Optimal Integer Delay-Budget Assignment on Directed Acyclic Graphs

Abstract

Abstract-Excess delay that each component of a design can tolerate under a given timing constraint is referred to as delay budget. Delay budgeting has been widely exploited to improve the design quality in VLSI CAD flow. The objective of the delay budgeting problem investigated in this paper is to maximize the total delay budget assigned to each node in a directed acyclic graph under a given timing constraint. Due to discreteness of the timing of the components in the libraries during design optimization flow, discrete solution for delay budgeting is essential. We present an optimal integer delay budgeting algorithm. We prove that the problem can be solved optimally in polynomial time. In addition, we look at different extensions of the delay budgeting problem, such as maximization of weighted summation of delay budgets assigned to the nodes with constraints on lower bound and upper bound on the delay budget allocated to each node. We prove that for both aforementioned extensions, our algorithm can produce an optimal integer solution in polynomial time. Our algorithm is generic and can be applied in different design tasks at different levels of abstraction. We applied our proposed optimal delay budgeting algorithm in library mapping during datapath synthesis on an FPGA platform ,using pre-optimized cores of FPGA libraries. For each application, we go through synthesis and place and route stages in order to obtain accurate results. Our optimal algorithm outperforms ZSA algorithm [4] in terms of area by ½¼± on average for all applications. In some applications, optimal delay budgeting can speedup runtime of place and route up to ¾ times.Index Terms-delay budgeting, timing constraint, core-based design implementations,integer programming.

Scheduling with integer time budgeting for low-power optimization

“…Based on equation (11), column (k + n/2) must not belong to partition J 1 . This implies that ∑ j∈J 1 Therefore, we prove that for any subset J of the columns, we can always derive an appropriate partition on J so that GhouilaHouri condition holds for all rows in A. According to Lemma 1, A is a totally unimodular matrix.…”

“…An optimal integral budget assignment algorithm was proposed in [1]. The maximum weighted sum of the delay budgets can be obtained using linear programming followed by a re-budgeting algorithm to optimally convert the solution with fractional values to integers.…”

“…In [1] the authors mention total unimodularity of the matrix in special cases when the input graph is a directed path. Authors in [5] show that the dual of the edge budgeting problem can be mapped to a min-cost network flow problem, which indirectly suggests that the constraint matrix of the dual problem is totally unimodular.…”

Slack Allocation and Routing to Improve FPGA Timing While Repairing Short-Path Violations

“…Techniques for optimally distributing slack using linear programming techniques and dual min-cost flow have recently been presented [8] [9]. These techniques support weights like Minimax-PERT, in addition to lower and upper bounds on budgets, and balanced budget distribution.…”

“…The techniques of [8] and [9] could be used instead of the Minimax-PERT algorithm to compute better delay budgets. However, the insensitivity to the weighting scheme used and the relatively aggressive stopping criterion imply that this application is unlikely to require this, and so the more computationally intensive approaches may not be appropriate.…”