Masashi Kiyomi

LCM ver.3: collaboration of array, bitmap and prefix tree for frequent itemset mining

For a transaction database, a frequent itemset is an itemset included in at least a specified number of transactions. To find all the frequent itemsets, the heaviest task is the computation of frequency of each candidate itemset. In the previous studies, there are roughly three data structures and algorithms for the computation: bitmap, prefix tree, and array lists. Each of these has its own advantage and disadvantage with respect to the density of the input database. In this paper, we propose an efficient way to combine these three data structures so that in any case the combination gives the best performance.

Swapping labeled tokens on graphs

Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to swap the tokens on adjacent vertices. We prove that every such puzzle is solvable in O ( n 2 ) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement. We give a polynomial-time 2-approximation algorithm for trees, and using this, obtain a polynomial-time 2α-approximation algorithm for graphs whose tree α-spanners can be computed in polynomial time. Finally, we show that the problem can be solved exactly in polynomial time on complete bipartite graphs.

Swapping Labeled Tokens on Graphs

Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to swap the tokens on adjacent vertices. We prove that every such puzzle is solvable in O(n 2) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement. We give a polynomial-time 2-approximation algorithm for trees, and using this, obtain a polynomial-time 2α-approximation algorithm for graphs whose tree α-spanners can be computed in polynomial time. Finally, we show that the problem can be solved exactly in polynomial time on complete bipartite graphs.

Depth-First Search Using O(n) Bits

We provide algorithms performing Depth-First Search (DFS) on a directed or undirected graph with \(n\) vertices and \(m\) edges using only \(O(n)\) bits. One algorithm uses \(O(n)\) bits and runs in \(O(m \log n)\) time. Another algorithm uses \(n+o(n)\) bits and runs in polynomial time. Furthermore, we show that DFS on a directed acyclic graph can be done in space \(n/2^{\varOmega (\sqrt{\log n})}\) and in polynomial time, and we also give a simple linear-time \(O(\log n)\)-space algorithm for the depth-first traversal of an undirected tree. Finally, we also show that for a graph having an \(O(1)\)-size feedback set, DFS can be done in \(O(\log n)\) space. Our algorithms are based on the analysis of properties of DFS and applications of the \(s\)-\(t\) connectivity algorithms due to Reingold and Barnes et al., both of which run in sublinear space.

Algorithmic Folding Complexity

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is O(lg2n). We also show that a lower bound of the complexity of the problems is Ω(lg2n/lg lg n). Next we show that almost all mountain-valley patterns require Ω(n/lg n) folds, which means that the uniform and pleat foldings are relatively easy problems. We also give a general algorithm for folding an arbitrary sequence of length n in O(n/lg n) folds, meeting the lower bound up to a constant factor.

On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints

We discuss the problems to list, sample, and count the chordal graphs with edge constraints. The objects we look at are chordal graphs sandwiched by a given pair of graphs where we assume at least one of the input pair is chordal. The setting is a natural generalization of chordal completions and deletions. For the listing problem, we give an efficient algorithm running in polynomial time per output with polynomial space. As for the sampling problem, we give two clues that seem to imply that a random sampling is not easy. The first clue is that we show #P-completeness results for counting problems. The second clue is that we give an instance for which a natural Markov chain suffers from an exponential mixing time. These results provide a unified viewpoint from algorithms theory to problems arising from various areas such as statistics, data mining, and numerical computation.

Generating Chordal Graphs Included in Given Graphs

A chordal graph is a graph which contains no chordless cycle of at least four edges as an induced subgraph. The class of chordal graphs contains many famous graph classes such as trees, interval graphs, and split graphs, and is also a subclass of perfect graphs. In this paper, we address the problem of enumerating all labeled chordal graphs included in a given graph. We think of some variations of this problem. First we introduce an algorithm to enumerate all connected labeled chordal graphs in a complete graph of n vertices. Next, we extend the algorithm to an algorithm to enumerate all labeled chordal graphs in a n-vertices complete graph. Then, we show that we can use, with small changes, these algorithms to generate all (connected or not necessarily connected) labeled chordal graphs in arbitrary graph. All our algorithms are based on reverse search method, and time complexities to generate a chordal graph are O(1), and also O(1) delay. Additionally, we present an algorithm to generate every clique of a given chordal graph in constant time. Using these algorithms we obtain combinatorial Gray code like sequences for these graph structures in which the differences between two consecutive graphs are bounded by a constant size.

Reconstruction of interval graphs

The graph reconstruction conjecture is a long-standing open problem in graph theory. There are many algorithmic studies related to it, such as DECK CHECKING, LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING. We study these algorithmic problems by limiting the graph class to interval graphs. Since we can solve GRAPH ISOMORPHISM for interval graphs in polynomial time, DECK CHECKING for interval graphs is easily done in polynomial time. Since the number of interval graphs that can be obtained from an interval graph by adding a vertex and edges incident to it can be exponentially large, developing polynomial time algorithms for LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING on interval graphs is not trivial. We present that these three problems are solvable in polynomial time on interval graphs.

Random Generation and Enumeration of Proper Interval Graphs

We investigate connected proper interval graphs without vertex labels. We first give the number of connected proper interval graphs of n vertices. Using it, a simple algorithm that generates a connected proper interval graph uniformly at random up to isomorphism is presented. Finally an enumeration algorithm of connected proper interval graphs is proposed. The algorithm is based on the reverse search, and it outputs each connected proper interval graph in

Reconstruction of Interval Graphs

\mbox{\cal O}(1)