Using well-solvable minimum cost exact covering for VLSI clock energy minimization

To save energy of VLSI systems flip-flops (FFs) are grouped in Multi-Bit Flip-Flop (MBFF), sharing a common clock driver. The energy savings strongly depends the grouping. For 2-bit MBFFs the optimal grouping turns into a minimum cost perfect graph matching problem. For k-bit MBFFs the optimal grouping turns into a minimum cost exact k-covering problem. We show that due to their special setting that is based on the FFs’ data toggling probabilities, those problems are well-solvable in O (n log n) time complexity. © 2014 Elsevier B.V. All rights reserved. 1. VLSI energy savings by multi-bit flip-flop grouping One of the major energy consumers in computing, communication and consumer electronics and other devices is the system’s clock signal, typically responsible for 30%–70% of the total switching energy [13]. Flip-flops (FFs) are the heart of digital systems, used to synchronize their operation and store the system’s state. To drive the FFs, a clock signal is distributed across the chip through a clocking network. FFs consume most of the clock energy. Within a FF, most of the energy is consumed by its internal clock driver. For simplicity, non-essential VLSI design details are ignored, and the interested reader can find those in any VLSI design textbook (e.g. [15]). k-bit data is usually stored in k individual FFs, where each of those has its own internal clock drivers. In an attempt to reduce the clock energy, a technique called Multi-Bit Flip-Flop (MBFF) has lately been adopted by the VLSI industry [10,4]. A k-bit MBFF combines several FFs integrated in a single entity, such that a common clock driver is used for all the k internal FFs rather than k drivers. The energy savings achieved by using MBFFs is considerable, and ∗ Corresponding author at: EE Dept., Technion – Israel Institute of Technology, Haifa 32000, Israel. Tel.: +972 3 5317208; fax: +972 3 7384051. E-mail addresses:[email protected], [email protected] (S. Wimer). http://dx.doi.org/10.1016/j.orl.2014.05.010 0167-6377/© 2014 Elsevier B.V. All rights reserved. may reach up to 20% of the entire system’s energy. The savings depend on the average (expected) data toggling probability p of the individual FFs, called data toggling probability, switching probability, or shortly probability. We use those terms interchangeably. By definition, there is 0 ≤ p ≤ 1, where p = 0 when the data is never toggling and p = 1 when the data is toggling at every clock cycle. Fig. 1 shows the energy ratio of two and four individual FFs to that of 2-bit and 4-bit MBFFs, respectively. To find the energy savings, we divide the energy difference between k individual FFs and k-bit MBFFs, by the energy of the k individual FFs. For small p it shows savings of (1.6 − 1) /1.6 = 35% for k = 2 and (2.2 − 1) /2.2 = 55% for k = 4. For high p the savings is (1.18 − 1) /1.18 = 15% for k = 2 and (1.3 − 1) /1.3 = 23% for k = 4. In typical VLSI systems 0 < p < 0.2, so high savings is expected. Combining MBFFs with Data-Driven Clock Gating (DDCG) considerably increases its energy savings. Ordinarily, FFs receive the clock signal regardless of whether or not their data will toggle in the next cycle. In DDCG the clock signal driving a FF is disabled (gated) when the FF’s state is not subject to change (toggle, switch) in the next clock cycle [7,17]. Due to the high hardware overhead involved in generating those signals, it was suggested to group several FFs and derive a joint disabling signal for those. The group size k yielding minimum energy depends on the toggling probabilities [17]. The problem of what FF should belong to what group so that the energy is minimized was studied in [18]. It was shown that under energy model based on the 0/1 toggling S. Wimer et al. / Operations Research Letters 42 (2014) 332–336 333 Fig. 1. Energy savings dependency on toggling probabilities of 2-bit and 4-bit MBFFs. correlation of the FFs, the problem is NP-hard, and a practical heuristic solution based onMinimum Cost Perfect GraphMatching (MCPM) was devised [16]. Applying DDCG in MBFF design methodology was proposed in [5]. However, the grouping in [5] and in other MBFF works [11,19,14] was not aware of the data togging probabilities and correlations, thus a big amount of potential energy savings was left untreated. The work in [16] used toggling correlation to derive the optimal FFs grouping for DDCG. It required huge data of 0/1 toggling vectors of all the FFs, obtained by simulations, which is a serious design burden. Furthermore, the corresponding optimization problem is NP-hard as mentioned before, and heuristic solution was thus proposed. In this paper we simplify the optimal grouping formulation by considering FF probabilities rather than their 0/1 toggling vectors. The simplification implies an optimization that is a kind ofminimal cost exact k-covering problem, where for k = 2 it turns into MCPM, formulated as follows. Given n real numbers (data toggling probabilities of FFs) pi ∈ [0, 1], n even, 1 ≤ i ≤ n, p1 ≤ p2 ≤ · · · ≤ pn, find a perfect matching  sj, tj  , 1 ≤ j ≤ n/2, of the integers 1, 2, . . . , n, minimizing the following energy loss expression (discussed in Section 3)