A Useful Skew Tree Framework for Inserting Large Safety Margins

Abstract

The construction of clock trees for modern designs is challenging because the clock trees need to be constructed with adequate safety margins such that the skew constraints are satisfied even under variations. The amount of safety margin required in a skew constraint is dependent on the distance of the corresponding sequential elements in the tree topology. In certain cases, the amount of safety margin that can be inserted may be limited. Consequently, the corresponding sequential elements should be placed close in the topology, i.e., the point of divergence to these elements is low in the clock tree, in order to reduce the influence of variations. By using safety margins and lowering the point of divergence, we present a framework for the construction of useful skew trees with large safety margins inserted in the skew constraints. The framework, called UST-LSM, first identifies tight skew constraints by the detection of negative cycles in a weighted skew constraint graph. Next, the corresponding sequential elements of these skew constraints are clustered early in tree topology. Compared to earlier studies, we can allow larger safety margins in skew constraints spanning between sequential elements within a subtree. This translates into an improvement of yield from 46.8% to 98.8% on a synthesized benchmark with 7, 674 sequential elements and 63, 440 skew constraints.

Clock Tree Construction based on Arrival Time Constraints

“…As no routing tree topologies are generated in [12], no interconnect delays can be computed, which is equivalent to setting w(k, i) = 0 and w(k, j) = 0 in both Eq (7) and Eq (8).…”

“…Clock trees with buffers are constructed by integrating the proposed constraints into a classical bottom-up tree construction framework, which is based on algorithms in [17,5,4,8,10]. An overview of the framework is shown in Figure 5.…”

“…Next, subtrees are iteratively pairwise merged to form larger subtrees while satisfying the arrival time constraints (see Section 4.2). Lastly, buffers are inserted to drive the constructed subtrees such that the transition time constraints are satisfied [4,8,10]. The iterative buffer stage construction process continues until only a tree remains.…”

“…The bounds of each constraint are defined by the length of two shortest paths in the SCG. Clock trees meeting the explicit skew constraints can be constructed based on iteratively merging subtrees (or sinks) while considering the dynamic implied skew constraints [17,6,8]. When a pair of sinks are merged, the skew between the sinks is specified and edge weights are required to be updated in the SCG, which in turn requires every implied skew constraint to be updated (with high run-time complexity).…”

“…In [17], the Greedy-UST/DME algorithm was proposed to construct useful skew trees based on implied skew constraints. However, only a limited number of topologies were explored, as it is costly in run-time to update the constraints [17,6,8].…”