This proves the theorem.
Apparently, the sequence (|Qn|)n∈N is given by sequence A012814 in the OEIS [Slo], which consists of every 5th entry of the Padovan sequence, see sequence A000931 in the OEIS. The growth rate of the Padovan sequence is given byρ:= √5
β1 which is also known as the plastic number. Hence, the running time of our algorithm can also be expressed asO
m
ρ5/2m .
Having very good lower and upper bounds has a high impact on the running time, so we carefully use any information available to update our bounds.
Assume now that we apply compute_opt to compute a table entry, i.e., to find an optimum circuit for the generalizedAnd-Orpathh(t0; Γ0)with input set I with delay at most D. Before starting our partitioning process (see Section 5.5.2), we compute several lower bounds as in the following section. If any of these is larger thanD, we know that there is no circuit with delay at mostDfor h(t0; Γ0) and need not start the partitioning process.
5.5.1 Lower Bounds
A basic lower bound that can be computed quickly for any generalized And-Or path h(t0; Γ0) arises from the lower bounds in Theorem 2.3.15 and Corollary 5.2.5, i.e.,
max (
log2W(t0) ,max
tmaxi∈P0
a(ti) + 1, max
ti∈Pb:b>0a(ti) + 2 )
,
where W(t0) = Pir−1−1
j=i0 2a(tij) as in Definition 2.3.16. as in Definition 2.5.6. Note that the first lower bound requires integral arrival times.
We use two other reducing lower boundsthat each consider a specific reduced generalized And-Or path h(t00; Γ00) of h(t0; Γ0) with similar structural complexity.
Forh(t00; Γ00), we recursively apply the algorithm with depth boundD. Either there is no solution, in which case D+ 1 is a lower bound on the optimum delay for h(t00; Γ00), thus also forh(t0; Γ0); otherwise, we know the optimum delay forh(t00; Γ00), which is a lower bound forh(t0; Γ0). This usually yields a strong lower bound, but is very time-consuming.
First, only in the special case of depth optimization, we consider the generalized And-Orpathh(t00; Γ00)arising fromh(t0; Γ0)by keeping only the largest input group in the signal partition completely and condensing each other input group to a single input (except for the last group, which keeps 2 inputs). In the case of depth optimization, only the input-group sizes matter, so there are only O(m3) of these generalizedAnd-Orpaths, and it is not harmful to solve them optimally.
Secondly, also in the case of delay optimization, we consider a reduced generalized And-Orpathh(t00; Γ00)that arises from removing a single input ofh(t0; Γ0)in a way that hopefully the optimum delay of any circuit for h(t00; Γ00) is the same as for h(t0; Γ0). Hence, among all inputs with the minimum arrival time, we remove an input of the largest input group. Empirically, we see that in the case of depth optimization, this lower bound is tight in 97%of its applications. This matches the observation that if we iteratively apply this lower bound m times, starting with a generalizedAnd-Orpath with optimum depth d, the optimum depth changes only dtimes, wheredm.
5.5.2 Partitioning the Same-Gate Inputs
For determining a solution with delay D for a generalized And-Orpathh(t0; Γ0) – if it exists –, we enumerate partitions S◦ = S1◦ ·∪S2◦ of its same-gate input set S◦ for all ◦ ∈ {And,Or} in line 13 of Algorithm 5.1. In our implementation, we first choose◦:=◦0 as the gate type of the input groupP0 as empirically, this more often yields a good circuit, and afterwards the other gate type. For both, we enumerate partitions of S◦ and recursively try to find a solution with delay at most D.
We avoid generating too many partitions of a setS◦by enumerating the partitions in a specific order. In a recursive approach, one by one, we assign the inputs to one
of the subsets of S◦. Here, just as in standard branch-and-bound algorithms, we follow the idea to make the most important decisions first. Recall from the proof of Theorem 5.3.1 that by convention, the last inputtir−1 is always contained inS2◦.
Now, we first enumerate the highest input index il for which input til goes in to the other part, S1◦. Once til is fixed, we have completely determined which of the inputs with different gate type than ◦ are contained in in both h(t0; Γ0)S◦
1 and h(t0; Γ0)S◦
2, or only in h(t0; Γ0)S◦
2. Based on this, we compute another lower bound, thecross-partition Huffman bound, by Huffman coding on all inputs ofh(t0; Γ0), where those inputs that are contained in both sub-functions are counted twice, and may stop when this lower bound exceedsD.
Astil is the input with the highest index inS1◦, we already know that all inputs ti ∈S◦ withi > ilmust be in S2◦. It remains to enumerate thoseti∈S◦ withi < il. They are assigned to the setsS1◦ andS2◦recursively, in the order of decreasing arrival time, and in case of ties, inputs with larger indices are considered first. For each input, we first put it into S2◦ and recursively continue with the other inputs; and then put it into S1◦ and go into recursion. This way, we in particular prioritize the construction of consecutive setsS1◦ and S2◦, which often allows finding an optimum solution quickly (cf. Section 6.3).
Now, assume that we try to compute a circuit for h(t0; Γ0) with delay at mostD via a fixed partitionS◦=S◦1 ·∪S2◦. Before computing a solution, we evaluate all lower bounds available for the two sub-instances, and stop if any of the lower bounds is larger thanD−1. Otherwise, we recursively compute the table entries ofh(t0; Γ0)S◦ 1
andh(t0; Γ0)S◦
2 with delay boundD−1. As already mentioned, based on whether we did find a solution or not, we may update the lower bound forh(t0; Γ0).
Note that the lower bound L on the best delay achievable for h(t0; Γ0) is also a lower bound for all generalizedAnd-Orpath on a superset of the inputsIofh(t0; Γ0).
Hence, if we have updated L for h(t0; Γ0), in lower bound propagation, we also update the lower bound for certain generalized And-Or paths whose inputs are a superset ofI. Doing this for all supersets would be to costly; so we only update lower bounds of supersets which are already contained in our dynamic programming table and arise from adding a single input. For those whose lower bounds are improved, we recurisvely repeat this procedure.
If we did not find a solution with delay at most D for the current partition, we might discard a part of our enumeration tree in subset enumeration pruning:
Consider the inputs ofS◦ in the ordertj0, . . . , tjl in which we enumerate whether to put them into S1◦ or S2◦. When considering an input tji, we have already assigned the inputs tj0, . . . , tji−1 to one of the two subsets. If we add tji to S2◦, the set S2◦ is minimal among all sets that will arise from enumerating assignments for the elements tji+1, . . . , tjl. The first assignment that will be tried for tji+1, . . . , tjl is to put them all into S1◦. Hence, when the computation of a solution with delay at most D for this generalizedAnd-Or path was not successful because theAnd-Or pathh(t0; Γ0)S◦
2 had too large delay, we already know that all other partitions with tj0, . . . , tji unchanged will also not lead to delay at mostD. Hence, we can skip this part of our enumeration tree. The same holds when addingtji to S1◦.
Finally, we note that the running time for the computation of a table entry highly depends onD. Hence, when computing a table entry with a lower bound of L, in delay probing, we in fact loop over all possible delays d∈ {L, . . . , D} with increasingd and try to find a solution with delay d. The first value dfor which a solution is found is then the optimum delay of any circuit forh(t0; Γ0).
Integral arrival times Fractional arrival times
# inputs With size opt. No size opt. With size opt. No size opt.
10 0.001 0.000 0.005 0.000
20 0.442 0.002 2.248 0.008
30 2374.234 0.012 5790.565 0.092
40 - 0.096 - 44.388
50 - 8.294 - 8.223
60 - 3.514 - ∗106.860
Table 5.3: Average running times of Algorithm 5.1 on 10 randomly generatedAnd-Orpath instances for each number of inputs. For frac-tional arrival times and the non-size-opt mode, we omit one instance with60inputs because there, the memory limit of 400 GB was reached and the run could not finish.