We investigate algorithmic problems for two-dimensional pattern matching. We improve upon known complexity results by introducing fine-grained reductions that lead to conditional lower bounds and new algorithms that have modestly more efficient runtimes. In terms of conditional lower bounds, we establish triangle finding (i.e. 3-clique) and 4-clique hardness results for previously unexplored cases of the Four Corners Problem over larger than binary alphabets. In terms of runtime improvements, we introduce a √(log (n)) factor improvement for detecting if a matrix contains a square block whose four corners are all 1’s.
A transformation between a pair of binary words u and v of the same length and Hamming distance d is a sequence of words w_0=u, w_1, w_2, … , w_d=v , such that w_i differs from w_i+1 only in one position. The transformation is said to be f-free, for a given word f, if f is absent in all w_i , i.e. if f is not a factor of any w_i . A word f is good if there exists an f-free transformation between any pair of f-free words of same length. Good words have been completely characterized using combinatorial methods on words. A word f is a minimal absent word for w if f is an absent word in w but all its proper factors occur in w. We consider f-free transformations between pairs of words u and v for which f is a minimal absent word and study how the property of including the proper factors of f can be maintained in the intermediate words of the transformation.
In this paper, we study the descriptional complexity of several variants of the shuffle operation on regular languages. In the perfect shuffle, words of equal length strictly alternate their symbols. If the words have different lengths, the initial literal shuffle performs the perfect shuffle while both words have symbols and then appends the remaining suffix of the longest word. Lastly, the literal shuffle is an extension of the previous operation, in which the initial shuffle may start at an arbitrary position in one of the two words. We analyze the number of states sufficient and necessary for nondeterministic and deterministic automata to recognize the resulting languages. For nondeterministic finite automata, the cost of the simulation is polynomial. In the worst case, a deterministic finite automaton requires exponentially many states to recognize the literal shuffle of two languages given also by deterministic finite automata, whereas a deterministic 1-limited automaton—namely, a one-tape Turing machine in which each cell may be rewritten only upon its first visit—requires only polynomial many states.
We study rectangulations of an axis-aligned rectangle in the integer lattice and the associated notions of weak and strong equivalence, which give rise to mosaic floorplans and floorplans. Using the representation via HV-matrices, we describe how such matrices generate rectangulations and investigate the correspondence between matrices, associated HV-words, and floorplans. Our main objectives are twofold. First, given a diagonal HV-matrix M, we provide a constructive procedure to generate all floorplans related to M and determine their cardinality. Second, given a family ℳ of diagonal HV-matrices, we enumerate the floorplans generated by its elements. To obtain these results, we develop suitable generating trees that describe recursively the structure of floorplans induced by diagonal HV-matrices and some of their subfamilies. This approach leads to explicit formulas for their enumeration.
In this article, we investigate the relationship between the decidability of the MSO-theory of the linear representation and the decidability of the MSO-theory of the base-k tree representation of an infinite word. The base-k tree representation is a representation of the word by an infinite tree, inspired by the theory of automatic sequences. We show that decidability of MSO on the linear representation and on the base-k tree representation are independent properties. In particular, we show that if one fixes a recursive set of positions in an infinite word whose gaps sizes are increasing sufficiently fast, then any recursive infinite word can be placed on those positions and the remaining positions can be filled to obtain a recursive infinite word with a decidable MSO-theory.
