We consider an edge-weighted graph G with a designated vertex v0 such that weights of edges incident to v0 may increase or decrease. We show that, with an O(mn+n2log n) time preprocessing, a minimum cut of the current G can be computed in O(log n) time per update of weight of any edge {v0,u}.

We consider a problem for constructing a minimum cost r-edge-connected multigraph in which degree d(v) of each vertex v ∈ V is specified. In this paper, we propose a 3-approximation algorithm for this problem under the assumption that edge cost is metric, r(u,v) ∈ {1,2} for each u,v ∈ V, and d(v) ≥ 2 for each v ∈ V. This problem is a generalization of metric TSP. We also propose an approximation algorithm for the digraph version of the problem.

In this paper, we give an algorithm for the node-to-set disjoint paths problem in a transposition graph. The algorithm is of polynomial order of n for an n-transposition graph. It is based on recursion and divided into two cases according to the distribution of destination nodes. The maximum length of each path and the time complexity of the algorithm are estimated theoretically to be O(n7) and 3n - 5, respectively, and the average performance is evaluated based on computer experiments.

A new edge-coloring algorithm for bipartite graphs is presented. This algorithm, based on the framework of the O(m log d + (m/d) log (m/d) log d) algorithm by Makino-Takabatake-Fujishige and the O(m log m) one by Alon, finds an optimal edge-coloring of a bipartite graph with m edges and maximum degree d in O(m log d + (m/d) log (m/d)) time. This algorithm does not require elaborate data structures, which the best known O(m log d) algorithm due to Cole-Ost-Schirra depends on.

Let G = (V,E) be a connected graph such that each edge e ∈ E and each vertex v ∈ V are weighted by nonnegative reals w(e) and h(v), respectively. Let r be a vertex designated as a root, and p be a positive integer. The minmax rooted-subtree cover problem (MRSC) asks to find a partition X = {X1,X2,...,Xp of V and a set of p subtrees T1,T2,...,Tp such that each Ti contains Xi∪{r} so as to minimize the maximum cost of the subtrees, where the cost of Ti is defined to be the sum of the weights of edges in Ti and the weights of vertices in Xi. Similarly, the minmax rooted-cycle cover problem (MRCC) asks to find a partition X = {X1,X2,...,Xp} of V and a set of p cycles C1,C2,...,Cp such that Ci contains Xi∪{r} so as to minimize the maximum cost of the cycles, where the cost of Ci is defined analogously with the MRSC. In this paper, we first propose a (3-2/(p+1))-approximation algorithm to the MRSC with a general graph G, and we then give a (6-4/(p+1))-approximation algorithm to the MRCC with a metric (G,w).

"Can one walk to infinity on Gaussian primes taking steps of bounded length?" We adopted computational techniques to probe into this open problem. We propose an efficient method to search for the farthest point reachable from the origin, which can be parallelized easily, and have confirmed the existence of a moat of width k =, whereas the best previous result was k = due to Gethner et al. The amount of computation needed for k = is about 5000 times larger than that for k =. A refinement of Vardi's estimate for the farthest distance reachable from the origin is proposed. The proposed estimate incorporates discreteness into Vardi's that is based on percolation theory.

Given a graph G, we give a fast algorithm for approximating the vertex connectivity κ of G. Our algorithm delivers a minimum vertex cut of G if κ δ/2, and returns a message "κ > δ/2" otherwise, where δ denotes the minimum degree of G. The algorithm runs in O(n2(1 + min {κ2, κ/δ)) time and O(n + m) space, where n and m denote the numbers of vertices and edges in G, respectively.

Subgraph isomorphism problems have various important applications, while generally being NP-complete. Though Ullmann and Konishi proposed the custom circuit designs to accelerate subgraph isomorphism problem, they require many hardware resources for large problems. This study describes the design of data-dependent circuits for subgraph isomorphism problem with evaluation results on an actual FPGA platform. Data-dependent circuits are logic circuits specialized in specific input data. Such circuits are smaller and faster than the original circuit, although it is not reusable and involves circuit generation for each input. In the present study, the circuits were implemented on Xilinx XC2V3000 FPGA, and they successfully operated at a clock frequency 25 MHz. In the case of graphs with 16 vertices, the average execution time is about 7.0% of the software executed on an up-to-date microprocessor (Athlon XP 2600+ of 2.1 GHz clock). Even if the circuit generation time is included, data-dependent circuits are about 14.4 times faster than the software (for random graphs with 16 vertices). This performance advantage becomes larger for larger graphs. Two algorithms (Ullmann's and Konishi's) were examined, and the data-dependent approach was found to be equally effective for both algorithms. We also examined two types of input graph sets, and found that the data-dependent approach shows advantage in both cases.

Graph data in large scientific/engineering applications are often too massive to fit inside the computer's main memory. The resulting input/output (I/O) costs could be a major performance bottleneck. This paper proposes an extension to extant multilevel graph partitioning algorithms with improved I/O-efficiency. The input graph is envisioned as the union of disjoint blocks (subgraphs) of almost the same size. Each block is coarsened in turn. Recursive matching and contraction are the operations in this phase. All the coarsened blocks are then merged in an iterative manner in order to ensure that the resulting graph fits in the main memory. This graph is then treated with an in-core multilevel graph partitioning algorithm in the usual way. Our experimental results show that the larger graph size is, the more dependent on the I/O-efficiency the performance is. And our modification can easily partition very large graphs. It also exhibits considerable improvement in I/O-complexity.

Given an edge-weighted graph G, the minimum maximal matching problem asks to find a minimum weight maximal matching. The problem is known to be NP-hard even if the graph is planar and unweighted. In this paper, we consider the problem in planar graphs. First, we prove a strong inapproximability for the problem in weighted planar graphs. Second, in contrast with the first result, we show that a polynomial time approximation scheme (PTAS) for the problem in unweighted planar graphs can be obtained by a divide-and-conquer method based on the planar separator theorem. For a given ε > 0, our scheme delivers in time a solution with size at most (1 + ε) times the optimal value, where n is the number of vertices in G and α is a constant number.

In this paper, we give an algorithm for the node-to-set disjoint paths problem in pancake graphs with its evaluation results. The algorithm is of polynomial order of n for an n-pancake graph. It is based on recursion and divided into two cases according to the distribution of destination nodes in classes into which all the nodes in a pancake graph are categorized. The sum of lengths of paths obtained and the time complexity of the algorithm are estimated and the average performance is evaluated based on computer simulation.

Although the subgraph isomorphism problem has various important applications, it is generally NP-complete and difficult to solve. Though a custom computing circuit can reduce the execution time substantially, it requires considerable hardware resources and is inapplicable to large problems. This paper examines the feasibility of data dependent designs, which are particularly suitable to a Field Programmable Gate Array (FPGA). The data dependent approach drastically reduces hardware requirements. For graphs of 32 vertices, the average logic scale of data dependent circuits is only 5% of the corresponding data independent circuit. The data dependent circuit is estimated to be maximally 460 times faster than the software. Even if the circuit generation time is included, a data dependent circuit is estimated to be 2.04 times faster than software for graphs of 32 vertices. The performance gain would increase for larger graphs.

This paper analyzes what structural features of graph problems allow efficient parallel algorithms. We survey some parallel algorithms for typical problems on three kinds of graphs, outerplanar graphs, trapezoid graphs and in-tournament graphs. Our results on the shortest path problem, the longest path problem and the maximum flow problem on outerplanar graphs, the minimum-weight connected dominating set problem and the coloring problem on trapezoid graphs and Hamiltonian path and Hamiltonian cycle problem on in-tournament graphs are adopted as working examples.

We sketch two algorithms that solve the undirected st-connectivity problem in a small amount of space. One is due to Nisan, Szemeredy and Wigderson, and takes space O(log3/2n), where n denotes the number of nodes in a give undirected graph. This is the first algorithm that overcame the Savitch barrier on the space complexity of the problem. The other is due to Tarui and this author, and takes space O(sw(G)2 log2 n), where sw(G) denotes the separation-width of a given graph G. Their result implies that the st-connectivity problem can be solved in logarithmic space for any class of graphs with separation-width bounded above by a predetermined constant. This class is one of few nontrivial classes for which the st-connectivity problem can be solved in logarithmic space.

Two algorithms for minimum cut linear arrangement of a class of graphs called p-q dags are proposed. A p-q dag represents the connection scheme of an adder tree, such as Wallace tree, and the VLSI layout problem of a bit slice of an adder tree is treated as the minimum cut linear arrangement problem of its corresponding p-q dag. One of the two algorithms is based on dynamic programming. It calculates an exact minimum solution within nO(1) time and space, where n is the size of a given graph. The other algorithm is an approximation algorithm which calculates a solution with O(log n) cutwidth. It requires O(n log n) time.

The maximal linear forest problem is to find, given a graph G = (V, E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n2 processors on a CRCW PRAM. The second one is deterministic and runs in O(log 2n) timeusing n4 processors on an EREW PRAM. The last one is deterministic and runs in O(log 5n) time using n3 processors on an EREW PRAM. The results put the problem in the class NC.

This paper presents a linear time algorithm for testing whether or not there is a path ,vm> of an undiercted tree T (|V(T)|n) that coincides with a string ss1sm (i.e., label(v1)label(vm)s1sm). Since any path of the tree is allowed, linear time substring matching algorithms can not be directly applied and a new method is developed. In the algorithm, O(n/m) vertices are selected from V(T) such that any path pf length more than m 2 must contain at least one of the selected vertices. A search is performed using the selected vertices as 'bases' and two tables of size O(m) are constructed for each of the selected vertices. A suffix tree, which is a well-known-data structure in string matching, is used effectively in the algorithm. From each of the selected vertices, a search is performed with traversing the suffix tree associated with s. Although the size of the alphabet is assumed to be bounded by a constant in this paper, the algorithm can be applied to the case of unbounded alphabets by increasing the time complexity to O(n log m).

This paper presents efficient algorithms for finding the largest tree S such that there are vertex disjoint subtrees S1, , S (k1) of T each of which is isomorphic to S and every leaf of T is a leaf of some Si. The algorithms are useful for learning a macro table.

An optimal algorithm for decomposing a special type of the Hasse diagram into a minimum set of disjoint paths is described. It is useful for testing the consistency of functional dependencies.