arXiv subjects

Tadao Takaoka

Publications and source records attributed to Tadao Takaoka.

6 recordsLinked to original sources

arXiv2015

O(1) Time Generation of Adjacent Multiset Combinations

We solve the problem of designing an O(1) time algorithm for generating adjacent multiset combinations in a different approach from Walsh. By the word adjacent, we mean that two adjacent multiset combinations are different at two places by one in their vector forms. Previous O(1) time algorithms for multiset combinations generated non-adjacent multiset combinations. Our algorithm in this paper can be derived from a general framework of combinatorial Gray code, which we characterise to suit our need for combinations and multiset combinations. The central idea is a twisted lexico tree, which is obtained from the lexicographic tree for the given set of combinatorial objects by twisting branches depending on the parity of each node. An iterative algorithm which traverses this tree will generate the given set of combinatorial objects in constant time as well as with a fixed number of changes from the present combinatorial object to the next.

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arXiv2015

Multi-level Loop-less Algorithm for Multi-set Permutations

We present an algorithm that generates multiset permutations in O(1) time for each permutation, that is, by a loop-less algorithm with O(n) extra memory requirement. There already exist several such algorithms that generate multiset permutations in various orders. For multiset permutations, we combine two loop-less algorithms that are designed in the same principle of tree traversal. Our order of generation is different from any existing order, and the algorithm is simpler and faster than the previous ones. We also apply the new algorithm to parking functions.

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arXiv2013

Combining the Shortest Paths and the Bottleneck Paths Problems

We combine the well known Shortest Paths (SP) problem and the Bottleneck Paths (BP) problem to introduce a new problem called the Shortest Paths for All Flows (SP-AF) problem that has relevance in real life applications. We first solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem on directed graphs with unit edge costs in $O(mn)$ worst case time bound. We then present two algorithms to solve SSSP-AF on directed graphs with integer edge costs bounded by $c$ in $O(m^2 + nc)$ and $O(m^2 + mn\log{(\frac{c}{m})})$ time bounds. Finally we extend our algorithms for the SSSP-AF problem to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem in $O(m^{2}n + nc)$ and $O(m^{2}n + mn^{2}\log{(\frac{c}{mn})})$ time bounds. All algorithms presented in this paper are practical for implementation.

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arXiv2013

Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems

We introduce a new problem that combines the well known All Pairs Shortest Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to compute the shortest paths for all pairs of vertices for all possible flow amounts. We call this new problem the All Pairs Shortest Paths for All Flows (APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with unit edge costs and real edge capacities in $\tilde{O}(\sqrt{t}n^{(ω+9)/4}) = \tilde{O}(\sqrt{t}n^{2.843})$ time, where $n$ is the number of vertices, $t$ is the number of distinct edge capacities (flow amounts) and $O(n^ω) < O(n^{2.373})$ is the time taken to multiply two $n$-by-$n$ matrices over a ring. Secondly we extend the problem to graphs with positive integer edge costs and present an algorithm with $\tilde{O}(\sqrt{t}c^{(ω+5)/4}n^{(ω+9)/4}) = \tilde{O}(\sqrt{t}c^{1.843}n^{2.843})$ worst case time complexity, where $c$ is the upper bound on edge costs.

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arXiv2013

Efficient Graph Algorithms for Network Analysis

The GC problem is to identify a pre-determined number of center vertices such that the distances or costs from (or to) the centers to (or from) other vertices is minimized. The bottleneck of a path is the minimum capacity of edges on the path. The Bottleneck Paths (BP) problem is to compute the paths that give us the maximum bottleneck values between pairs of vertices. The Graph Bottleneck (GB) problem is to find the minimum bottleneck value out of bottleneck paths for all possible pairs of vertices. We give two similar algorithms that are based on binary search to solve the 1-center GC problem and the GB problem on directed graphs with unit edge costs. We achieve $\tilde{O}(n^{2.373})$ worst case time complexity for both the 1-center GC problem and the GB problem, where $n$ is the number of vertices in the graph. This is better than the straightforward methods of solving the two problems in $O(n^{2.575})$ and $O(n^{2.688})$ time bounds, respectively. We then combine the Bottleneck Paths (BP) problem with the well known Shortest Paths (SP) problem to compute the shortest paths for all possible flow values. We call this problem the Shortest Paths for All Flows (SP-AF) problem. We show that if the flow demand is uncertain, but between two consecutive capacity values, the unique shortest path can be computed to push that flow. If the uncertainty stretches over two intervals, we need to prepare two shortest paths to accommodate the uncertainty, etc. In introducing this new problem, we define a new semi-ring called the distance/flow semi-ring, and show that the well known algorithm by Floyd can be used over the distance/flow semi-ring to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem.

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arXiv2013

Some Extensions of the All Pairs Bottleneck Paths Problem

We extend the well known bottleneck paths problem in two directions for directed unweighted (unit edge cost) graphs with positive real edge capacities. Firstly we narrow the problem domain and compute the bottleneck of the entire network in $O(n^ω\log{n})$ time, where $O(n^ω)$ is the time taken to multiply two $n$-by-$n$ matrices over ring. Secondly we enlarge the domain and compute the shortest paths for all possible flow amounts. We present a combinatorial algorithm to solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem in $O(mn)$ worst case time, followed by an algorithm to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem in $O(\sqrt{d}n^{(ω+9)/4})$ time, where $d$ is the number of distinct edge capacities. We also discuss real life applications for these new problems.

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