nodes of the edge during part of the time period and with the other node during the remainder of the time period. The progression of the disease at each node is determined via compartment models, that include vaccination r...
Consider a transportation problem with sets of sources and sinks. There are profits and prices on... more Consider a transportation problem with sets of sources and sinks. There are profits and prices on the edges. The goal is to maximize the profit while meeting the following constraints; the total flow going out of a source must not exceed its capacity and the total price of the incoming flow on a sink must not exceed its budget. This problem is closely related to the generalized flow problem. We propose an auction based primal dual approximation algorithm to solve the problem. The complexity is $O(\epsilon^{-1}(n^2+ n\log{m})m\log U)$ where $n$ is the number of sources, $m$ is the number of sinks, $U$ is the ratio of the maximum profit/price to the minimum profit/price. We also show how to generalize the scheme to solve a more general version of the problem, where there are edge capacities and/or the profit function is concave and piece-wise linear. The complexity of the algorithm depends on the number of linear segments, termed ${\cal L}$, of the profit function.
We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a si... more We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a sink vertex $t$ in the presence of obstacles in $\Re^2$. Our algorithm takes $O(T+m(\lg{m})(\lg{n}))$ time and $O(n)$ space. Here, $O(T)$ is the time to triangulate the polygonal region, $m$ is the number of obstacles, and $n$ is the number of vertices. This bound is close to the known lower bound of $O(n+m\lg{m})$ time and $O(n)$ space. Our approach involve progressing shortest path wavefront as in continuous Dijkstra-type method, and confining its expansion to regions of interest.
Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ d... more Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $\Omega$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $\epsilon$. We utilize $\Omega$ to compute a $(1+\epsilon)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+\delta} + (\frac{1}{\epsilon}\lg{\frac{h}{\epsilon}})))$ time. Here, $\epsilon$ is a user parameter, and $\delta$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a $(2+\epsilon)$-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.
Network performance is a critical aspect of highperformance computers, and improving its performa... more Network performance is a critical aspect of highperformance computers, and improving its performance is a major goal in the design of future systems; this work proposes to improve network performance through new routing algorithms, leveraging the rich multi-path topologies of multi-dimensional torus networks commonly found in supercomputers built in the past fifteen years. Virtually all torus networks in production today utilize the dimension order routing algorithm, which is essentially a static and deterministic routing strategy to allow internode communication. This static routing strategy has significant load balancing implications, leading to sub-par performance. We propose a new Random Distance Routing algorithm, which randomly distributes packets to different neighboring nodes that are closer to the destination, leading to global load balanced network. Through the CODES/ROSS [4] simulator, we show that the proposed randomized multi-path routing algorithm can increases through...
In a typical optimization problem in a network, the goal is to optimize an incurred cost, while m... more In a typical optimization problem in a network, the goal is to optimize an incurred cost, while meeting a structural requirement. For example, in steiner tree problem, the structural requirement is to connect all the terminals and the incurred cost is the sum of the cost of all the edges included in the solution. However, we sometimes have to optimize two different measures. This kind of optimization is called Bicriteria Optimization. In order to efficiently solve such problems we use Approximation Algorithms that are guaranteed to provide a solution close to the optimum within a certain bound. We consider optimization problems where, apart from the structural requirement, an additional constraint has been placed. This additional constraint is in the form of a budget on a secondary cost. We consider four sets of such problems in this thesis. The first is the Bounded Diameter Minimum Cost Steiner Tree (BDMST) problem, and its various special cases. It is NP-hard to even approximate t...
This paper describes a quantum algorithm for finding the maximum among N items. The classical met... more This paper describes a quantum algorithm for finding the maximum among N items. The classical method for the same problem takes O(N) steps because we need to compare two numbers in one step. This algorithm takes O(sqrt(N)) steps by exploiting the property of quantum states to exist in a superposition of states and hence performing an operation on a number of elements in one go. A tight upper bound of 6.8(sqrt(N)) for the number of steps needed using this algorithm was found. These steps are the number of queries made to the oracle.
In this draft we consider dynamic multipath routing and introduce two methods that use additive i... more In this draft we consider dynamic multipath routing and introduce two methods that use additive increase and multiplicative decrease for flow control, similar to TCP. Our first method allows for congestion control and re-routing flows as users join in or leave the network. As the number of applications and services supported by the Internet grows, bandwidth requirements increase dramatically so it is imperative to design methods to ensure not only that network throughput is maximized but also to ensure a level of fairness in network resource allocation. Our second method provides fairness over multiple streams of traffic. We drive the multiplicative decrease part of the algorithm with link queue occupancy data provided by an enhanced routing protocol.
The spread of the COVID-19 virus has had an enormous impact on the world’s health and socioeconom... more The spread of the COVID-19 virus has had an enormous impact on the world’s health and socioeconomic system. While lockdowns, which severely limit the movement of the population, have been implemented in March 2020 and again recently, the psychological and economic cost are severe. Removal of these restrictions occurred with varying degrees of success.To study the resurgence of the virus in some communities we consider an epidemiological model, SIR-SD-L, that incorporates introduction of new population due to the removal of lockdown and identify parameters that impact the spread of the virus. This compartmental model of the epidemic incorporates a social distance metric based on progression of the infections; it models the dynamic propensity of infection spread based on the current infections relative to the susceptible population. The model is validated using data on growth of infections, hospitalizations and death, considering 24 counties in multiple US states and a categorization ...
The COVID-19 Pandemic has impacted the world's socio-economic system, resulting in a serious ... more The COVID-19 Pandemic has impacted the world's socio-economic system, resulting in a serious health crisis and lockdowns around the world. As the growth of infection slows, decisions regarding easing of lockdown restrictions are required, keeping in view the capacity of the health-care system. In this paper, we quantify the impact of a multi-phased release of the population from lockdown. Using the SIR model for epidemic spread, we design and implement a method to determine the earliest time of release from lockdown restrictions, constrained by a specified threshold on the subsequent peaks of infection. Trade-offs between the threshold and the earliest times of removing lockdown restrictions are illustrated. Additionally, we consider alternative policy decisions where the population is released gradually from lockdown restrictions and illustrate the trade-offs between the rate of release of population and number of active infections.
Given a set P of non-intersecting polygonal obstacles in R^2 defined with n vertices, we compute ... more Given a set P of non-intersecting polygonal obstacles in R^2 defined with n vertices, we compute a sketch Ω of P whose size is independent of n. We utilize Ω to devise an algorithm to compute a (1+ϵ)-approximate Euclidean shortest path between two points given with the description of P. When P comprises of convex polygonal obstacles, we devise a (2+ϵ)-approximation algorithm to efficiently answer two-point Euclidean distance queries.
In this paper we study the problem of market equilibrium. Firstly we consider the model (Fischer ... more In this paper we study the problem of market equilibrium. Firstly we consider the model (Fischer model) where there is a supply of money associated with each buyer and a quantity associated with each item. The market equilibrium problem is to compute a price vector which ensures market clearing, i.e. the demand of a good equals its supply, and subject to his endowment, each buyer maximizes his utility. Each item has a utility for a given buyer. We assume that the utility function is an increasing, differentiable, concave function. We show that under the assumptions of gross substitutibility, an auction algorithm can determine approximate market clearing. Not only does this algorithm extends the class of utility functions for which market equilibrium can be determined polynomially, the auction algorithm is efficient and in this model O(n) faster than the auction algorithm for the case of linear utilities in the Arrow-Debreu model. We outline an extension of our method to the Arrow-De...
Abstract—In this paper we address the issue of designing multipath routing algorithms. Multi-path... more Abstract—In this paper we address the issue of designing multipath routing algorithms. Multi-path routing has the potential of improving the throughput but requires buffers at the destination. Our model assumes a network with capacitated edges and a delay function associated with the network links (edges). We consider the problem of establishing a specified throughput from multiple source to destination pairs in the network, given bounds on the buffer sizes available at the network intermediate nodes and a bound on the maximum delay that the paths are allowed to have. We formulate the problem using a linear programming model. We also present practical heuristics and present the experimental results on an existing network topology. The results are promising and demonstrate the effectiveness of simultaneous multipaths in the multi-commodity scenario. I.
Many interactive 3D graphics applications like manufacturing, design, networked-gaming, need to a... more Many interactive 3D graphics applications like manufacturing, design, networked-gaming, need to access 3D data remotely. Transfer of large data sets consumes lot of bandwidth and space. In this paper we describe a lossless scheme for compressing 3D-surface data, represented by triangular meshes. We present a new algorithm which exploits geometry (vertex) information for encoding connectivity. The experimental results obtained are better than Edgebreaker, the existing best method.
We devise a polynomial-time approximation scheme for the classical geometric problem of finding a... more We devise a polynomial-time approximation scheme for the classical geometric problem of finding an approximate short path amid weighted regions. In this problem, a triangulated region P comprising of n vertices, a positive weight associated with each triangle, and two points s and t that belong to P are given as the input. The objective is to find a path whose cost is at most (1+epsilon)OPT where OPT is the cost of an optimal path between s and t. Our algorithm initiates a discretized-Dijkstra wavefront from source s and progresses the wavefront till it strikes t. This result is about a cubic factor (in n) improvement over the Mitchell and Papadimitriou '91 result, which is the only known polynomial time algorithm for this problem to date. Further, with polynomial time preprocessing of P, a set of data structures are computed which allow answering approximate weighted shortest path queries in polynomial time.
In this paper we investigate Object Oriented Binary Space Partitioning. The Binary Space Partitio... more In this paper we investigate Object Oriented Binary Space Partitioning. The Binary Space Partition Tree (BSP-Trees) is a widely used and e ective data structure for solid modeling and hidden surface removal. We present algorithms for eÆciently constructing Object BSP Trees(OBSP) in 2-Dimensions. The term object used arises since the construction of our tree utilizes the property of hierarchical ellipsoid covers. Besides being object-oriented, the major advantage of the OBSP trees is the reduction in the fragmentation of the scene. This paper formulates algorithms and data-structures for representing polyhedral objects and operations such as composition, deletion and motion of objects based on Object Hierarchy using the modied version of the traditional BSP tree. We improve on the design presented in [11]. Implementation details are presented which show increased eÆciency resulting through our approach.
1. ABSTRACT Torus Networks are an important and widely used architecture in modern supercomputers... more 1. ABSTRACT Torus Networks are an important and widely used architecture in modern supercomputers. However, these supercomputers frequently employ static routing protocols, which have poor worst case performance under heavy network traffic. In this paper, we first characterize bottlenecking behavior under the current standard protocol, then demonstrate that even simplistic dynamic routing protocols offer lower, more scalable latencies over a 3D torus network. As supercomputing moves towards exascale, such scalability will be necessary to cope with unprecedented workloads. These results were generated using the CODES highly parallel simulator, built on top of the ROSS parallel discrete event simulator.
The crisis induced by the Coronavirus pandemic has severely impacted educational institutes. Even... more The crisis induced by the Coronavirus pandemic has severely impacted educational institutes. Even with vaccination efforts underway, it is not clear that sufficient confidence will be achieved for schools to reopen soon. This paper considers the impact of vaccination rates and testing rates to reduce infections and hospitalizations and evaluates strategies that will allow educational institute in urban settings to reopen. These strategies are also applicable to businesses and would help plan reopening in order to help the economy. Our analysis is based on a graph model where nodes represent population groups and edges represent population exchanges due to commuting populations. The commuting population is associated with edges and is associated with one of the end nodes of the edge during part of the time period and with the other node during the remainder of the time period. The progression of the disease at each node is determined via compartment models, that include vaccination r...
Consider a transportation problem with sets of sources and sinks. There are profits and prices on... more Consider a transportation problem with sets of sources and sinks. There are profits and prices on the edges. The goal is to maximize the profit while meeting the following constraints; the total flow going out of a source must not exceed its capacity and the total price of the incoming flow on a sink must not exceed its budget. This problem is closely related to the generalized flow problem. We propose an auction based primal dual approximation algorithm to solve the problem. The complexity is $O(\epsilon^{-1}(n^2+ n\log{m})m\log U)$ where $n$ is the number of sources, $m$ is the number of sinks, $U$ is the ratio of the maximum profit/price to the minimum profit/price. We also show how to generalize the scheme to solve a more general version of the problem, where there are edge capacities and/or the profit function is concave and piece-wise linear. The complexity of the algorithm depends on the number of linear segments, termed ${\cal L}$, of the profit function.
We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a si... more We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a sink vertex $t$ in the presence of obstacles in $\Re^2$. Our algorithm takes $O(T+m(\lg{m})(\lg{n}))$ time and $O(n)$ space. Here, $O(T)$ is the time to triangulate the polygonal region, $m$ is the number of obstacles, and $n$ is the number of vertices. This bound is close to the known lower bound of $O(n+m\lg{m})$ time and $O(n)$ space. Our approach involve progressing shortest path wavefront as in continuous Dijkstra-type method, and confining its expansion to regions of interest.
Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ d... more Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $\Omega$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $\epsilon$. We utilize $\Omega$ to compute a $(1+\epsilon)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+\delta} + (\frac{1}{\epsilon}\lg{\frac{h}{\epsilon}})))$ time. Here, $\epsilon$ is a user parameter, and $\delta$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a $(2+\epsilon)$-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.
Network performance is a critical aspect of highperformance computers, and improving its performa... more Network performance is a critical aspect of highperformance computers, and improving its performance is a major goal in the design of future systems; this work proposes to improve network performance through new routing algorithms, leveraging the rich multi-path topologies of multi-dimensional torus networks commonly found in supercomputers built in the past fifteen years. Virtually all torus networks in production today utilize the dimension order routing algorithm, which is essentially a static and deterministic routing strategy to allow internode communication. This static routing strategy has significant load balancing implications, leading to sub-par performance. We propose a new Random Distance Routing algorithm, which randomly distributes packets to different neighboring nodes that are closer to the destination, leading to global load balanced network. Through the CODES/ROSS [4] simulator, we show that the proposed randomized multi-path routing algorithm can increases through...
In a typical optimization problem in a network, the goal is to optimize an incurred cost, while m... more In a typical optimization problem in a network, the goal is to optimize an incurred cost, while meeting a structural requirement. For example, in steiner tree problem, the structural requirement is to connect all the terminals and the incurred cost is the sum of the cost of all the edges included in the solution. However, we sometimes have to optimize two different measures. This kind of optimization is called Bicriteria Optimization. In order to efficiently solve such problems we use Approximation Algorithms that are guaranteed to provide a solution close to the optimum within a certain bound. We consider optimization problems where, apart from the structural requirement, an additional constraint has been placed. This additional constraint is in the form of a budget on a secondary cost. We consider four sets of such problems in this thesis. The first is the Bounded Diameter Minimum Cost Steiner Tree (BDMST) problem, and its various special cases. It is NP-hard to even approximate t...
This paper describes a quantum algorithm for finding the maximum among N items. The classical met... more This paper describes a quantum algorithm for finding the maximum among N items. The classical method for the same problem takes O(N) steps because we need to compare two numbers in one step. This algorithm takes O(sqrt(N)) steps by exploiting the property of quantum states to exist in a superposition of states and hence performing an operation on a number of elements in one go. A tight upper bound of 6.8(sqrt(N)) for the number of steps needed using this algorithm was found. These steps are the number of queries made to the oracle.
In this draft we consider dynamic multipath routing and introduce two methods that use additive i... more In this draft we consider dynamic multipath routing and introduce two methods that use additive increase and multiplicative decrease for flow control, similar to TCP. Our first method allows for congestion control and re-routing flows as users join in or leave the network. As the number of applications and services supported by the Internet grows, bandwidth requirements increase dramatically so it is imperative to design methods to ensure not only that network throughput is maximized but also to ensure a level of fairness in network resource allocation. Our second method provides fairness over multiple streams of traffic. We drive the multiplicative decrease part of the algorithm with link queue occupancy data provided by an enhanced routing protocol.
The spread of the COVID-19 virus has had an enormous impact on the world’s health and socioeconom... more The spread of the COVID-19 virus has had an enormous impact on the world’s health and socioeconomic system. While lockdowns, which severely limit the movement of the population, have been implemented in March 2020 and again recently, the psychological and economic cost are severe. Removal of these restrictions occurred with varying degrees of success.To study the resurgence of the virus in some communities we consider an epidemiological model, SIR-SD-L, that incorporates introduction of new population due to the removal of lockdown and identify parameters that impact the spread of the virus. This compartmental model of the epidemic incorporates a social distance metric based on progression of the infections; it models the dynamic propensity of infection spread based on the current infections relative to the susceptible population. The model is validated using data on growth of infections, hospitalizations and death, considering 24 counties in multiple US states and a categorization ...
The COVID-19 Pandemic has impacted the world's socio-economic system, resulting in a serious ... more The COVID-19 Pandemic has impacted the world's socio-economic system, resulting in a serious health crisis and lockdowns around the world. As the growth of infection slows, decisions regarding easing of lockdown restrictions are required, keeping in view the capacity of the health-care system. In this paper, we quantify the impact of a multi-phased release of the population from lockdown. Using the SIR model for epidemic spread, we design and implement a method to determine the earliest time of release from lockdown restrictions, constrained by a specified threshold on the subsequent peaks of infection. Trade-offs between the threshold and the earliest times of removing lockdown restrictions are illustrated. Additionally, we consider alternative policy decisions where the population is released gradually from lockdown restrictions and illustrate the trade-offs between the rate of release of population and number of active infections.
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