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Revised version May 2021; We consider the problem of querying the existence of hyperedges in hypergraphs. More formally, we are given a hypergraph, and we need to answer queries of the form "does the following set of vertices form a hyperedge in the given hypergraph?''. Our aim is to set up data structures based on hashing to answer these queries as fast as possible. We propose an adaptation of a well-known perfect hashing approach for the problem at hand.We analyze the space and run time complexity of the proposed approach, and experimentally compare it with the state of the art hashing-based solutions. Experiments demonstrate that the proposed approach has the shortest query response time than the other considered alternatives, while having a larger construction time than some of the alternatives.; Nous considérons le problème de requêtes d'existences d'hyperarêtes dans des hypergraphes. Plus formellement, pour un hypergraphe donné, nous devons répondre à des requêtes de la forme "est-ce que l'ensemble de sommets suivant forme une hyperarête de l'hypergraphe?". Notre objectif est de mettre en place une structure de donnée basée sur du hachage pour répondre à ces requêtes le plus rapidement possible. Nous proposons une adaptation d'une approche bien connue de hachage parfait pour notre problème. Nous analysons la complexité en temps et en espace de cette approche, et nous la comparons expérimentalement à des solutions de l'état de l'art basées sur le hachage. Les expériences démontrent que cette approche a un temps de réponse aux requêtes plus court que les alternatives considérées, pour un temps de construction plus long que certaines des alternatives.

This HDR investigates three classes of problems at the interplay of discrete algorithms, combinatorial optimization, and numerical methods. The three problem classes are that of partitioning, matching, and ordering.Partitioning problems arise in task decomposition for parallel computing, where load balance and low communication cost are two objectives. We discuss acyclic partitioning of directed acyclic graphs and hypergraphs. While our contributions for the former problem concern combinatorial tools for the desired partitioning objectives and constraints, those for the second problem concern the use of hypergraph partitioning tools for efficient tensor decomposition in distributed memory systems.Matching problems arise in settings where agents compete for exclusive access to resources. We present approximation algorithms for matchings in graphs and effective heuristics for finding matchings in hypergraphs. We also investigate the problem of finding Birkhoff--von Neumann decompositions with a small number of permutation matrices and present complexity results and theoretical insights into this problem. Ordering problems arise when one wants to permute sparse matrices and tensors into desirable forms. We propose heuristics to permute sparse matrices into special forms to reduce the height of the resulting elimination tree. We also propose heuristics to cluster nonzeros of a given sparse tensor around the diagonal in order to improve the performance of certain tensor operations.The general research area is called combinatorial scientific computing (CSC). In CSC, the contributions have practical and theoretical flavor. For all problems discussed in this document, we have the design, analysis, and implementation of algorithms along with many experiments. The theoretical results are included in this document, some with proofs; the reader is invited to the original papers for the omitted proofs. A similar approach is taken for presenting the experiments. While most results for observing theoretical findings in practice are included, the reader is referred to the original papers for some other results (e.g., run time analysis).

IV ÖZET SEYREK DİKDÖRTGENSEL MATRİSLERİN AATx'W PARALEL İŞLEMCİLERDE HESAPLANABİLMESİ İÇİN PARÇALANMASI Bora Uçar Bilgisayar ve Enformatik Mühendisliği, Yüksek Lisans Tez Yöneticisi: Doc. Dr. Cevdet Aykanat Eylül, 1999 Birçok bilimsel uygulama, bir seyrek dikdörtgensel matrisin bir vektör ve de aynı matrisin transpozunun bir başka vektör ile çarpılmasını içermektedir. Şimdiye kadar kullanılan çizge ve hiperçizge parçalanması algoritmalarında toplam haberleşme hacmi azaltılırken işlemciler arasındaki işlemsel denge, ma trisin tek boyutlu ayrıştırılması ile yakalanmaya çalışılmıştır. Bu tezde, toplam haberleşme sayısı ve hacmi iki yaklaşımla azaltılmaya çalışılmıştır. Genel haberleşme yaklaşımındaki birleşik haberleşme hacminin azaltılması problemi, matrislerin tek sınırlı çarpraz bloklar haline getirilmesi problemine dönüştü rülmüştür. Bu yaklaşımdaki birleşik ve toplam haberleşme sayısı paralel iş lemcilerin bağlantısına bağlı olup değiştirilememektedir. Kişiselleştirilmiş ha berleşme yaklaşımındaki toplam haberleşme sayısı ve hacmi iki aşamada azal tılmıştır. Birinci aşamada, toplam haberleşme hacmi azaltılırken işlemsel denge sağlanılmıştır. ikinci aşamada, önerilen haberleşme hiperçizgesi yardımı ile toplam haberleşme sayısı azaltılırken işlemciler arasında haberleşme işi dengesi sağlanmaya çalışılmıştır. Sunulan yöntemler birçok matris üzerinde sınanmış ve iyi yönde sonuçlar elde edilmiştir. Anahtar kelimeler: Seyrek dikdörtgensel matrisler, hesap hiperçizgesi, haber leşme hiperçizgesi, hiperçizge parçalama. Ill ABSTRACT PARTITIONING SPARSE RECTANGULAR MATRICES FOR PARALLEL COMPUTING OF AATx Bora Uçar M.S. in Computer Engineering and Information Science Supervisor: Asoc. Prof. Dr. Cevdet Aykanat Spetember, 1999 Many scientific applications involve repeated sparse matrix- vector and matrix- transpose- vector product computations. Graph and hypergraph partitioning based approaches used in the literature aim at minimizing the total commu nication volume while maintaining computational load balance through one dimensional partitioning of sparse matrices. In this thesis, we consider two approaches which consider minimizing both the total message count and com munication volume while maintaining balance on the communication loads of the processors. Two communication schemes are investigated for the fold and expand operations needed in the parallel algorithm. For the global communica tion scheme, we show that the problem of minimizing concurrent communica tion volume can be formulated as the problem of permuting the sparse matrix into a singly-bordered block-diagonal form, where the total and concurrent message count is determined by the interconnection topology. For the person alized communication scheme, a two stage approach is proposed. In the first stage, the total communication volume is minimized while maintaining balance on the computational loads of the processors. In the second stage, a novel com munication hypergraph model is proposed which enables the minimization of the total message count while maintaining balance on the communication loads of the processors through hypergraph-partitioning-like methods. The solution methods are tested on various matrices and results, which are quite attractive in terms of solution quality and running times, are obtained. Key words: Sparse Rectangular Matrices, Computational Hypergraph Model, Communication Hypergraph Model, Hypergraph Partitioning. 74