Despite the possibility of increasing the skew variation by inserting a crosslink, the authors in 12 propose an effective algorithm that. An algorithm for zeroskew clock tree routing with buffer. A tree is a nonlinear data structure, compared to arrays, linked lists, stacks and queues which are linear data structures. We give the first constantfactor approximation algorithms for this problem and its. When m is the l 1 plane, we refer to the l 1 variants of these problems. An exact zero skew clock routing algorithm using the elmore delay model is presented. Usefulskew tree ust algorithm based on deferredmerge embedding dme. A stretched tree consisting of an arborescence, a mapping such that is a onetoone mapping between the leaves of and and a cost function such that for every edge of, and furthermore, for. We proposed a lineartime optimal tree decomposition algorithm. Their proposed htree algorithm is based on the construction of. In this pap er, w e prop ose a zeroskew routing algorithm that is muc h simpler and practically more e cien t than the clusteringbased algorithm. Dmebased zeroskew algorithms can be adapted to handle the. Zeroskew clock routing trees with minimum wirelength kenneth d.
Elec 7770 advanced vlsi design spring 2014 zeroskew clock routing vishwani d. An exact zeroskew clock routing algorithm ieee xplore. The algorithm can be applied to singlestaged clock trees, multistaged clock trees, and multichip system clock trees. Integer is if haschildren node then result sk ew clo c k tree with resp ect to the. Power efficient treebased crosslinks for skew reduction. The algorithm always yields exact zero skew trees with respect to the appropriate. An exact zeroskew clock routing algorithm computeraided design of in tegrated circuits and systems, ieee transactions on author. The methodology has been tested using a real design database and resulting in a significant improvement in the through put time required to complete the clock tree generation. The corresponding graph is nearly the lengthnline graph. Recursively in a bottomup fashion, two zeroskewed subtrees are merged into a.
Practical approximation algorithms for zero and boundedskew. In summary, our k ey con tributions in this pap er are. If the skew of ts is zero then t s is a zeroskew tree zst. Advanced vlsi design agrawal 1 zeroskew clock routing ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ff ck. A clocksynthesis methodology for spiceaccurate skew optimization with tolerance to voltage variations was proposed in 14. Recursively in a bottomup fashion, two zeroskewed subtrees are merged into a new tree with zero skew. Several algorithms have been proposed which all achieve zero clock skew at the cost of varying complexity, wire lengths, and other measures. Pdf practical approximation algorithms for zero and. The skew of an edgeweighted rooted tree is the maximum difference between any two roottoleaf path weights. Symmetrical buffer placement in clock trees for minimal skew immune to global onchip variations renshen wang 1. It will still have an eigenvaluezero eigenvector supported on the root if and only if the formula evaluates to zero. With its advantages in wirelength reduction and routing flexibility compared with conventional manhattan routing, x architecture has been proposed and applied to modern ic design.
Clocktree routing realizing a clockschedule for semi. Nanoscale vlsi clock routing module based on usefulskew. Compared with the industrystrength gated clock tree synthesis, experimental data show that our approach can significantly reduce the clock skew in every process corner with a small penalty on the clock tree area and the clock tree. This algorithm is foundational to all the latest developments in clock. The routing algorithm used in this tool is based on the exact zero skew routing algorithm. Practical approximation algorithms for zero and boundedskew trees alexander z. The remainder of the pap er is organized as follo ws.
A good definition for a skew tree is a binary tree such that all the nodes except one have one and only one child. A stretched tree t is a zeroskew tree zst if all roottoleaf paths in t have. The pathlength skew of a tree is defined to be the maximum difference of the pathlengths in tbe tree from the root to any two leaves. Were upgrading the acm dl, and would like your input. A zeroskew clock tree optimization algorithmfor clock delay and power optimization is proposed. An exact zeroskew clock routing algorithm computer. Construction algorithm an overview sciencedirect topics. Another good definition is a binary tree of n nodes such that its depth is n1. The ustbp algorithm of 17 was the first work that incorpo rated a flavor of simultaneous clock scheduling and routing. Algorithmic tuning of clock trees and derived nontree. The operation of most digital circuits is synchronized by a periodic.
It searched for a good, easytorealize clock layout, called the use ful skew tree ust, under general clock skew constraints. In what follows, we address the zero skew clock routing problem. A single layer zero skew clock routing in x architecture. The greedybstdme algorithm very closely matches the best known heuristics for the zeroskew case and for the unboundedskew case i. An algorithm for zeroskew clock tree routing with buffer insertion. Clock skew sometimes called timing skew is a phenomenon in synchronous digital circuit systems such as computer systems in which the same sourced clock signal arrives at different components at different times.
This algorithm was improedv to calculate exact equidistant routings 3,4 11. Though it is easy to produce zero skew clock routing trees see e. In this scenario our algorithm produces a clock routing tree that is embedded in the channel intersection graph 10 of an arbitrary buildingblock layout. The clock routing trees produced by our method attain almost zero skew with only modest wirelength penalty. Basically, taco migrates the tree nodes to new locations to reduce the worst case clock skew under thermal variation. For a given topology, it outputs the locations of steiner points achieving zero skew and optimal wirelength. We study the problem of multistage zero skew clock tree construction for minimizing clock phase delay and wirelength.
The instantaneous difference between the readings of any two clocks is called their skew. The deferredmerge embedding dme algorithm was introduced for zeroskew clocktree routing independently by edahiro 9, 11, chao et al. In linear time, it embeds any given connection topology into the manhattan plane to create a clock tree with zero skew while minimizing total wire length. The optimized clock trees are zeroskew, or free from designinherited clock skews, and their processinduced clock skews are reduced by clock delay minimization. Practical approximation algorithms for zero and bounded. Finally, 22 further enhance the bst algorithm and proposed an incremental scheduling technique to compute the feasibleskew range for constructing a usefulskew tree ust for general skew constraints. Typematching clock tree for zero skew clock gating. The simplest approach was first proposed by dhar, franklin, and wang 2. A clusteringbased optimization algorithm in zeroskew. Wong 1996, an algorithm for zeroskew clock tree routing with buffer insertion, in european design and test conference, ieee proceedings.
The exact zero skew clock routing algorithm, proposed by tsay in 1993 is still used in many clock tree synthesis tools. However, the algorithm will require n time to detect this. The problem is to construct a clock tree with zero skew to achieve synchronicity and minimal sum of edge lengths so that circuit area and clock tree capacitance are minimized. Given a set s of sink locations, construct a zst ts with minimum cost. The buffered clock tree construction algorithm in vittal 1995 is an extension of the greedydme algorithm to consider the insertion of intermediate clock buffers during the construction of a zeroskew routing tree. Equivalence between zeroskew tree zst and hierarchical clustering hc on points p p1,p2,p3,p4. For a prescribed topology, bstdme constructs a boundedskew tree bst in two phases.
In existing approaches clock buffers are inserted only after clock tree is constructed. A clock tree router for general skew constraints cecs. Clock tree synthesis for timing convergence and timing. Pdf an exact zero skew clock routing algorithm using the elmore delay model is. Fast clock tree generation using exact zero skew clock. With this new insight, we designed an effective onlogntime o1approximation algorithm and an optimal dynamic programming for zst construction. The time complexity of our algorithm is linear on the a v erage. In each merging step, three sets of possible locations for embedding an internal node of the abstract topology are computed. Moreover, zeroskew and boundedskew sched ules may not always be legitimate.
Minimum wirelength zero skew clock routing trees with. Our boundedskew routing algorithm, called the bstdme algorithm, extends the dme algorithm for exact zeroskew trees via the concept of a merging region. We propose a clusteringbased algorithm which uses shortest. A tree can be empty with no nodes or a tree is a structure consisting of one node called the root and zero or one or more subtrees. The skew of ts is the maximum value of its0, s, ts0, s,l over all sink pairs s3 e s. An example of a tree of merging segments with sinks s1s8. Details of algorithm can be found here i managed to understand overall concept but failing to match it with provided implementation and hence not. Our algorithm only needs o1 space at each node, overall on 17 implementations in real implementations, some more issues. Experimental results show that the pathlength skew of our routing tree. Abstract the skew of an edgeweighted rooted tree is the maximum difference between any two roottoleaf path weights.
Tree height general case an on algorithm, n is the number of nodes in the tree require node. I am trying to understand implementation of linear time suffix array creation algorithm by karkkainen, p. Skewed trees relation to binary search tree stack overflow. A rc network model was applied to these algorithms, so that exact zero skew routings were nally achieved 16. Although the clokc skew in equidistant routings is usually eryv small 3, equidistant routings are not exact zero skew in general. Understanding implementation of dc3skew algorithm to. A survey on buffered clock tree synthesis for skew. A tree is called a perfect pathlength balanced tree if its pathlength skew is zero. Clock tree synthesis based on wire length minimization. An exact zero skew clock routing algorithm using elmore delay model is presented. Mathematically, an unordered tree or algebraic tree can be.
Zero or boundedskew trees are needed for achieving synchronization in many applications, including network multicasting g. In this section, we present our temperature aware clocktree optimization algorithm, taco, which incrementally optimizes a given zero skew clock tree constructed under the uniform thermal condition. Symmetrical buffer placement in clock trees for minimal. Then, based on the idea of typematching clock tree, we propose a zero skew gated clock tree synthesis algorithm. Additionally, wirelength of clock routing trees should be. The dynamic nearestneighbor algorithm dnna to generate tree topology and the walksegment breadth first. Minimizing wirelength in zero and bounded skew clock trees.