Minimizing Channel Density with Movable Terminals

We give algorithms to minimize density for VLSI channel routing problems with terminals that are movable subject to certain constraints. The main cases considered are channels with linear order constraints, channels with linear order constraints and separation constraints, channels with movable modules containing fixed terminals, and channels with movable modules and terminals. In each case, we improve previous results for running time and space by a factor of L/\lgn and L, respectively, where L is the channel length, and n is the number of terminals

Feasible Offset and Optimal Offset for Single-Layer Channel Routing

The paper provides an efficient method to find all feasible offsets for a given separation in a VLSI channel routing problem in one layer. The prior literature considers this task only for problems with no single-sided nets. When single-sided nets are included, the worst-case solution time increases from Theta(n) to Omega(n^2), where n is the number of nets. But, if the number of columns c is O(n), one can solve the problem in time O(n^{1.5}lg n ), which improves upon a `naive\u27 O(cn) approach. As a corollary of this result, the same time bound suffices to find the optimal offset (the one that minimizes separation). Better running times are obtained when there are no two-sided nets or all single-sided nets are on one side to the channel. The authors also give improvements upon the naive approach for c≠O(n), including an algorithm with running time independent of c

Finding a Maximum-Density Planar Subset of a Set of Nets in a Channel

We present efficient algorithms to find a maximum-density planar subset of n 2-pin nets in a channel. The simplest approach is to make repeated usage of Supowit\u27s dynamic programming algorithm for finding a maximum-size planar subset, which leads to O(n^3) time to find a maximum-density planar subset. But we also provide an algorithm whose running time is dependent on other problem parameters and is often more efficient. A simple bound on the running time of this algorithm is O(nlgn+n(t+1)w), where t is the number of two-sided nets, and w is the number of nets in the output. Though the worst-case running time is still O(n^3), this algorithm achieves better results when either t or w is of modest magnitude. In the very special case when there are no two-sided nets, the bound stated above becomes O(nlgn+nw); this bound can also be achieved in the case of no single-sided nets. In addition, the bounds stated so far can be strengthened by incorporating into the running time the number of edges in certain interval overlap and interval containment graphs

Parallel Algorithms for Single-Layer Channel Routing

We provide efficient parallel algorithms for the minimum separation, offset range, and optimal offset problems for single-layer channel routing. We consider all the variations of these problems that are known to have linear- time sequential solutions rather than limiting attention to the river-routing context, where single-sided connections are disallowed. For the minimum separation problem, we obtain O(lgN) time on a CREW PRAM or O(lgN / lglgN) time on a (common) CRCW PRAM, both with optimal work (processor- time product) of O(N), where N is the number of terminals. For the offset range problem, we obtain the same time and processor bounds as long as only one side of the channel contains single-sided nets. For the optimal offset problem with single-sided nets on one side of the channel, we obtain time O(lgN lglgN) on a CREW PRAM or O(lgN / lglgN) time on a CRCW PRAM with O(N lglgN) work. Not only does this improve on previous results for river routing, but we can obtain an even better time of O((lglgN)^2) on the CRCW PRAM in the river routing context. In addition, wherever our results allow a channel boundary to contain single-sided nets, the results also apply when that boundary is ragged and N incorporates the number of bendpoints

Adaptive fault-tolerant wormhole routing algorithms for hypercube and mesh interconnection networks

In this paper, we present adaptive fault-tolerant deadlock-free routing algorithms for hypercubes and meshes by using only 3 virtual channels and 2 virtual channels respectively. Based on the concept of unsafe nodes, we design a routing algorithm for hypercubes that can tolerate at least n, 1 node faults and can route a message via a path of length no more than the Hamming distance between the source and destination plus four. We also develop a routing algorithm for meshes that can tolerate any block faults, as long as the distance between any two nodes in di erent faulty blocks is at least 2 in each dimension.

Feasible Offset and Optimal Offset for General Single-Layer Channel Routing

. This paper provides an efficient method to find all feasible offsets for a given separation in a VLSI channel routing problem in one layer. The prior literature considers this task only for problems with no single-sided nets. When single-sided nets are included, the worst-case solution time increases from \Theta(n) to \Omega\Gamma n 2 ), where n is the number of nets. But if the number of columns c is O(n), the problem can be solved in time O(n 1:5 lg n), which improves upon a "naive" O(cn) approach. As a corollary of this result, the same time bound suffices to find the optimal offset (the one that minimizes separation). Better running times result when there are no two-sided nets or all single-sided nets are on one side of the channel. This paper also gives improvements upon the naive approach for c<F NaN> 6= O(n), including an algorithm with running time independent of c. An interesting algorithmic aspect of the paper is a connection to discrete convolution. Key words. VLSI..

Minimizing Channel Density with Movable Terminals

We give algorithms to minimize density for VLSI channel routing problems with terminals that are movable subject to certain constraints. The main cases considered are channels with linear order constraints, channels with linear order constraints and separation constraints, channels with movable modules containing fixed terminals, and channels with movable modules and terminals. In each case, we improve previous results for running time and space by a factor of L= lg n and L, respectively, where L is the channel length, and n is the number of terminals. 1 Introduction The channel routing problem has received a great deal of attention in VLSI layout design. In the usual model, terminals lie on grid points along two horizontal line segments which delimit the channel. Each terminal is labeled with a net number, and the problem is to connect terminals belonging to the same net, using horizontal and vertical wire segments in a grid of two layers, one reserved for horizontal wires and one f..

Single-Layer Channel Routing and Placement with Single-Sided Nets

This paper considers the optimal offset, feasible offset, and optimal placement problems for a more general form of single-layer VLSI channel routing than has usually been considered in the past. Most prior works require that every net has exactly one terminal on each side of the channel. As long as only one side of the channel contains multiple terminals of the same net, we provide linear-time solutions to all three problems. Such results are implausible if the placement of terminals is entirely unrestricted; in fact, the size of the output for the feasible offset problem may be \Omega\Gamma n 2 ). The linear-time results also hold with a ragged boundary on the side of the channel with multiple connections to the same net. 1 Introduction We are given two horizontal lines, whose positions may be adjusted to form the top and bottom boundaries (sides) of a rectilinear grid, and a set of n nets. Each net consists of terminals located at grid points on the two sides, and we refer to the..

Parallel algorithms for single-layer channel routing

We provide efficient parallel algorithms for the minimum separation, offset range, and optimal offset problems for single-layer channel routing. We consider all the variations of these problems that have linear-time sequential solutions rather than limiting attention to the “river-routing” context, where single-sided connections are disallowed. For the minimum separation problem, we obtain O(lg N) time on a CREW PRAM or O(lg N/lg lg N) time on a CRCW PRAM, both with optimal work (processor-time product) of O(N), where N is the number of terminals. For the offset range problem, we obtain the same time and processor bounds as long as only one side of the channel contains single-sided nets. For the optimal offset problem with single-sided nets on one side of the channel, we obtain time O(lg N lg lg N) on a CREW PRAM or O(lg N) time on a CRCW PRAM with O(N lg lg N) work. Not only does this improve on previous results for river routing, but we can obtain an even better time of O((lg lg N)^2) on the CRCW PRAM in the river routing context