ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.
In addition to the electronic theses found in the ASU Digital Repository, ASU Theses and Dissertations can be found in the ASU Library Catalog.
Dissertations and Theses granted by Arizona State University are archived and made available through a joint effort of the ASU Graduate College and the ASU Libraries. For more information or questions about this collection contact or visit the Digital Repository ETD Library Guide or contact the ASU Graduate College at gradformat@asu.edu.
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The two major issues concerning covering arrays are explicit construction of a covering array, and exact or approximate determination of the covering array number---the minimum size of a covering array. Although these problems have been investigated extensively for the last couple of decades, in this thesis we present significant improvements on both of these questions using tools from the probabilistic method and randomized algorithms.
First, a series of improvements is developed on the previously known upper bounds on covering array numbers. An estimate for the discrete Stein-Lovász-Johnson bound is derived and the Stein- Lovász -Johnson bound is improved upon using an alteration strategy. Then group actions on the set of symbols are explored to establish two asymptotic upper bounds on covering array numbers that are tighter than any of the presently known bounds.
Second, an algorithmic paradigm, called the two-stage framework, is introduced for covering array construction. A number of concrete algorithms from this framework are analyzed, and it is shown that they outperform current methods in the range of parameter values that are of practical relevance. In some cases, a reduction in the number of tests by more than 50% is achieved.
Third, the Lovász local lemma is applied on covering perfect hash families to obtain an upper bound on covering array numbers that is tightest of all known bounds. This bound leads to a Moser-Tardos type algorithm that employs linear algebraic computation over finite fields to construct covering arrays. In some cases, this algorithm outperforms currently used methods by more than an 80% margin.
Finally, partial covering arrays are introduced to investigate a few practically relevant relaxations of the covering requirement. Using probabilistic methods, bounds are obtained on partial covering arrays that are significantly smaller than for covering arrays. Also, randomized algorithms are provided that construct such arrays in expected polynomial time.
This dissertation defines a partially-ordered set on equivalence classes of certain saturated chains of T(n) called the Tamari Block poset, TB(lambda). It further proves TB(lambda) is a graded lattice. It then shows for lambda = (n-1,...,2,1) TB(lambda) is anti-isomorphic to the Higher Stasheff-Tamari orders in dimension 3 on n+2 elements. It also investigates enumeration questions involving TB(lambda), and proves other structural results along the way.
A few years ago, Bergeron and Préville-Ratelle generalized the Tamari lattices to the m-Tamari lattices. The original Tamari lattices T(n) are the case m=1. I establish a bijection between maximum length chains in the m-Tamari lattices and standard m-shifted Young tableaux. Using Thrall’s formula, I thus derive the formula for the number of maximum length chains in T(n).
For each i greater or equal to -1 and for all n greater or equal to 1, I define C(i,n) to be the set of maximal chains of length n+i in T(n). I establish several properties of maximal chains (treated as tableaux) and identify a particularly special property: each maximal chain may or may not possess a plus-full-set. I show, surprisingly, that for all n greater or equal to 2i+4, each member of C(i,n) contains a plus-full-set. Utilizing this fact and a collection of maps, I obtain a recursion for the number of elements in C(i,n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in T(n) is n choose 3.
I discuss current work and future plans involving certain equivalence classes of maximal chains in the Tamari lattices. If a maximal chain may be obtained from another by swapping a pair of consecutive edges with another pair in the Hasse diagram, the two maximal chains are said to differ by a square move. Two maximal chains are said to be in the same equivalence class if one may be obtained from the other by making a set of square moves.