Enumerate Permutations in a Ball

In this section, we give a recursive algorithm to enumerate all frequency permutations in B(d, e). For convenience, let x_i denote the i-th entry of x in the rest of this section. First, we investigate e closely.

i · · · kλ − λ kλ − λ + 1 · · · kλ kλ + 1 · · · e(i) · · · k− 1 k · · · k k + 1 · · ·

Observe that symbol k appears at the (kλ− λ + 1)-th, . . . , (kλ)-th positions in e. Therefore, x is d-close to e if and only if x_kλ−λ+1, . . . , x_kλ ∈ [k − d, k + d]. (Note that we simply ignore the non-positive values when k ≤ d.) In other words, ℓ∞(e, x) ≤ d if and only if symbol k

only appear in the (kλ− dλ − λ + 1)-th, . . . , (kλ + dλ)-th positions of x. This observation leads us to define the shift operator ⊕.

Definition 3.2.1. For an integer set S and an integer z, define S⊕ z = {s + z : s ∈ S}.

Fact 3.2.2. If x = (x1, . . . , xn) is d-close to e, then xi = k implies

i∈ [−dλ + 1, dλ + λ] ⊕ (kλ − λ).

The above fact is useful to capture the frequency permutations that are d-close to e.

Note that the set S = [−dλ + 1, dλ + λ] is independent of k. We give a recursive algorithm EnumV_λ,n,d in Figure 3.1. It enumerates all frequency permutations x ∈ Sn^λ in B(d, e). It is a depth-first-search style algorithm. Basically, it first tries to put 1’s into λ proper vacant positions of x. Then, it tries to put 2’s, . . . , m’s into the partial frequency permutations recursively. According to fact 3.2.2, symbol k is assigned to positions of indices in [−dλ + 1, dλ + λ]⊕ (kλ − λ), and these positions are said to be valid for k.

EnumV_λ,n,d(k, P ) 1. if k≤ m then

2. for each partition (X, X^′) of P with |X| = λ do 3. if X^′ ∩ (−∞, −dλ + λ] = ∅ then

// Make sure it is a proper partition 4. for i∈ X ⊕ (kλ − λ) do

5. x_i ← k; // Assign k to the i-th position 6. Y ← (X^′⊕ (−λ)) ∪ [dλ + 1, dλ + λ];

7. EnumV_λ,n,d(k + 1, Y );

8. for i∈ X ⊕ (kλ − λ) do

9. x_i ← 0; // Reset xi to be vacant 10. else

11. if P = [dλ + λ] then output (x₁, . . . , x_n);

Figure 3.1: EnumV_λ,n,d(k, P )

EnumV_λ,n,d takes an integer k and a subset P of [−dλ + 1, dλ + λ] as its input, and EnumV_λ,n,duses an (n + 2dλ)-dimensional vector x as a global variable. For convenience, we extend the index set of x to [−dλ + 1, n + dλ] and every entry of x is initialized to 0, which indicates that the entry is vacant. We use P to trace the indices of valid vacant positions for symbol k, and the set of such positions is exactly P ⊕ (kλ − λ).

We call EnumV_λ,n,d(1, [dλ+λ]) to enumerate all frequency permutations which are d-close to e. During the enumeration, EnumV_λ,n,d(k, P ) assigns symbol k into some λ positions, indexed by X ⊕ (kλ − λ), of a partially assigned frequency permutation, then it recursively invokes EnumV_λ,n,d(k + 1, X^′⊕ (−λ)) ∪ [dλ + 1, dλ + λ]), where X and X^′ form a partition of P and|X| = λ. After the recursive call is done, EnumVλ,n,d(k, P ) reset positions indexed by X ⊕ (kλ − λ) as vacant. Then, it repeats to search another choice of λ positions until all possible combinations of λ positions are investigated. For k = m + 1, EnumV_λ,n,d(k, P ) outputs x if P = [dλ + λ]. Given n = mλ, k is initialized to 1 and P to [dλ + λ], we have the following claims.

Claim 3.2.3. In each of the recursive call of EnumVλ,n,d, in line 6 we have max(Y ) = dλ+λ.

Proof. By induction, it is clear for k = 1. Suppose max(P ) = dλ + λ. Since

Y = (X^′ ⊕ (−λ)) ∪ [dλ + 1, dλ + λ]

and max(X^′)≤ dλ + λ, we have max(Y ) = dλ + λ.

Claim 3.2.4. In line 6, for each k ∈ [m+1] and each i ∈ Y ⊕((k +1)λ−λ), we have xi = 0.

Proof. We prove it by induction on k. It is clear for k = 1. Assume the claim is true up to k < m + 1, i.e., for each i ∈ P ⊕ (kλ − λ), xi = 0. Now, consider the following scenario, EnumV_λ,n,d(k, P ) invokes EnumV_λ,n,d(k + 1, Y ).

Since Y = (X^′⊕ (−λ)) ∪ [dλ + 1, dλ + λ], we have

Y ⊕ ((k + 1)λ − λ) = (X^′⊕ (kλ − λ)) ∪ [kλ + dλ + 1, kλ + dλ + λ].

While X^′ ⊂ P and [kλ + dλ + 1, kλ + dλ + λ] are new vacant positions, it is clear xi = 0 in these entries.

Claim 3.2.5. In each recursive call of EnumV_λ,n,d(k, P ), P must be a subset of [−dλ + 1, dλ + λ] of cardinality dλ + λ. It implies, |ei− k| ≤ d for i ∈ P ⊕ (kλ − λ).

Proof. We prove it by induction on k. For k = 1, P is [dλ + λ], and the claim is obvious.

Assume the claim is true up to k, and EnumV_λ,n,d(k, P ) invokes EnumV_λ,n,d(k + 1, Y ). Thus

Y = (X^′⊕ (−λ)) ∪ [dλ + 1, dλ + λ]. Due to the constraint on X^′ in line 3 and the induction hypothesis, we have X^′⊕ (−λ) ⊆ [−dλ + 1, dλ]. We conclude that

Y ⊆ [−dλ + 1, dλ] ∪ [dλ + 1, dλ + λ]

and |Y | = |X^′| + λ = dλ + λ. Since [−dλ + 1, dλ + λ] ⊕ (kλ) = [kλ − dλ + 1, kλ + dλ + λ], we know e has values from [k− d + 1, k + d + 1] in these positions. I.e., |ei− (k + 1)| ≤ d for i∈ Y ⊕ (kλ). Hence, the claim is true.

Claim 3.2.6. At the beginning of the invocation of EnumV_λ,n,d(k, P ), i ∈ P ⊕ (kλ − λ) implies i > 0.

Proof. It is clear for k = 1. Observe that min(Y )≥ min(P ) − λ. Since

(min(P )− λ) ⊕ (kλ) = min(P ) ⊕ (kλ − λ),

the claim holds for k > 1.

Claim 3.2.7. For k∈ [m], when EnumVλ,n,d(k, P ) invoke EnumV_λ,n,d(k + 1, Y ) in line 7, there are exactly λ entries of x equal i for i∈ [k − 1].

Proof. It is implied by lines 4 and 5.

Lemma 3.2.8. At the beginning of the execution of EnumV_λ,n,d(k, P ),

P ⊕ (kλ − λ) = {i : i > 0 ∧ xi = 0∧ i ∈ [−dλ + 1, dλ + λ] ⊕ (kλ − λ)}.

Proof. The lemma holds by claims 3.2.4, 3.2.5, and 3.2.6.

In order to investigate the behavior of EnumV_λ,n,d, we define partial frequency permu-tations. A partial frequency permutation can be derived from a frequency permutation in S_n^λ with some entries replaced with ∗. The symbol ∗ does not contribute to the dis-tance. I.e., Chebyshev distance between two k-dimensional partial frequency permutations, y and y^′, is defined as ℓ_∞(y, y^′) = max_{i∈[k],yi̸=∗,y}^′

i̸=∗|yi− y^′i|. Now, we begin to prove that EnumV_λ,n,d(1, [dλ + λ]) only outputs the frequency permutations d-close to e.

Lemma 3.2.9. Let τ_k be a partial frequency permutation d-close to e and with each symbol 1, . . . , k− 1 appearing exactly λ times in τk for some k∈ [m + 1]. If the entries of the global variable x are configured as

x_i =





(τ_k)_i , (τ_k)_i ̸= ∗, 0 , otherwise,

then every frequency permutation y in S_n^λ, generated by EnumV_λ,n,d(k, P ), satisfies the following conditions:

1. y is consistent with τ_k over the entries with symbols 1, . . . , k− 1.

2. y is d-close to e.

Proof. We prove it by reverse induction. First, we consider the case k = m + 1. By claim 3.2.7, every symbol appears exactly λ times in τ_m+1 and x. By claim 3.2.4 and claim 3.2.6, we know that all of x1, . . . , xn are nonzero if and only if P = [dλ + λ]. There are two possible cases:

• P = [dλ + λ]: By claim 3.2.5, (x1, . . . , xn) is the only frequency permutation in S_n^λ satisfying both conditions.

• P ̸= [dλ + λ]: Then there is some i ∈ [n] such that xi = 0. But there is no frequency permutation y in S_n^λ with y_i = m + 1 > m. This means x is not well assigned.

Note that EnumV_λ,n,d(k, P ) outputs x only if P = [dλ + λ], otherwise there is no output.

Hence, the claim is true for k = m + 1. Assume the claim is true down to k + 1. For k, by lemma 3.2.8, P ⊕ (kλ − λ) is exactly the set of all positions which are vacant and valid for k. In order to enumerate frequency permutations satisfying the second condition, we must assign k’s into the valid positions. Therefore, we just need to try all possible choices of λ-element subset X⊕ (kλ − λ) ⊆ P ⊕ (kλ − λ). Line 3 of EnumVλ,n,d(k, P ) ensures that X is properly selected. Then symbol k is assigned to X⊕ (kλ − λ) and we have a new partial frequency permutation τ_k+1 where

(τ_k+1)_i =





x_i , x_i ̸= 0,

∗ , otherwise.

By induction hypothesis, it is clear that the generated frequency permutations satisfy both conditions.

We have the following theorem as an immediate result of lemma 3.2.9.

Theorem 3.2.10. EnumV_λ,n,d(1, [dλ+λ]) enumerates exactly all the frequency permutations d-close to e in S_n^λ.

Proof. By lemma 3.2.9, EnumV_λ,n,d(1, [dλ + λ]) only outputs frequency permutations d-close to e. The rest is to show that every frequency permutation d-close to e will be enumerated.

Let y be a frequency permutation d-close to e in S_n^λ. We define x^<1, . . . , x^<m+1 as

x^<k_i =





yi , yi < k, 0 , y_i ≥ k.

Note that x^<1 = (0, . . . , 0), x^<m+1 = y, and for k∈ [m], EnumVλ,n,d(k, P ) with x initialized to x^<k must invokes EnumV_λ,n,d(k + 1, Y ) with x initialized to x^<k+1 in some iteration of the for-loop. Thus, y will be enumerated eventually.