One of the difficulties in routing problem is in wirability which is to guarantee a physical connection of a given topological route. Wirability often fails since the width of a wire is relatively large compared with the size of modules. As a possible solution, this paper proposes an incremental wiring algorithm which generates wires net-by-net without overlapping other pre-placed circuit elements. The idea is to divide a wire into a series of rectangles and handles them as modules with additional constraints to keep the shape of the wire. The algorithm was implemented and experimented on a small instance to show its promising performance.

The family Pk of graphs with proper-path-width at most k is minor-closed. It is known that the number of minimal forbidden minors for a minor-closed family of graphs is finite, but we have few such families for which all the minimal forbidden minors are listed. Although the minimal acyclic forbidden minors are characterized for Pk, all the minimal forbidden minors are known only for P1. This paper lists 36 minimal forbidden minors for P2, and shows that there exist no other minimal forbidden minors for P2.

A graph G is said to be universal for a family F of graphs if G contains every graph in F as a subgraph. A minimum universal graph for F is a universal graph for F with the minimum number of edges. This paper considers a minimum universal graph for the family Fkn of graphs on n vertices with path-width at most k. We first show that the number of edges in a universal graph Fkn is at least Ω(kn log(n/k)). Next, we construct a universal graph for Fkn with O(kn log(n/k)) edges, and show that the number of edges in a minimum universal graph for Fkn is Θ(kn log(n/k)) .

It is known that the clock-period can be shorter than the maximum of signal-delays between registers if the clock arrival time to each register is properly scheduled. The algorithm to design an optimal clock-schedule was given. In this paper, we propose a clock-tree routing algorithm that realizes a given clock-schedule using the Elmore-delay model. Following the deferred-merge-embedding (DME) framework, the algorithm generates a topology of the clock-tree and simultaneously determines the locations and sizes of intermediate buffers. The experimental results showed that this method constructs a clock-tree with moderate wire length for a random layout of scheduled registers. Notably, the required wire length for a gentle layout of scheduled registers was shown to be almost equal to that of zero-skew clock-trees.

The three dimensional (3D) packing problem is to arrange given rectangular boxes in a rectangular box of the minimum volume without overlapping each other. As an approach, this paper introduces the system of three sequences of the box labels, the sequence-triple, to encode the topology of the 3D-packing. The topology is the system of relative relations in pairs of boxes such as right-of, above, front-of, etc. It will be proved that the sequence-triple represents the topology of the tractable 3D-packings which is a 3D-packing such that there is an order of the boxes along which all the boxes are extracted one by one in a certain fixed direction without disturbing other remaining boxes. The idea is extended to the system of five ordered sequences, the sequence-quintuple. A decoding rule is given by which any 3D-packing is represented. These coding systems are applied to design heuristic algorithms by simulated annealing which search the codes for better 3D-packings. Experimental results were very convincing its usefulness as automated packing algorithms.

The switch-block architecture of FPGAs is discussed to see a good balance between programmable-switch resources and routability. For the purpose, FPGAs are assumed to have certain extremal structures, whose switch-blocks consist of parallel or complete switch-sets where a switch-set is a set of switches between two sides of the switch-block. A polynomial time detailed-routing algorithm for a given global-routing is presented if the switch-block consists of two or less parallel switch-sets or three that form a cycle. For other FPGAs, the corresponding decision problem is proved to be -complete. A best compromise between switch resources and routability is offered.

We introduce the interval set of a graph G which is a representation of the proper-path-decomposition of G, and show a linear time algorithm to construct an optimal interval set for any tree T. It is shown that a proper-path-decomposition of T with optimal width can be obtained from an optimal interval set of T in O(n log n) time.

Given a switch-box, let C be a connection requirement. If there is a polynomial time algorithm (router) to complete C, C is said to be tractable by the algorithm. There have been proposed a number of switch-box routers but none that makes clear its tractable problems. We propose a switch-box router, or rather a principle, BOX-PEELER with a simple characterization of a class of tractable problems. BOX-PEELER is developed to be an underlying concept in switch-box routing as LEFT-EDGE method has been in 2-side channel routing.

A challenge to an automated layout of analog ICs starts with the insight into high quality placements crafted by experts. We observe first that matched devices or elemental functions such as input, output, amplifiers, etc are clustered. Second, devices in the same cluster are located faithfully to the drawn schema. Third, these two features are simultaneously fulfilled in a well-compacted placement. This paper proposes a novel device-level placement that simulates the above features based on Sequence-Pair. A slight modification of the meaning, say, of relation "A is left-of B" to relation "A is not right-of B" enlarges the freedom and allows a neater compaction of clusters allowing zigzag border curves. As the consequence, clusters are placed faithfully to relative position in the schema. We tested our algorithm for industrial instances and compared results with those by manual design. The results showed better features in performance figures than the those of manual designs by, on average, 13.5% and 21.2% with respect to the area and total net-length.

The success in topdown design of recent huge system LSIs is in a seamless transfer of the information resulted from the high level design to the lower level of floorplanning. For the purpose, we introduce a new concept abstract floorplan which is included in the output of high level design. From the abstract floorplan, the pillar blocks are derived which are critical sets of blocks that are expected to determine the width and height of the chip, named the frame. Since the frame and pillar blocks are obtained in the high level stage, they are useful to keep the consistency in the low level physical design if we apply optimization regarding them as constraints. Experiments to MCNC benchmarks showed that abstract floorplanning by pillar blocks output a placement faithful to the one physically optimized block placement with respect to the chip area and the wire-length.

A floorplan specifies the layout of modules in very large scale integration (VLSI) design, and a new code, called the EQ-sequence, for representing a floorplan is presented in this paper. The EQ-sequence is based on a Q-sequence. The EQ-sequence can preserve the adjacent relationships of rooms on a floorplan, but the Q-sequence cannot. The algorithms for encoding, moving and decoding of an EQ-sequence are introduced. With the EQ-sequence, we can check whether two modules abut each other on a floorplan. It has been proved that any floorplan of n rooms is uniquely encoded by an EQ-sequence and any EQ-sequence is uniquely decoded to a floorplan, both in O(n) time.

Lately, time-multiplexed I/Os for multi-device implementations (e.g., multi-FPGA systems), have come into practical use. They realize multiple I/O signal transmissions between two devices in one system clock cycle using one I/O wire between the devices and multiple I/O clock cycles. Though they ease the limitation of the number of I/O-pins of each device, the system clock period becomes much longer approximately in proprotion to the maximum number of multiplexed I/Os on a signal path. There is no conventional partitioning algorithm considering the effect of time-multiplexed I/Os directly. We introduce a new cost function for evaluating the suitability of a bipartition for multi-device implementations with time-multiplexed I/Os. We propose a performance-driven bipartitioning method VIOP which minimizes the value of the cost function. Our method VIOP combines three algorithms, such that i) min-cut partitioning, ii) coarse performance-driven partitioning, iii) fine performance-driven partitioning. For min-cut partitioning and coarse performance-driven partitioning, we employ a well-known conventional bipartitioning algorithms CLIP-FM and DUBA, respectively. For fine performance-driven partitioning for the final improvement of a partition, we propose a partitioning algorithm CAVP. By our method VIOP, the average cost was improved by 10.4% compared with the well-known algorithms.

In this paper, we propose an improved network-flow based multi-way circuit partitioning algorithm whose objective is to minimize the number of sub-circuits. It iteratively extracts a size-maximal feasible sub-circuit one at a time. In our approach, two devices are applied. One is in the use of an exact min-cut graph, and the other is in the idea of keeping the number of I/O pins of the residual circuit as small as possible after one-time extraction. We implemented our algorithm in C for experiments, and tested it with several industrial cases and MCNC benchmarks. Compared to the known approach, we observed more than 10% reduction in average of the sub-circuit number.

The most crucial factor that degrades a high speed VLSI is the signal propagation delay in a routing tree. It is estimated by the sum of the delay caused by the source-to-sink path length and by the total length. To design a routing tree in which these two are both small and balanced, we propose an algorithm to construct such a spanning tree, based on the idea of constructing a tree combining the minimum-spanning-tree and shortest-path-tree algorithms. This idea is extended to finding a rectilinear Steiner tree. Experiments are presented to illustrate how the source-to-sink path length and total length can be ballanced and small.

The shape-based routing needs a routing architecture with a geometrical computation framework on it. This paper introduces a novel routing architecture, Oct-Touched Tile (OTT), with a geometrical computation method along the horizontal- and vertical-constraints. The architecture is represented by the tiles spreading over the 2-D plane. Each tile is flexible to satisfy the constraints imposed for non-overlapping and sizing request. In this framework, path finding and shape-based sizing are executed on the same architecture. In experiments, our system demonstrates the performance comparable to a commercial tool. In addition, we show potential of OTT by introducing several ideas of extensions to analog layout constraints.

A floorplan is a partition of a rectangle into subrectangles, each of which is associated with a module. Zero-wasted-area layouts are known to exist when the height and width of modules are constrained only by the area, and several methods have been proposed for deriving such layouts. However, because these methods are global and indirect, they are inherently slow. We propose a new algorithm which simulates the air-pressure mechanics. It begins with a layout, which is not necessarily feasible, and iterates the movement of one wall at a time to the force-balancing position. The key issue is that it is guaranteed that every movement makes a current layout approach a zero-wasted-area layout by the measure of energy which is defined here. Experimental results on the example in several literatures and artificially made complex examples showed very fast convergence. The algorithm is evolved to methods which move all the walls simultaneously, resulting in a further speed enhancement.

The packing problem is to pack given items into given containers as efficiently as possible under various constraints. It is fundamental and significant with variations and applications. The Set-Bin-Packing (SBP) is a class of packing problems: Pack given items into as few bins which have the same capacity where every item is a set and a bin can contain items as long as the number of distinct elements in the union of the items equals to or less than the capacity. One of applications is in FPGA technology mapping, which is our initial motivation. In this paper, the computational complexity of SBP is studied with respect to three parameters α, γ, and δ which are the capacity, the upper bound of the number of elements in an item, and the upper bound of the number of items having an element, respectively. In contrast that the well known Integer-Bin-Packing (IBP) is NP-hard but is proved that even a simplest heuristics First-Fit-Decreasing (FFD) outputs exact solutions as long as α 6, our result reveals that SBP remains NP-hard for a small values of these parameters. The results are summarized on a 3D map of computational complexities with respect to these three parameters.

Given a plane graph G, a trail of G is said to be dual if it is also a trail in the geometric dual of G. We show that the problem of partitioning the edges of G into the minimum number of dual trails is NP-hard.

We improve the algorithm to obtain the min-cut graph of a hyper-graph and show an application to the sub-network extraction problem. The min-cut graph is a directed acyclic graph whose directed cuts correspond one-to-one to the min-cuts of the hyper-graph. While the known approach trades the exactness of the min-cut graph for some speed improvement, our proposed algorithm gives an exact one without substantial computation overhead. By using the exact min-cut graph, an exhaustive algorithm finds an optimal sub-circuit that is extracted by a min-cut from the circuit. By experiments with the industrial data, the proposing method showed a performance enough for practical use.

For a given 2-edge-connected graph G and a spanning tree T of G, the graph augmentation problem 2ECA (T,G) is to find a minimum edge set AE (G) such that T A is 2-edge-connected. This note proves that 2ECA (T, G) is solvable in polynomial time if G is series-parallel.