We discuss applications of quantum computation to geometric data processing. Especially, we give efficient algorithms for intersection problems and proximity problems. Our algorithms are based on Brassard et al. 's amplitude amplification method, and analogous to Buhrman et al. 's algorithm for element distinctness. Revealing these applications is useful for classifying geometric problems, and also emphasizing potential usefulness of quantum computation in geometric data processing. Thus, the results will promote research and development of quantum computers and algorithms.

Betweenness centrality is one of the most significant and commonly used centralities, where centrality is a notion of measuring the importance of nodes in networks. In 2001, Brandes proposed an algorithm for computing betweenness centrality efficiently, and it can compute those values for all nodes in O(nm) time for unweighted networks, where n and m denote the number of nodes and links in networks, respectively. However, even Brandes' algorithm is not fast enough for recent large-scale real-world networks, and therefore, much faster algorithms are expected. The objective of this research is to theoretically improve the efficiency of Brandes' algorithm by introducing graph decompositions, and to verify the practical effectiveness of our approaches by implementing them as computer programs and by applying them to various kinds of real-world networks. A series of computational experiments shows that our proposed algorithms run several times faster than the original Brandes' algorithm, which are guaranteed by theoretical analyses.

Two new algorithms for improving the speed of the LZ77 compression are proposed. One is based on a new hashing algorithm named two-level hashing that enables fast longest match searching from a sliding dictionary, and the other uses suffix sorting. The former is suitable for small dictionaries and it significantly improves the speed of gzip, which uses a naive hashing algorithm. The latter is suitable for large dictionaries which improve compression ratio for large files. We also experiment on the compression ratio and the speed of block sorting compression, which uses suffix sorting in its compression algorithm. The results show that the LZ77 using the two-level hash is suitable for small dictionaries, the LZ77 using suffix sorting is good for large dictionaries when fast decompression speed and efficient use of memory are necessary, and block sorting is good for large dictionaries.

In [5], the following pyramid construction problem was proposed: Given nonnegative valued functions ρ and µ in d variables, we consider the optimal pyramid maximizing the total parametric gain of ρ against µ. The pyramid can be considered as the optimal unimodal approximation of ρ relative to µ, and can be applied to hierarchical data segmentation. In this paper, we give efficient algorithms for a couple of two-dimensional pyramid construction problems.

When we search from a huge amount of documents, we often specify several keywords and use conjunctive queries to narrow the result of the search. Though the searched documents contain all keywords, positions of the keywords are usually not considered. As a result, the search result contains some meaningless documents. It is therefore effective to rank documents according to proximity of keywords in the documents. This ranking is regarded as a kind of text data mining. In this paper, we propose two algorithms for finding documents in which all given keywords appear in neighboring places. One is based on plane-sweep algorithm and the other is based on divide-and-conquer approach. Both algorithms run in O(n log n) time where n is the number of occurrences of given keywords. We run the algorithms on a large collection of html files and verify its effectiveness.

We present a new data structure called the packed compact trie (packed c-trie) which stores a set S of k strings of total length n in nlog σ+O(klog n) bits of space and supports fast pattern matching queries and updates, where σ is the alphabet size. Assume that α=logσn letters are packed in a single machine word on the standard word RAM model, and let f(k,n) denote the query and update times of the dynamic predecessor/successor data structure of our choice which stores k integers from universe [1,n] in O(klog n) bits of space. Then, given a string of length m, our packed c-tries support pattern matching queries and insert/delete operations in $O(rac{m}{alpha} f(k,n))$ worst-case time and in $O(rac{m}{alpha} + f(k,n))$ expected time. Our experiments show that our packed c-tries are faster than the standard compact tries (a.k.a. Patricia trees) on real data sets. We also discuss applications of our packed c-tries.