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Probabilistic Methods in Discrete Applied Mathematics
Fink, Jiří ; Loebl, Martin (advisor) ; Koubek, Václav (referee) ; Sereni, Jean-Sébastein (referee)
One of the basic streams of modern statistical physics is an effort to understand the frustration and chaos. The basic model to study these phenomena is the finite dimensional Edwards-Anderson Ising model. We present a generalization of this model. We study set systems which are closed under symmetric differences. We show that the important question whether a groundstate in Ising model is unique can be studied in these set systems. Kreweras' conjecture asserts that any perfect matching of the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle. We prove this conjecture. The {\it matching graph} $\mg{G}$ of a graph $G$ has a vertex set of all perfect matchings of $G$, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph $\mg{Q_n}$ is bipartite and connected for $n \ge 4$. This proves Kreweras' conjecture that the graph $M_n$ is connected, where $M_n$ is obtained from $\mg{Q_n}$ by contracting all vertices of $\mg{Q_n}$ which correspond to isomorphic perfect matchings. A fault-free path in $Q_n$ with $f$ faulty vertices is said to be \emph{long} if it has length at least $2^n-2f-2$. Similarly, a fault-free cycle in $Q_n$ is long if it has length at least $2^n-2f$. If all faulty vertices are...
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Applications of Gray codes in cache-oblivious algorithms
Mička, Ondřej ; Fink, Jiří (advisor) ; Gregor, Petr (referee)
Modern computers employ a sophisticated hierarchy of caches to decrease the latency of memory accesses. This led to the development of cache-oblivious algorithms that strive to achieve the best possible performance on such memory hierarchies with minimal knowledge of the exact parameters of the hierarchy. A common technique used in the design of cache-oblivious algorithms is a recursion-based divide-and-conquer method. In this work, we show an alternative technique based on the Gray codes. We use the binary reflected Gray code to traverse arrays in the cache-oblivious way, allowing us to design algorithms for problems such as matrix transposition, naive matrix multiplication or naive convolution that match the asymptotic performance of their recursion-based counterparts. The advantage is that our algorithms can be implemented without recursion (or a stack that simulates it) by using a loopless algorithm. We also introduce a variant of the binary reflected Gray code tuned to certain applications of our technique and an almost loopless algorithm to generate it. Apart from the theoretical analysis of our technique's performance, we also examine its practical performance on the problem of matrix transposition.
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Probabilistic Methods in Discrete Applied Mathematics
Fink, Jiří ; Loebl, Martin (advisor) ; Koubek, Václav (referee) ; Sereni, Jean-Sébastein (referee)
One of the basic streams of modern statistical physics is an effort to understand the frustration and chaos. The basic model to study these phenomena is the finite dimensional Edwards-Anderson Ising model. We present a generalization of this model. We study set systems which are closed under symmetric differences. We show that the important question whether a groundstate in Ising model is unique can be studied in these set systems. Kreweras' conjecture asserts that any perfect matching of the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle. We prove this conjecture. The {\it matching graph} $\mg{G}$ of a graph $G$ has a vertex set of all perfect matchings of $G$, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph $\mg{Q_n}$ is bipartite and connected for $n \ge 4$. This proves Kreweras' conjecture that the graph $M_n$ is connected, where $M_n$ is obtained from $\mg{Q_n}$ by contracting all vertices of $\mg{Q_n}$ which correspond to isomorphic perfect matchings. A fault-free path in $Q_n$ with $f$ faulty vertices is said to be \emph{long} if it has length at least $2^n-2f-2$. Similarly, a fault-free cycle in $Q_n$ is long if it has length at least $2^n-2f$. If all faulty vertices are...
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