In many systems—like factory machines, sensor networks, or digital communication—tasks must be performed regularly, without long gaps. The pinwheel scheduling (PS) problem [4] models this need: each task has a time limit within which it must be repeated, and the goal is to build a schedule that performs one task per unit time (or “day” herein) while satisfying all these limits. Specifically, each task \(i \in [k] = \{1, \dots, k\}\) must be scheduled at least once every \(a _i\) days, where \(a _i\) is the period of task \(i\). Thus, an instance of PS is an array \((a _i) _{i \in [k]} = (a _1, \dots, a _k)\) of positive integers, which we assume to be arranged in non-decreasing order, and we seek to find pairwise disjoint sets \(S _1\), …, \(S _k \subseteq \mathbb{Z}\) (with \(S _i\) specifying the days on which to perform task \(i\)) satisfying, for all \(i \in [k]\), the frequency condition:

for each \(m \in \mathbb{Z}\), we have \([m, m + a _i) \cap S _i \neq \emptyset\).

An instance for which such a schedule \((S _i) _{i \in [k]}\) exists is said to be schedulable. For example, the instances \((3, 3, 3)\), \((2, 4, 8, 8)\), and \((3, 4, 5, 8)\) are schedulable (see Figure 1 for the last one), but all become non-schedulable if any one period \(a _i\) is replaced by \(a _i - 1\).

For an instance \(A = (a _i) _{i \in [k]}\) to be schedulable, the condition that its density \[\mathrm D(A) = \sum _{i \in [k]} \frac{1}{a _i}\] does not exceed \(1\) is clearly necessary, but not sufficient. For example, \((2, 3, a _3)\) is non-schedulable for all values of \(a _3\). On the other hand, it is relatively easy to see that every instance with a density of at most \(\frac{1}{2} = 0.5\) is schedulable [4]. The challenge of improving this sufficient condition has been taken up by several authors, who succeeded in increasing the bound to \(0.66\ldots {}\) [5], to \(0.7\) [6], and then to \(0.75\) [7]. It has been conjectured [5] that this value could be increased to \(\frac{5}{6} = 0.83\ldots \;\), which is the best possible because of the instance \((2, 3, a _3)\) mentioned above. This conjecture has been confirmed in a number of special cases: when \(A\) has three (or fewer) distinct period values [8], when the smallest period \(a _1\) is \(2\) [7], when for each period value there are at least five tasks having that period [9], and when the number \(k\) of tasks is \(\leq 12\) [10]. We resolve this conjecture affirmatively:

The reason for the explicit restriction to integers is that, in the following section, we extend the PS problem and allow periods to be positive real numbers, not just integers. This extension plays a key role in our proof of Theorem 1.

Following the preliminary announcements of this paper [1], [3], Fujiwara, Miyagi and Ouchi [11] conjectured, and confirmed for \(k \leq 3\), that in fact Theorem 1 holds for possibly non-integer instances as well.

Further Results↩︎

The condition \(\mathrm D(A) \leq 1\), which is obviously necessary for any instance \(A\), has been shown also to be sufficient for instances \(A\) containing only two distinct periods [12]. Our idea of fractional periods mentioned above yields a simple proof of this statement (for possibly non-integer periods as well). That is, we (re)prove:

Bamboo garden trimming (BGT) [13] is an optimization version of PS: in a grove of \(k\) bamboo plants, plant \(i \in [k]\) grows in height at a daily rate \(h _i \in \mathbb{N}\); each day (at a fixed time of day) we select one plant to be trimmed (i.e., reduced to height \(0\)) with the goal of keeping the highest plant in the grove as low as possible. Clearly, a trimming schedule achieves height \(G\) (i.e., ensures that no plant ever grows beyond it) for this instance if and only if it satisfies the PS instance \((\lfloor G / h _i \rfloor)_{i \in [k]}\). Note that the smallest such \(G\) is always an integer. Efficient algorithms to estimate this optimal \(G\) have been proposed, with approximation ratios \(2\) [14] (see also [15] for the discussion of how simple algorithms achieve or do not achieve this ratio), \(1.88\ldots\) [16], \(1.71\ldots\) [17], \(1.60\ldots\) [13], and \(1.42\ldots\) [18]. We use Theorem 1, and ideas used for it, to obtain an improved approximation ratio \(\frac{4}{3} = 1.33\ldots {}\):

Although Theorem 3 is stated in terms of finding the optimal value approximately, it will be easy to see that our algorithm can be turned into one that finds an underlying schedule. But it is not clear whether it does so in “polynomial time,” because our schedule is an infinite object. See the comments below on “fast online schedulers”.

After the preliminary announcements of Theorem 3 were made [2], [3], Mishra [19] published further improvement on the approximation ratio.

Related Work↩︎

The class of problems including ours and its variants below, where we seek perpetual schedules of recurring tasks (such as monitoring, maintenance, or replenishment) that should be performed with sufficient frequencies, has been also called windows scheduling or periodic scheduling. These problems have found applications such as packet scheduling in communication networks [4], [20], machine maintenance [13], [21], security games [22], broadcasting commercials [23], and spaced repetition of review material in education [24].

Our PS problem is perhaps the most basic among such settings. Some generalizations immediately present themselves: one might suppose that multiple tasks can be performed each day [20], [25], [26], that different tasks take different amounts of time to perform [23], [27], [28], or that the required spacing between tasks changes over time [24]. There are also variants with continuous flavour, where the tasks are placed on metric spaces so that moving between them takes some time or cost [13], [29], or even those problems, sometimes called patrolling, where we look for continuous movements of agents so that all points in the terrain are visited with sufficient frequency [30]–[33]. There are also various ways, besides BGT, to turn PS settings into optimization problems by introducing different notions of cost to measure deviation from the task’s requisite frequency [23], [34].

The frequency condition for PS is that task \(i\) must be scheduled at least once in the interval \([m, m + a _i)\) starting on any day \(m\). We could consider a somewhat simpler problem [9], [35], [36] where instead we require this condition only when the starting day \(m\) is an integer multiple of \(a _i\). One may also consider variants in which (there can be days to which no task is assigned, and) each task \(i\) must be scheduled precisely once (rather than at least once) in \(a _i\) days [21], [37], [38].

The packing-style problem considered in this paper (in which the tasks are packed into \(\mathbb{Z}\)) is complementary to a covering-style variant of the problem, in which task \(i\) may only be scheduled at most once in \(a_i\) days [33], [39]. For the covering problem, there was a conjecture similar to Theorem 1 (namely, that any instance of integer-valued periods with a density of at least \(1.26\ldots\) suffices to cover \(\mathbb{Z}\)) [33]. After the preliminary announcements of our Theorem 1 [1], [3] were made, this conjecture has also been confirmed, independently in two papers [19], [40], by partly applying our techniques.

Organization of the paper↩︎

In the next section, we generalize PS to non-integer periods, and note some basic properties of this general problem, which, as we pause to note, allow a simple proof of Theorem 2. After that, we reduce our main theorem (Theorem 1) to checking the schedulability of finitely many instances. The techniques for this exhaustive scheduling are then explained. Finally, we discuss how our bound for PS can be used to establish a new approximation ratio for BGT.

Fractional Periods↩︎

As noted above, an essential component of our argument is to extend the allowed values of periods to real numbers. We do so by requiring that a task \(i\) with possibly non-integer period \(a _i > 0\) be performed at least \(\lfloor r / a _i \rfloor\) times during any \(r\)-day interval. The new frequency condition for task \(i\) is thus

for each \(r \in \mathbb{N}\) and \(m \in \mathbb{Z}\), the set \([m, m + r) \cap S _i\) has at least \(\lfloor r / a _i \rfloor\) elements,

or equivalently that

for each \(s \in \mathbb{N}\) and \(m \in \mathbb{Z}\), the set \([m, m + s \cdot a _i) \cap S _i\) has at least \(s\) elements.

For integer-valued \(a_i\) this reduces to the original condition, but for non-integer \(a_i\) it acquires new significance. For example, a period \(a _i = \frac{7}{2}\) requires that task \(i\) be performed at least once every \(4\) days and at least twice every \(7\) days; this could be achieved if we schedule task \(i\) every week on Monday and Thursday, but not if we schedule it on Monday and Wednesday, or if we simply schedule it every \(4\) days.

One advantage of this extension is that it does not affect the validity of the basic properties of schedulability stated in the following lemma, which have been used, explicitly or implicitly, in previous studies as well (for integer periods). We write \(A \sqcup B\) for the instance consisting simply of tasks in \(A\) and tasks in \(B\); for example, \((6, 6) \sqcup (4, 4, 6) = (4, 4, 6, 6, 6)\).

This is the only preparation we need to give a concise proof of Theorem 2, which we pause to state here (though it is not needed for Theorem 1).

Proof of Theorem 2.. By the splitting property of Lemma 1, we may assume we have just one task with each period value, i.e., our instance is \((a _1, a _2)\) with \(1 / a _1 + 1 / a _2 \leq 1\). By the weakening property, we may further assume \(1 / a _1 + 1 / a _2 = 1\).

Consider a circle with circumference \(1\), and two particles \(i \in [2]\) moving along it in opposite directions, each with daily speed \(1 / a _i\) (Figure 2),

so that they meet each other every midnight, which is possible because \(1 / a _1 + 1 / a _2 = 1\). Fix any point on the circle and call it the reference point. During each day, the two particles together cover the circumference exactly once, and thus exactly one of them sees the reference point, under the convention (just in case the particles happen to meet at the reference point) that midnight belongs to the next day for particle \(1\), and to the previous day for particle \(2\). This allows us to schedule each task \(i \in [2]\) on days when particle \(i\) sees the reference point. This schedule satisfies the frequency condition for both tasks \(i\), because the time at which particle \(i\) is at the reference point and the time at which it comes back there after going around the circle \(s \in \mathbb{N}\) times are \(s \cdot a _i\) apart, and thus they belong to days that are at most \(\lceil s \cdot a _i \rceil\) apart. ◻

The fact that we obtain two complementary sets of integers in this way is sometimes referred to as Rayleigh’s or Beatty’s theorem, especially when \(a _1\) and \(a _2\) are irrational.

Proof of the Main Theorem↩︎

Let \(\mathscr A _{5 / 6}\) be the set of instances in Theorem 1, i.e., those with integer periods and density \(\leq \frac{5}{6}\). This is an infinite set, and we want to prove all its instances schedulable by reducing them to finitely many instances so that schedulability can be checked exhaustively. For this purpose, we repeatedly apply the procedures of Lemma 1 in reverse to each instance in \(\mathscr A _{5 / 6}\), obtaining an instance that has only small periods. Consider the following process which can be applied to any instance whose largest period \(a\) exceeds a threshold \(\theta > 0\):

• if the instance contains only one task with period \(a\), decrease its period to the second largest period \(> \theta\), or to \(\theta\) in case \(a\) is the only period \(> \theta\);

• if the instance contains multiple tasks with period \(a\), replace two of them with a single task with period \(a / 2\).

Given an instance \(A\) and \(\theta > 0\), every second application of this process strictly decreases the number of tasks with period \(> \theta\), so we eventually arrive at an instance with all periods \(\leq \theta\). We write \(\mathit{fold}_\theta (A)\) for the resulting instance (because Gąsieniec, Smith and Wild [10] call a variant of this process folding). For example, \(\mathit{fold}_{22} (3, 4, 8, 17, 42, 55, 72) = (3, 4, 8, \frac{55}{4}, 17)\).

Thus, given \(A \in \mathscr A _{5 / 6}\), the resulting instance \(\mathit{fold}_\theta (A)\) belongs to the set \(\mathscr B _{5 / 6} ^\theta\) of instances \(B\) that have periods in \(((0, \theta / 2] \cap \mathbb{N}) \cup (\theta / 2, \theta]\) and density \(\mathrm D(B) \leq \frac{5}{6} + 1 / \theta\). But \(\mathscr B _{5 / 6} ^\theta\) is still an infinite set. Given \(B \in \mathscr B _{5 / 6} ^\theta\), we define \(\mathit{round}_\theta (B)\) to be the instance obtained from \(B\) by replacing every period \(b > \theta / 2\) by \(\lceil b \rceil - 1\), the largest integer less than \(b\). For example, \(\mathit{round}_{22} (3, 4, 8, \frac{55}{4}, 17) = (3, 4, 8, 13, 16)\). Again by the weakening property of Lemma 1, \(B\) is schedulable if \(\mathit{round}_\theta (B)\) is. Note that \(\mathrm D' (\mathit{round}_\theta (B)) \leq \mathrm D(B)\), where \[\mathrm D' \bigl( (c _i) _{i \in [k]} \bigr) = \sum _{i \in [k]} \begin{cases} \dfrac{1}{c _i} & \text{for c _i < \biggl\lfloor \dfrac \theta 2 \biggr\rfloor}, \\[10pt] \dfrac{1}{c _i + 1} & \text{for c _i \geq \biggl\lfloor \dfrac \theta 2 \biggr\rfloor}. \end{cases}\] We have thus reduced the schedulability of \(A\) to that of \(\mathit{round}_\theta (\mathit{fold}_\theta (A))\), which belongs to the set \(\mathscr C _{5 / 6} ^\theta\) of instances \(C\) that consist of integer periods \(< \theta\) and satisfy \(\mathrm D' (C) < \frac{5}{6} + 1 / \theta\). Theorem 1 follows if we manage to schedule all instances in the finite set \(\mathscr C _{5 / 6} ^\theta\). Of course, there is a danger that our reduction has been too sloppy, so that \(\mathit{round}_\theta (\mathit{fold}_\theta (A))\) may be unschedulable despite \(A\) being schedulable. But in fact, choosing \(\theta = 22\) proves this strategy successful: we have verified the following lemma (and hence Theorem 1) by exhaustive computer search as we explain in the next section.

Scheduling Small Instances (Verifying Lemma 3)↩︎

Although we have reduced Theorem 1 to Lemma 3, which can be verified exhaustively in principle, we still face a substantial computational challenge. The previously best experimental result of this kind [10] has shown that every instance with \(12\) tasks and density \(\leq 5 / 6\) is schedulable. We need a significant improvement over it, not only because our set \(\mathscr C _{5 / 6} ^{22}\) simply has far more instances, but also because these instances are typically longer and denser and hence are harder to schedule. In this section, we describe some techniques we used to carry out the computation in a reasonable amount of time. A computer program implementing these ideas is publicly available on GitHub at

https://github.com/a7kawamura/pinwheel_solver .

State Transition Graphs↩︎

Although solutions to the PS problem are infinite schedules, the following discussion shows [4] that any schedulable instance \(A = (a _i) _{i \in [k]}\) has a schedule that repeats a finite sequence of tasks (perhaps justifying the name of “pinwheel” scheduling).

The elements of \([a _1] \times \dots \times [a _k]\) are called states; if, at the end of a day, we are in a state \((u _i) _{i \in [k]}\), this means that each task \(i\) must be performed no later than \(u_i\) days from now. Performing task \(j \in [k]\) on the next day will bring us into state \(u' = (u' _i) _{i \in [k]}\), where \[u' _i = \begin{cases} a _i & \text{if i = j}, \\ u _i - 1 & \text{otherwise}. \end{cases}\] We write \(u \vdash u'\) if this is the case for some \(j\). An instance \(A\) is thus schedulable if and only if its state transition graph, i.e., the directed graph with states as nodes and the relation \(\mathord\vdash\) as edges, admits an infinite walk, or equivalently, contains a cycle. This is why any schedulable instance has a cyclic schedule, and also why schedulability can be checked in finite amount of time by simply searching this graph for a cycle.

The challenge is that the number of states in the graph is exponential in the size of the input \(A\), and there indeed exist families of schedulable instances whose graphs have only exponentially long cycles. Thus, although the problem of deciding schedulability is in \(\mathsf{PSPACE}\) [4] by Savitch’s theorem, we do not know whether it is in, say, \(\mathsf{NP}\). There is evidence [41] that it is not likely in \(\mathsf{P}\), and some even conjecture [42] that it is \(\mathsf{PSPACE}\)-complete.

This lack of a notion of a solution of polynomial length (certificate of schedulability) gives rise to some subtlety in discussing algorithms that are supposed to “output” a schedule. One (somewhat informal) way to formulate efficient scheduling in this context is to require that we can, given an instance, generate in polynomial time a fast online scheduler (FOLS), i.e., a program that efficiently computes which task to perform each day [4]. Many of the results about PS or BGT, including our Theorem 3, give such algorithms.

Techniques for Fast Scheduling↩︎

We thus have to search an extremely large graph, but there are some ways to reduce the computational cost by exploiting the way this graph is built.

One useful observation is that if the state transition graph has a cycle, then the cycle is reachable from the state \((a _i) _{i \in [k]}\). Thus, we can simply start a depth-first search at this initial state, while maintaining the path to the current state until this path makes a cycle. We also keep the set of dead states found so far, i.e., those that never lead to a cycle: for any \(i \in \mathbb{N}\), a state with more than \(i\) entries \(\leq i\) is dead (this is a benefit of our “urgency” formulation of states [10]), and a state that only leads to dead states is dead.

Another improvement comes from the observation that, since our instances have periods up to \(21\), and the hard instances are those with \(12\)–\(18\) tasks, they typically have multiple tasks with the same period. We thus identify states up to permuting them. This reduces not just the number of states, but also the out-degree of each state, because we may assume that among multiple tasks with the same period, we always perform the least recently performed one. This significantly reduces the number of states and edges, and also often the length of the sought repeating pattern. For example, instead of saying that the instance \((2, 4, 8, 8)\) has a schedule that repeats the \(8\)-day pattern \((1, 2, 1, 3, 1, 2, 1, 4)\) of tasks, we can more economically say that this schedule just repeats the \(4\)-day sequence \((2, 4, 2, 8)\) of periods, with the understanding that the two period-\(8\) tasks always take turns.

Reducing the Number of Instances to Check↩︎

Given that scheduling each individual instance can take time, it is important to cut down on computation so that we do not have to check all the \(25\,592\,972\) instances in \(\mathscr C _{5 / 6} ^{22}\).

A trivial but useful idea to do so is that, by the weakening property of Lemma 1, we can skip instances that componentwise dominate another instance in \(\mathscr C _{5 / 6} ^{22}\). In our program, we call the remaining \(676\,225\) instances critical.

We may also hope to reduce the number of instances by using a value smaller than \(22\) in Lemma 3. Unfortunately, we cannot: for example, the instance \((3, 4, 7, 10, 15) \in \mathscr C _{5 / 6} ^{20}\) is not schedulable, even though \(\frac{1}{3} + \frac{1}{4} + \frac{1}{7} + \frac{1}{11} + \frac{1}{16} < \frac{5}{6} + \frac{1}{20}\) (it is still possible, for all we know, that all instances in \(\mathscr B _{5 / 6} ^{20}\) are schedulable). But such instances are not so common: most instances in \(\mathscr C _{5 / 6} ^{22}\) remain schedulable even if we “apply folding a bit further”. For an instance \(C = (c _i) _{i \in [k]}\) of length \(k > 1\), define \(C' = (c' _i) _{i \in [k - 1]}\) as the result of applying to \(C\) a single step of folding and then rounding down: \(c' _i = c _i\) for \(i = 1\), …, \(k - 2\) and \(c' _{k - 1} = \lfloor c _{k - 1} / 2 \rfloor\). Again by Lemma 1, \(C\) is schedulable if \(C'\) is. Since \(C'\) has fewer tasks than \(C\), it is often easier to schedule, although of course there is a risk that \(C'\) may not be schedulable while \(C\) is. Thus, to check the schedulability of \(C\), we run the cycle-detecting algorithm above on instances \(C\), \(C'\), \(C''\), …(some of which may not be in \(\mathscr C _{5 / 6} ^{22}\)) in parallel, until one of them turns out schedulable.

This idea of continued folding causes many instances to be proved schedulable via a common short instance, thereby reducing the number of instances that we need to actually schedule. We ended up completing the cycle-detecting algorithm only about \(60\,000\) times (this number fluctuates depending on which of the parallel searches finish first).

Approximation Algorithm for BGT↩︎

We will provide a polynomial-time algorithm \(M\) that, given an integer PS instance \((a _i) _{i \in [k]}\), either accepts or rejects so that

• if \(M\) rejects, \((a _i) _{i \in [k]}\) is not schedulable, and

• if \(M\) accepts, \(\bigl( \bigl\lfloor \frac{4}{3} \cdot a _i \bigr\rfloor \bigr) _{i \in [k]}\) is schedulable;

note that for some instances, both answers are allowed. This implies Theorem 3: given a BGT instance \((h _i) _{i \in [k]}\), we can binary search for an integer height \(G\) such that \(M\) accepts \((\lfloor G / h _i \rfloor) _{i \in [k]}\) but not if \(G\) is replaced by \(G - 1\).

Such an algorithm \(M\) comes easily if all periods \(a _i\) are \(> 2\): simply accept \((a _i) _{i \in [k]}\) if and only if it has density \(\leq 1\). When \(M\) accepts, the instance \(\bigl( \bigl\lfloor \frac{4}{3} \cdot a _i \bigr\rfloor \bigr) _{i \in [k]}\) has density \(\leq \frac{5}{6}\) (unless \(k = 1\)) and is hence schedulable by Theorem 1. This is because elementwise we have \(\bigl\lfloor \frac{4}{3} \cdot a \bigr\rfloor \geq \frac{6}{5} \cdot a\), as can be verified by \[\biggl\lfloor \frac{4}{3} \cdot a \biggr\rfloor \geq \frac{4}{3} \cdot a - \frac{2}{3} \geq \frac{4}{3} \cdot a - \frac{2}{15} \cdot a = \frac{6}{5} \cdot a\] for \(a \geq 5\) and individually for \(a = 3\), \(4\).

Since this inequality does not hold for \(a = 2\), We need special treatment for period \(2\). But a period-\(2\) task is actually easy to deal with: we can safely schedule it regularly every other day, since performing the same task for two days in a row would only make scheduling other tasks harder. Precisely,

Using this fact, our algorithm \(M\) eliminates (recursively if needed) period \(2\) before scheduling other tasks. That is, given a PS instance \(A = (a _i) _{i \in [k]}\):

• If \(A\) is empty, accept.

• If \(a _1 = 2\), recursively apply \(M\) itself to the instance \(\bigl( \bigl\lfloor \frac{1}{2} \cdot a _{i + 1} \bigr\rfloor \bigr) _{i \in [k - 1]}\).

• If \(a _1 \neq 2\), accept if \(A\) has density \(\leq 1\), and reject otherwise.

Note that the first case is correct by Lemma 4: the rejecting case is the “only if” direction of the lemma, and the accepting case is because the schedulability of the instance \(\bigl( \bigl\lfloor \frac{4}{3} \cdot \bigl\lfloor \frac{1}{2} \cdot a _{i + 1} \bigr\rfloor \bigr\rfloor \bigr) _{i \in [k - 1]}\) implies that of the elementwise dominating instance \(\bigl( \bigl\lfloor \frac{1}{2} \cdot \bigl\lfloor \frac{4}{3} \cdot a _{i + 1} \bigr\rfloor \bigr\rfloor \bigr) _{i \in [k - 1]}\), and hence of \(\bigl( \bigl\lfloor \frac{4}{3} \cdot a _i \bigr\rfloor \bigr) _{i \in [k]}\) by the “if” direction of the lemma.

Conclusion↩︎

Our Theorem 1 thus enters the list of simply stated mathematical conjectures that have been confirmed by a computer experiment while defying attempts at human-readable proofs for decades. This list includes classics such as the Four-Colour Theorem [43] and Kepler Conjecture [44], as well as more recent examples in enumerative combinatorics reported in this journal [45], [46]. As the authors of both these last papers remark in their final paragraphs, we are still left with the intriguing human challenge of finding a short and elegant “proof from The Book”, although they both point out the possibility that such a short explanation may not exist.

Let us end with a conjecture that came out of the computer experiment for Theorem 1 but which seems to require a better understanding of its proof. In the course of our experiment, it turned out that the bound \(\frac{5}{6}\) was supported by very few instances: there is a positive constant \(\varepsilon\) such that the only unschedulable instances with density \(< \frac{5}{6} + \varepsilon\) are of the form \((2, 3, a _3)\) or \((3, 4, 4, a _4)\). Instances of the form \((a _1, a _1 + 1, \dots, a _1 + 1, a _{a _1 + 1})\) are likewise unschedulable, but have higher density for \(a _1 > 3\). We suspect that these are the sparsest unschedulable instances starting with each fixed value of \(a _1\):

Since the results in this paper were announced [3], some authors have studied such relations between the density bound and the smallest period \(a _1\) [13], [39], [47].

Acknowledgements↩︎

The author is grateful to Hiroshi Fujiwara, Leszek Gąsieniec, Yusuke Kobayashi, Igor Potapov, Benjamin Smith and Sebastian Wild, as well as the anonymous referees, for valuable comments on earlier versions of this work. He also thanks Keita Hiroshima for helping to verify the list of schedules in Lemma 3. This research was supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research) JP23K28036 and by the Kyoto University Foundation.

References↩︎