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**11 - 19**of**19**### Parallel Processing Letters, ❢c World Scientific Publishing Company SYSTEMATIC DERIVATION OF TREE CONTRACTION ALGORITHMS ∗

, 2004

"... While tree contraction algorithms play an important role in efficient tree computation in parallel, it is difficult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving efficient tree contraction algorithm ..."

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While tree contraction algorithms play an important role in efficient tree computation in parallel, it is difficult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving efficient tree contraction algorithms from recursive functions on trees. We identify a general recursive form that can be parallelized into efficient tree contraction algorithms, and present a derivation strategy for transforming general recursive functions to the parallelizable form. We illustrate our approach by deriving a novel parallel algorithm for the maximum connected-set sum problem on arbitrary trees, the tree-version of the well-known maximum segment sum problem.

### ABSTRACT

"... A set of vertices S in a graph G =(V�E) is called a dominating set of G if every vertex in the set (V n S) is adjacent tosomevertex in S. For arbitrary graphs, the problem of computing smallest dominating set is NP-complete [3]. A slightly more general version of this problem is called \mixed domina ..."

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A set of vertices S in a graph G =(V�E) is called a dominating set of G if every vertex in the set (V n S) is adjacent tosomevertex in S. For arbitrary graphs, the problem of computing smallest dominating set is NP-complete [3]. A slightly more general version of this problem is called \mixed domination &quot; problem [1]. In this paper we present new parallel NC algorithm to nd smallest mixed dominating set in trees. The algorithm is based on tree compression techniques that have been traditionally used to evaluate linear arithmetic expressions in parallel [2], [4], [5]. In this respect the paper generalizes the application of tree compression techniques to solve combinatorial problems in trees. The model of parallel computation used is the CRCW P-RAM (Concurrent Read Concurrent Write Parallel RAM), where more than one processor can concurrently read from or write into the same memory location during the same memory cycle. Writing con icts are resolved in a non-deterministic fashion. The algorithm requires O(n) processors and runs in O(log n) time on a CRCW P-RAM.

### TABLE OF CONTENTS

, 2003

"... Dynamic computational complexity is the study of resource-bounded ongoing compu-tational processes. We consider the general problem of processing a sequence of inputs, instead of a single input. We introduce a new model for dynamic computation, and inves-tigate the computational complexity of variou ..."

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Dynamic computational complexity is the study of resource-bounded ongoing compu-tational processes. We consider the general problem of processing a sequence of inputs, instead of a single input. We introduce a new model for dynamic computation, and inves-tigate the computational complexity of various dynamic problems. The field of computational complexity has previously studied static computation, which takes a single fixed input and computes the desired result. We define a dynamic problem to be the function mapping a stream of data to the desired stream of output, and we inves-tigate the complexity of the dynamic computation required to compute that function. We describe complexity classes of dynamic problems, reductions between dynamic problems, and complete problems for dynamic complexity classes.

### Mixed Domination in Trees: A Parallel Algorithm

"... A set of vertices S in a graph G = (V; E) is called a dominating set of G if every vertex in the set (V n S) is adjacent to some vertex in S. For arbitrary graphs, the problem of computing smallest dominating set is NP-complete [3]. A slightly more general version of this problem is called "mix ..."

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A set of vertices S in a graph G = (V; E) is called a dominating set of G if every vertex in the set (V n S) is adjacent to some vertex in S. For arbitrary graphs, the problem of computing smallest dominating set is NP-complete [3]. A slightly more general version of this problem is called "mixed domination" problem [1]. In this paper we present new parallel NC algorithm to find smallest mixed dominating set in trees. The algorithm is based on tree compression techniques that have been traditionally used to evaluate linear arithmetic expressions in parallel [2], [4], [5]. In this respect the paper generalizes the application of tree compression techniques to solve combinatorial problems in trees. The model of parallel computation used is the CRCW P-RAM (Concurrent Read Concurrent Write Parallel RAM), where more than one processor can concurrently read from or write into the same memory location during the same memory cycle. Writing conflicts are resolved in a non-deterministic fashion. The algorithm requires O(n) processors and runs in O(log n) time on a CRCW P-RAM. Keywords: NC algorithm, domination, tree, graph 1

### Optimal and Sublogarithmic Time

"... Abstract.We assume a parallel RAM model which allows both concurrent reads and concurrent writes of a global memory. Our main result is an optimal randomized parallel algorithm for INTE-GER SORT (i.e., for sorting n integers in the range [1,n]). Our algorithm costs only logarithmic time and is the f ..."

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Abstract.We assume a parallel RAM model which allows both concurrent reads and concurrent writes of a global memory. Our main result is an optimal randomized parallel algorithm for INTE-GER SORT (i.e., for sorting n integers in the range [1,n]). Our algorithm costs only logarithmic time and is the first known that is optimal: the product of its time and processor bounds is upper bounded by a linear function of the input size. We also give a deterministic sub-logarithmic time algorithm for prefix sum. In addition we present a sub-logarithmic time algorithm for obtaining a random permutation of n elements in parallel. And finally, we present sub-logarithmic time algorithms for GENERAL SORT and INTEGER SORT. Our sublogarithmic GENERAL SORT algorithm is also optimal.

### CERTIFICATE

, 2010

"... It is certified that the work contained in this thesis, titled “Exploring Irregular Memory Access Ap-plications on the GPU ” by Mohammed Suhail Rehman, has been carried out under our supervision and is not submitted elsewhere for a degree. Date Advisor: Dr. P. J. Narayanan Advisor: Dr. Kishore Kotha ..."

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It is certified that the work contained in this thesis, titled “Exploring Irregular Memory Access Ap-plications on the GPU ” by Mohammed Suhail Rehman, has been carried out under our supervision and is not submitted elsewhere for a degree. Date Advisor: Dr. P. J. Narayanan Advisor: Dr. Kishore Kothapalli To my parents, for supporting my (un)healthy obsession with computers ' / 1Æz/¿Î 1Ë 1 '
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/Ơ I ́ //»v,ÌÈ 1Ǯ 1 12 /Æ0 ¿ /)K0Ò0 Ü /) /ÂÜ/Ç1 IØ¨À0Û “High above all is God, the King, the Truth! Be not in haste with the Qur’an before its revelation to thee is completed, but say, “O my Lord! advance me in knowledge””

### Graph Invariants and Graph Isomorphism

"... Abstract — In graph theory, Graph Isomorphism is an important issue. Information in the database can be stored in the form of graph. Graph represents the structural information in an efficient way. Graph Isomorphism problem is to determine if there exists one to one correspondence between the struct ..."

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Abstract — In graph theory, Graph Isomorphism is an important issue. Information in the database can be stored in the form of graph. Graph represents the structural information in an efficient way. Graph Isomorphism problem is to determine if there exists one to one correspondence between the structures of two graphs. Graph isomorphism problem arises in many fields such as chemistry, switching theory, information retrieval, social networks, etc. Graph Invariants are used to determine the isomorphism between two graphs. Graph Invariant is only the necessary condition for graph Isomorphism. Most of the researchers believe that isomorphism problem is NP complete problem and the most difficult problem.Graph isomorphism is a NP complete or not is always a hot issue for the researchers to study. In this paper we discuss the graph invariants and graph isomorphism techniques.

### ADAPTIVE INFERENCE FOR GRAPHICAL MODELS

, 2012

"... Many algorithms and applications involve repeatedly solving a variation of the same statistical inference problem. Adaptive inference is a technique where the previous computations are leveraged to speed up the computations after modifying the model parameters. This approach is useful in situations ..."

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Many algorithms and applications involve repeatedly solving a variation of the same statistical inference problem. Adaptive inference is a technique where the previous computations are leveraged to speed up the computations after modifying the model parameters. This approach is useful in situations where a slow-to-compute statistical model needs to be re-run after some minor manual changes or in situations where the model is changing over time in minor ways; for example while studying the effects of mutations on proteins, one often constructs models that change slowly as mutations are introduced. Another important application of adaptive inference is in situations where the model is being used iteratively; for example in approximate inference we may want to decompose the problem into simpler inference subproblems that are solved repeatedly and iteratively using adaptive updates. In this thesis we explore both exact inference and iterative approximate inference approaches using adaptive updates. We rfist present algorithms for adaptive exact inference on general graphs that can be used to efficiently compute marginals and update MAP configurations under arbitrary changes to the input factor graph and its associated elimination tree. We then apply them to approximate inference using a framework called dual decomposition. The key