On the uniform generation of random derangements

1 Introduction

Derangements are permutations $σ = σ_{1} \dots σ_{n}$ on $n \geq 2$ labels such that $σ_{i} \neq i$ for all $i = 1, \dots, n$ . Besides being useful as permutations, derangements are important per se in a number of applications like in the testing of software branch instructions and random paths and data randomization and experimental design (Sedgewick & Flajolet, 2013; Edgington & Onghena, 2007). A recent review on the generation of random permutations appeared in Bacher et al. (2017). A well known algorithm to generate random derangements is Sattolo’s algorithm, that outputs a random cyclic derangement on $n$ labels in $O (n)$ time (Sattolo, 1986; Gries & Xue, 1988; Prodinger, 2002). An explicit $O (2 n)$ algorithm to generate random derangements in general (not only cyclic derangements) has been given in Panholzer et al. (2004) and Martínez et al. (2008). Algorithms to generate all $n$ -derangements are also known (Baril & Vajnovszki, 2004; Korsh & LaFollette, 2004; Wilson, 2009).

In this letter we propose and test two procedures to generate random derangements with the expected distribution of cycle lengths: one based on the randomization of derangements and the other based on a simple sequential importance sampling scheme. Simulations show that the randomized algorithm samples a derangement uniformly in $O (n log n)$ time while the sequential importance sampling algorithm does it in $O (n)$ time but with a small probability $\sim O (1 / n)$ of failing. The proposed algorithms do not use pre-calculated quantities or auxiliary data structures, being straighforward to understand and implement.

2 Mathematical preliminaries

Let us briefly review some notation and terminology on permutations; for details see Charalambides (2002) and James & Kerber (1981).

We denote the set (that forms a group under the operation of composition) of all permutations on $n \geq 2$ labels ${1, \dots, n}$ by $S_{n}$ . We write an $n$ -permutation in one-line notation as $σ = σ_{1} \dots σ_{n}$ , where $σ_{i} = σ (i)$ . A cycle of length $k \leq n$ in a $n$ -permutation $σ$ is a sequence of indices $i_{1}, \dots, i_{k}$ such that $σ_{i_{1}} = i_{2}$ , …, $σ_{i_{k - 1}} = i_{k}$ , and $σ_{i_{k}} = i_{1}$ , completing the cycle. Fixed points are $1$ -cycles, while transpositions are $2$ -cycles. An $n$ -permutation with $a_{k}$ cycles of length $k$ , $1 \leq k \leq n$ , is said to be of type $(a_{1}, \dots, a_{n})$ , with $\sum_{k} k a_{k} = n$ . For example, the $9$ -permutation $174326985 = (8) (6) (34) (2795) (1)$ has $5$ cycles and is of type $(3, 1, 0, 1)$ , where we have omitted the trailing $a_{5} = a_{6} = \dots = a_{9} = 0$ .

The number of $n$ -permutations with $k$ cycles is given by the unsigned Stirling number of the first kind $[\frac{n}{k}]$ . We have $[\frac{n}{n}] = 1$ , counting just the identity permutation $(1) (2) \dots (n)$ , $[\frac{n}{n - 1}] = (\frac{n}{2})$ , counting $n$ -permutations of $n - 2$ fixed points, that can be taken in $(\frac{n}{n - 2}) = (\frac{n}{2})$ different ways, plus a transposition of the remaining two labels, and $[\frac{n}{1}] = (n - 1)!$ , the number of $1$ -cycle (or cyclic) $n$ -derangements. It can be shown that $[\frac{n}{2}] = (n - 1)! H_{n - 1}$ , where $H_{n}$ is the $n$ -th harmonic number. Other useful formulae involving Stirling numbers of the first kind are $[\frac{0}{0}] = 1$ , $[\frac{n}{0}] = 0$ , and the recursion relation

[\frac{n}{k}] = (n - 1) [\frac{n - 1}{k}] + [\frac{n - 1}{k - 1}] .

(1)

Obviously, $[\frac{n}{1}] + \dots + [\frac{n}{n}] = n!$ .

Let us denote the set (that does not form a group) of all $n$ -derangements by $D_{n}$ . It is well known that

| D_{n} | = n! (1 - \frac{1}{1!} + \dots + \frac{(- 1)^{n}}{n!}) = ⌊ \frac{n! + 1}{e} ⌋, n \geq 1,

(2)

the so-called rencontres numbers (OEIS A000166). Let us also denote the set of $k$ -cycle $n$ -derangements, irrespective of their type, by $D_{n}^{(k)}$ . Note that $D_{n}^{(k)} = \emptyset$ for $k > ⌊ n / 2 ⌋$ . If we want to generate random $n$ -derangements uniformly over $D_{n} = D_{n}^{(1)} \cup \dots \cup D_{n}^{(⌊ n / 2 ⌋)}$ , we must be able to generate $k$ -cycle random $n$ -derangements with probabilities

(3)

To calculate these probabilities we need to determine $| D_{n}^{(k)} |$ . Perusal of the inclusion-exclusion principle furnishes

| D_{n}^{(k)} | = k \sum j = 0 (- 1)^{j} (\frac{n}{j}) [\frac{n - j}{k - j}] .

(4)

Equation (4) recovers $| D_{n}^{(1)} | = [\frac{n}{1}] = (n - 1)!$ , while we find that $| D_{n}^{(2)} | = (n - 1)! (H_{n - 2} - 1)$ for $n \geq 2$ . Accordingly, already for small $n$ (say, $n \geq 8$ ) we have $P (σ \in D_{n}^{(1)}) ≃ e / n$ and $P (σ \in D_{n}^{(2)}) ≃ (H_{n - 2} - 1) e / n$ .

3 Generating random derangements by random transpositions

Our first approach to generate random $n$ -derangements correctly distributed over $D_{n}$ consists in taking an initial cyclic $n$ -derangement and to scramble it by random restricted transpositions enough to obtain the required distribution. By restricted transpositions we mean swaps $σ_{i} \leftrightarrow σ_{j}$ avoiding pairs for which $σ_{i} = j$ or $σ_{j} = i$ . Algorithm 1 describes the generation of random $n$ -derangements according to this idea, where $m i x \geq n / 2$ is a constant establishing the amount of restricted transpositions to be attempted and $r n d$ is a computer generated pseudorandom uniform deviate in $(0, 1)$ .

The initial derangement in Algorithm 1 does not need to be cyclic, but this minimizes the risk of a careless implementation botching up the algorithm. We always start with the cycle $23 \dots n 1$ . The minimum number of transpositions necessary to turn a cyclic $n$ -derangement into a $k$ -cycle $n$ -derangement is $k - 1$ , $1 \leq k \leq ⌊ n / 2 ⌋$ , since transpositions of labels that belong to the same cycle split it into two cycles,

(a b) (i_{1} \dots i_{a - 1} i_{a} i_{a + 1} \dots i_{b - 1} i_{b} i_{b + 1} \dots i_{k}) = (i_{1} \dots i_{a - 1} i_{b} i_{b + 1} \dots i_{k}) (i_{a + 1} \dots i_{b - 1} i_{a}),

(5)

and, conversely, transpositions involving labels of different cycles join them into a single one.

Remark 1.

Algorithm 1 is applicable only for $n \geq 4$ , as it is not possible to connect the even permutations $231$ and $312$ by a single transposition.

0: Cyclic

n

-derangement

σ_{1} \dots σ_{n}

m i x \leftarrow

number of restricted transpositions to attempt

2: for

s = 1

m i x

i \leftarrow ⌈ r n d \cdot n ⌉

j \leftarrow ⌈ r n d \cdot n ⌉

4: if

(σ_{i} \neq j) \land (σ_{j} \neq i)

then

5: swap

σ_{i} \leftrightarrow σ_{j}

6: end if

7: end for

Algorithm 1 Random derangements by random restricted transpositions

We run Algorithm 1 for different values of $m i x \geq n / 2$ and collect data. Our results appear in Table 1. We choose $n = 64$ because the difference between $2 n = 128$ and $n log n = 266$ is significant in this case. From Table 1 we clearly see that $m i x = n / 2$ random restricted transpositions are unable to lead the initial cyclic derangement into higher $k$ -cycle derangements—there is an excess of probability mass in the lower $k$ -cycle sets with $k = 1, 2$ , and $3$ . The same imbalance can be noted, although less clearly, with $m i x = n$ random restricted transpositions. Figures for derangements of higher cycle number fluctuate more due to the finite size of the sample. However, while the difference between trying to scramble the initial cyclic $n$ -derrangement by $n$ and $2 n$ restricted transpositions is significant, the difference between attempting $2 n$ or $n log n$ restricted transpositions is much less pronounced. Our data suggest that Algorithm 1 can efficiently generate a random $n$ -derangement correctly distributed on $D_{n}$ in $O (2 n)$ time employing of the order of $4 n$ pseudorandom numbers in the process. This is further discussed in Section 5.

Cycles	Algorithm 1, nr. restricted transpositions attempted				Algorithm 2	Exact
$k$	$n / 2$	$n$	$2 n$	$n log n$	—	Eqs. (2)–(4)
$1$	$0.048 055$	$0.042 933$	$0.042 479$	$0.042 473$	$0.042 475$	$0.042 473$
$2$	$0.160 153$	$0.158 395$	$0.157 691$	$0.157 679$	$0.157 684$	$0.157 677$
$3$	$0.278 164$	$0.260 129$	$0.258 787$	$0.258 765$	$0.258 788$	$0.258 772$
$4$	$0.241 317$	$0.252 739$	$0.253 304$	$0.253 305$	$0.253 306$	$0.253 301$
$5$	$0.167 413$	$0.167 189$	$0.167 621$	$0.167 639$	$0.167 622$	$0.167 635$
$6$	$0.070 203$	$0.079 498$	$0.080 390$	$0.080 402$	$0.080 389$	$0.080 400$
$7$	$0.026 470$	$0.028 825$	$0.029 192$	$0.029 195$	$0.029 196$	$0.029 200$
$8$	$0.006 457$	$0.008 087$	$0.008 269$	$0.008 274$	$0.008 272$	$0.008 274$
$9$	$0.001 498$	$0.001 821$	$0.001 868$	$0.001 869$	$0.001 868$	$0.001 869$
$10$	${2.317}_{- 4}$	${3.292}_{- 4}$	${3.416}_{- 4}$	${3.418}_{- 4}$	${3.412}_{- 4}$	${3.417}_{- 4}$
$11$	${3.523}_{- 5}$	${4.914}_{- 5}$	${5.109}_{- 5}$	${5.120}_{- 5}$	${5.103}_{- 5}$	${5.116}_{- 5}$
$12$	${3.619}_{- 6}$	${5.997}_{- 6}$	${6.322}_{- 6}$	${6.301}_{- 6}$	${6.354}_{- 6}$	${6.326}_{- 6}$
$13$	${3.639}_{- 7}$	${6.215}_{- 7}$	${6.493}_{- 7}$	${6.301}_{- 7}$	${6.507}_{- 7}$	${6.499}_{- 7}$
$14$	${2.53}_{- 8}$	${4.83}_{- 8}$	${5.40}_{- 8}$	${5.57}_{- 8}$	${5.44}_{- 8}$	${5.569}_{- 8}$
$15$	${1.2}_{- 9}$	${4.6}_{- 9}$	${3.1}_{- 9}$	${3.0}_{- 9}$	${4.1}_{- 9}$	${3.989}_{- 9}$
$16$	$2_{- 10}$	$4_{- 10}$	$1_{- 10}$	$3_{- 10}$	$1_{- 10}$	${2.390}_{- 10}$
runtime (sec)	$9357$	$11298$	$15068$	$25238$	$18954$	–

Table 1: Proportion of

n

-derangements in

D_{n}^{(k)}

observed in

10^{10}

samples generated by Algorithms 1 and 2 for

n = 64

. Data for Algorithm 2 are based on a run that performed with a ratio of completed/attempted derangements of

0.985 471

. The notation

n_{- a}

reads

n \times 10^{- a}

. The last line of the table gives runtimes for comparison.

Remark 2.

It is a classic result that $O (n log n)$ transpositions are needed before a shuffle becomes “sufficiently random” (Aldous & Fill, 2002; Levin & Peres, 2017; Diaconis, 1988). A similar analysis for random restricted transpositions over derangements is complicated by the fact that derangements do not form a group. Recently, the analysis of the spectral gap of the Markov transition kernel of the process of restricted transpositions over derangements provided the bound $m i x > C n + a n log n^{2}$ , with $a > 0$ and $C \geq 0$ a decreasing continuous function (Smith, 2015)

. This bound results from involved estimations and approximations and may not be very accurate. Related results appear in the remarkable (and difficult) paper by

Hanlon (1996). We are not aware of other rigorous results on this particular problem.

4 Sequential importance sampling of derangements

Algorithm 2

describes a sequential importance sampling (SIS) algorithm to generate random derangements inspired by the analogous problem of sampling contingency tables with restrictions

(Diaconis et al., 2001; Chen et al., 2005) as well as the problem of estimating the permanent of a matrix (Rasmussen, 1994; Kuznetsov, 1996; Jerrum et al., 2004)—namely, the permanent of the

n \times n

matrix with

0

on the diagonal and

1

elsewhere.

J \leftarrow {1, \dots, n}

d \leftarrow 0

2: for

i = 1

n - 1

3: choose

j \in J ∖ {i}

uniformly at random

σ_{i} \leftarrow j

J \leftarrow J ∖ {j}

d \leftarrow d + 1{1} {j = n}

7: end for

8: if

d > 0

then

σ_{n} \leftarrow

the remaining label

j \in J

10: else

11: fail

12: end if

Algorithm 2 Random derangements by sequential importance sampling

In the $i$ -th iteration of the loop in Algorithm 2 (lines 2–7), $σ_{i}$ can pick (lines 3–4) one of

| J_{i} | = n - i + i - 1 \sum j = 1 1{1} {σ_{j} = n}

(6)

available labels, where the indicator function $1{1} {P} = 1$ if $P$ is true and $0$ otherwise—i. e., $σ_{i}$ can choose among either $n - i$ or $n - i + 1$ labels, depending on whether in the $i$ -th iteration label $i$ itself has already been picked. Note that $J_{i}$ is never empty during the execution of the algorithm. This guarantees the construction of the $n$ -derangement till the last but one element $σ_{n - 1}$ . The $n$ -derangement will be completed only if the last remaining label $j \neq n$ , such that $σ_{n}$ does not pick $n$ . Variable $d$ (line 6) monitors this event: if after $n - 1$ choices no one picked label $n$ , $d = 0$ and the derangement failed. The probability that Algorithm 2 fails is thus given by

P (σ_{n} = n) = P (σ_{1} \neq n) P (σ_{2} \neq n ∣ σ_{1} \neq n) \dots P (σ_{n - 1} \neq n ∣ σ_{1} \neq n, \dots, σ_{n - 2} \neq n) .

(7)

Now, $P (σ_{i} \neq n ∣ \dots) = 1 - P (σ_{i} = n ∣ \dots)$ with (Algorithm 2, line 3)

(8)

and since

E (| J_{i} |) = E (n - i + i - 1 \sum j = 1 1{1} {σ_{j} = n}) = n - i + \frac{i - 1}{n}

(9)

we deduce that Algorithm 2 fails with probability

P (σ_{n} = n ∣ \dots) = n - 1 \prod i = 1 \frac{(n - 1) (n - i) - 1}{n (n - i) + i - 1} \sim O (\frac{1}{n}) .

(10)

According to (10), for $n = 64$ Algorithm 2 fails with probability $0.014 492$ ; compare this figure with the observed failure rate $1 - 0.985 471 = 0.014 529$ given in Table 1.

5 Mixing times of the restricted transpositions shuffle

To shed some light on the question of how many random restricted transpositions are necessary to generate random derangements uniformly over $D_{n}$ , we investigate the convergence of Algorithm 1 numerically. This can be done by monitoring the evolution of the empirical probabilities observed along the run of the algorithm towards the exact probabilities given by (3)–(4).

Let $ν$ be the measure that puts mass on the set $D_{n}^{(k)}$ and $μ_{t}$ be the empirical measure

μ_{t} (k) = \frac{1}{t} t \sum s = 1 1{1} {σ_{s} \in D_{n}^{(k)}},

(11)

where $σ_{s}$ is the derangement obtained after attempting $s$ restricted transpositions by Algorithm 1 on a given initial derangement $σ_{0}$ . The chi-square distance between $ν$ and $μ_{t}$ is given by

d (t) = ∥ μ_{t} - ν ∥_{2, ν} = ⌊ n / 2 ⌋ \sum k = 1 \frac{[μ_{t} (k) - ν (k)]^{2}}{ν (k)} .

(12)

Distance $d (t)$ allows us to define $τ_{m i x} (ε)$ of the process as the time it takes for $μ_{t}$ to fall within distance $ε$ of $ν$ ,

τ_{m i x} (ε) = min {t \geq 0 : d (t) < ε} .

(13)

It is usual to define the mixing time $τ_{m i x}$ by setting $ε = \frac{1}{4}$ or $ε = e^{- 1}$

, a figure reminiscent of the spectral analysis of Markov chains. In what follows we set

ε = \frac{1}{4}

Figure 1: Chi-square distance $⟨ d (t) ⟩$ (averaged over $10^{6}$ runs) between the empirical measure $μ_{t}$ and the stationary measure $ν$ of the process defined by Algorithm 1 for $n = 128$ with $μ_{0} (1) = 1$ . The dotted line indicates the level $\frac{1}{4}$ , that meets $⟨ d (t) ⟩$ at $t = τ_{m i x} = 861$ .

Starting with a cyclic derangement, i. e., with $μ_{0} (1) = 1$ and all other $μ_{0} (k) = 0$ , we run Algorithm 1 and measure $d (t)$ for some time. Figure 1 displays the average $⟨ d (t) ⟩$ over $10^{6}$ runs for $n = 128$ . The behavior of $⟨ d (t) ⟩$ does not show any sign of the cutoff phenomenon (Diaconis, 1988; Aldous & Fill, 2002; Levin & Peres, 2017). Our data indicate that

τ_{m i x} \sim O (n log n),

(14)

which roughly agrees with the bound given in Smith (2015). Table 2 lists data for derangements of larger sizes; all seem to behave like $O (n log n)$ to leading order.

$n$	$τ_{m i x}$	$n log n$	$τ_{m i x} / n log n$
$64$	$543$	$266$	$2.04$
$128$	$861$	$621$	$1.39$
$256$	$1396$	$1420$	$0.98$
$512$	$2085$	$3194$	$0.65$
$1024$	$3347$	$7098$	$0.47$

Table 2: Measured mixing time

τ_{m i x}

compared to

n log n

6 Summary

While a simple acception-rejection algorithm generates random derangements in $O (n)$ with an acceptance rate of $\sim e^{- 1}$ $≃ 0.367$ , thus being $O (e \cdot n)$ (the cost of verifying if the permutation generated is a derangement is negligible), Sattolo’s $O (n)$ algorithm only generates cyclic derangements, and Martínez-Panholzer-Prodinger algorithm, with guaranteed uniformity, is $2 n + O ({log}^{2} n)$ , we described two procedures that are competitive for the efficient generation of random derangements. We found, empirically (Tables 1 and 2), that $O (n log n)$ random restricted transpositions suffice to spread an initial $n$ -derangement correctly over $D_{n}$ with the expected distribution of cycle lengths. In terms of the amount of pseudorandom numbers employed, Algorithm 1 employs of the order of $2 n log n$ pseudorandom numbers and Algorithm 2 (SIS) employs $n + O (1)$ pseudorandom numbers to generate an $n$

-derangement uniformly distributed over

D_{n}

. The advantage of the SIS algorithm is obvious.

Acknowledgments

The author acknowledges partial financial support from FAPESP (Brazil) through grant No. 2017/22166-9.

References

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On the uniform generation of random derangements

Efficient uniform generation of random derangements with the expected distribution of cycle lengths

An attempt to trace the birth of importance sampling

Randomized sequential importance sampling for estimating the number of perfect matchings in bipartite graphs

Symmetrized importance samplers for stochastic differential equations

Sequential Importance Sampling With Corrections For Partially Observed States

On estimating the alphabet size of a discrete random source

Sequential importance sampling for multi-resolution Kingman-Tajima coalescent counting

1 Introduction

2 Mathematical preliminaries

3 Generating random derangements by random transpositions

Remark 1.

Remark 2.

4 Sequential importance sampling of derangements

5 Mixing times of the restricted transpositions shuffle

6 Summary

Acknowledgments

References

References

On the uniform generation of random derangements

Related Research

Efficient uniform generation of random derangements with the expected distribution of cycle lengths

An attempt to trace the birth of importance sampling

Randomized sequential importance sampling for estimating the number of perfect matchings in bipartite graphs

Symmetrized importance samplers for stochastic differential equations

Sequential Importance Sampling With Corrections For Partially Observed States

On estimating the alphabet size of a discrete random source

Sequential importance sampling for multi-resolution Kingman-Tajima coalescent counting

1 Introduction

2 Mathematical preliminaries

3 Generating random derangements by random transpositions

Remark 1.

Remark 2.

4 Sequential importance sampling of derangements

5 Mixing times of the restricted transpositions shuffle

6 Summary

Acknowledgments

References

References