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/base/combinatorics.jl

# This file is a part of Julia. License is MIT: https://julialang.org/license

# Factorials

const _fact_table64 = Vector{Int64}(undef, 20)
_fact_table64[1] = 1
for n in 2:20
    _fact_table64[n] = _fact_table64[n-1] * n
end

const _fact_table128 = Vector{UInt128}(undef, 34)
_fact_table128[1] = 1
for n in 2:34
    _fact_table128[n] = _fact_table128[n-1] * n
end

function factorial_lookup(n::Integer, table, lim)
    n < 0 && throw(DomainError(n, "`n` must not be negative."))
    n > lim && throw(OverflowError(string(n, " is too large to look up in the table; consider using `factorial(big(", n, "))` instead")))
    n == 0 && return one(n)
    @inbounds f = table[n]
    return oftype(n, f)
end

factorial(n::Int128) = factorial_lookup(n, _fact_table128, 33)
factorial(n::UInt128) = factorial_lookup(n, _fact_table128, 34)
factorial(n::Union{Int64,UInt64}) = factorial_lookup(n, _fact_table64, 20)

if Int === Int32
    factorial(n::Union{Int8,UInt8,Int16,UInt16}) = factorial(Int32(n))
    factorial(n::Union{Int32,UInt32}) = factorial_lookup(n, _fact_table64, 12)
else
    factorial(n::Union{Int8,UInt8,Int16,UInt16,Int32,UInt32}) = factorial(Int64(n))
end


# Basic functions for working with permutations

@inline function _foldoneto(op, acc, ::Val{N}) where N
    @assert N::Integer > 0
    if @generated
        quote
            acc_0 = acc
            Base.Cartesian.@nexprs $N i -> acc_{i} = op(acc_{i-1}, i)
            return $(Symbol(:acc_, N))
        end
    else
        for i in 1:N
            acc = op(acc, i)
        end
        return acc
    end
end

"""
    isperm(v) -> Bool

Return `true` if `v` is a valid permutation.

# Examples
```jldoctest
julia> isperm([1; 2])
true

julia> isperm([1; 3])
false
```
"""
isperm(A) = _isperm(A)

function _isperm(A)
    n = length(A)
    used = falses(n)
    for a in A
        (0 < a <= n) && (used[a] ⊻= true) || return false
    end
    true
end

isperm(p::Tuple{}) = true
isperm(p::Tuple{Int}) = p[1] == 1
isperm(p::Tuple{Int,Int}) = ((p[1] == 1) & (p[2] == 2)) | ((p[1] == 2) & (p[2] == 1))

function isperm(P::Tuple)
    valn = Val(length(P))
    _foldoneto(true, valn) do b,i
        s = _foldoneto(false, valn) do s, j
            s || P[j]==i
        end
        b&s
    end
end

isperm(P::Any32) = _isperm(P)

# swap columns i and j of a, in-place
function swapcols!(a::AbstractMatrix, i, j)
    i == j && return
    cols = axes(a,2)
    @boundscheck i in cols || throw(BoundsError(a, (:,i)))
    @boundscheck j in cols || throw(BoundsError(a, (:,j)))
    for k in axes(a,1)
        @inbounds a[k,i],a[k,j] = a[k,j],a[k,i]
    end
end

# swap rows i and j of a, in-place
function swaprows!(a::AbstractMatrix, i, j)
    i == j && return
    rows = axes(a,1)
    @boundscheck i in rows || throw(BoundsError(a, (:,i)))
    @boundscheck j in rows || throw(BoundsError(a, (:,j)))
    for k in axes(a,2)
        @inbounds a[i,k],a[j,k] = a[j,k],a[i,k]
    end
end

# like permute!! applied to each row of a, in-place in a (overwriting p).
function permutecols!!(a::AbstractMatrix, p::AbstractVector{<:Integer})
    require_one_based_indexing(a, p)
    count = 0
    start = 0
    while count < length(p)
        ptr = start = findnext(!iszero, p, start+1)::Int
        next = p[start]
        count += 1
        while next != start
            swapcols!(a, ptr, next)
            p[ptr] = 0
            ptr = next
            next = p[next]
            count += 1
        end
        p[ptr] = 0
    end
    a
end

function permute!!(a, p::AbstractVector{<:Integer})
    require_one_based_indexing(a, p)
    count = 0
    start = 0
    while count < length(a)
        ptr = start = findnext(!iszero, p, start+1)::Int
        temp = a[start]
        next = p[start]
        count += 1
        while next != start
            a[ptr] = a[next]
            p[ptr] = 0
            ptr = next
            next = p[next]
            count += 1
        end
        a[ptr] = temp
        p[ptr] = 0
    end
    a
end

"""
    permute!(v, p)

Permute vector `v` in-place, according to permutation `p`. No checking is done
to verify that `p` is a permutation.

To return a new permutation, use `v[p]`. This is generally faster than `permute!(v, p)`;
it is even faster to write into a pre-allocated output array with `u .= @view v[p]`.
(Even though `permute!` overwrites `v` in-place, it internally requires some allocation
to keep track of which elements have been moved.)

See also [`invpermute!`](@ref).

# Examples
```jldoctest
julia> A = [1, 1, 3, 4];

julia> perm = [2, 4, 3, 1];

julia> permute!(A, perm);

julia> A
4-element Vector{Int64}:
 1
 4
 3
 1
```
"""
permute!(v, p::AbstractVector) = (v .= v[p])

function invpermute!!(a, p::AbstractVector{<:Integer})
    require_one_based_indexing(a, p)
    count = 0
    start = 0
    while count < length(a)
        start = findnext(!iszero, p, start+1)::Int
        temp = a[start]
        next = p[start]
        count += 1
        while next != start
            temp_next = a[next]
            a[next] = temp
            temp = temp_next
            ptr = p[next]
            p[next] = 0
            next = ptr
            count += 1
        end
        a[next] = temp
        p[next] = 0
    end
    a
end

"""
    invpermute!(v, p)

Like [`permute!`](@ref), but the inverse of the given permutation is applied.

Note that if you have a pre-allocated output array (e.g. `u = similar(v)`),
it is quicker to instead employ `u[p] = v`.  (`invpermute!` internally
allocates a copy of the data.)

# Examples
```jldoctest
julia> A = [1, 1, 3, 4];

julia> perm = [2, 4, 3, 1];

julia> invpermute!(A, perm);

julia> A
4-element Vector{Int64}:
 4
 1
 3
 1
```
"""
invpermute!(v, p::AbstractVector) = (v[p] = v; v)

"""
    invperm(v)

Return the inverse permutation of `v`.
If `B = A[v]`, then `A == B[invperm(v)]`.

See also [`sortperm`](@ref), [`invpermute!`](@ref), [`isperm`](@ref), [`permutedims`](@ref).

# Examples
```jldoctest
julia> p = (2, 3, 1);

julia> invperm(p)
(3, 1, 2)

julia> v = [2; 4; 3; 1];

julia> invperm(v)
4-element Vector{Int64}:
 4
 1
 3
 2

julia> A = ['a','b','c','d'];

julia> B = A[v]
4-element Vector{Char}:
 'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
 'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
 'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
 'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)

julia> B[invperm(v)]
4-element Vector{Char}:
 'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
 'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
 'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
 'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
```
"""
function invperm(a::AbstractVector)
    require_one_based_indexing(a)
    b = zero(a) # similar vector of zeros
    n = length(a)
    @inbounds for (i, j) in enumerate(a)
        ((1 <= j <= n) && b[j] == 0) ||
            throw(ArgumentError("argument is not a permutation"))
        b[j] = i
    end
    b
end

function invperm(p::Union{Tuple{},Tuple{Int},Tuple{Int,Int}})
    isperm(p) || throw(ArgumentError("argument is not a permutation"))
    p  # in dimensions 0-2, every permutation is its own inverse
end

function invperm(P::Tuple)
    valn = Val(length(P))
    ntuple(valn) do i
        s = _foldoneto(nothing, valn) do s, j
            s !== nothing && return s
            P[j]==i && return j
            nothing
        end
        s === nothing && throw(ArgumentError("argument is not a permutation"))
        s
    end
end

invperm(P::Any32) = Tuple(invperm(collect(P)))

#XXX This function should be moved to Combinatorics.jl but is currently used by Base.DSP.
"""
    nextprod(factors::Union{Tuple,AbstractVector}, n)

Next integer greater than or equal to `n` that can be written as ``\\prod k_i^{p_i}`` for integers
``p_1``, ``p_2``, etcetera, for factors ``k_i`` in `factors`.

# Examples
```jldoctest
julia> nextprod((2, 3), 105)
108

julia> 2^2 * 3^3
108
```

!!! compat "Julia 1.6"
    The method that accepts a tuple requires Julia 1.6 or later.
"""
function nextprod(a::Union{Tuple{Vararg{Integer}},AbstractVector{<:Integer}}, x::Real)
    if x > typemax(Int)
        throw(ArgumentError("unsafe for x > typemax(Int), got $x"))
    end
    k = length(a)
    v = fill(1, k)                    # current value of each counter
    mx = map(a -> nextpow(a,x), a)   # maximum value of each counter
    v[1] = mx[1]                      # start at first case that is >= x
    p::widen(Int) = mx[1]             # initial value of product in this case
    best = p
    icarry = 1

    while v[end] < mx[end]
        if p >= x
            best = p < best ? p : best  # keep the best found yet
            carrytest = true
            while carrytest
                p = div(p, v[icarry])
                v[icarry] = 1
                icarry += 1
                p *= a[icarry]
                v[icarry] *= a[icarry]
                carrytest = v[icarry] > mx[icarry] && icarry < k
            end
            if p < x
                icarry = 1
            end
        else
            while p < x
                p *= a[1]
                v[1] *= a[1]
            end
        end
    end
    # might overflow, but want predictable return type
    return mx[end] < best ? Int(mx[end]) : Int(best)
end

1	# This file is a part of Julia. License is MIT: https://julialang.org/license
2
3	# Factorials
4
5	const _fact_table64 = Vector{Int64}(undef, 20)
6	_fact_table64[1] = 1
7	for n in 2:20
8	_fact_table64[n] = _fact_table64[n-1] * n
9	end
10
11	const _fact_table128 = Vector{UInt128}(undef, 34)
12	_fact_table128[1] = 1
13	for n in 2:34
14	_fact_table128[n] = _fact_table128[n-1] * n
15	end
16
17	function factorial_lookup(n::Integer, table, lim)	15✔
18	n < 0 && throw(DomainError(n, "`n` must not be negative."))	15✔
19	n > lim && throw(OverflowError(string(n, " is too large to look up in the table; consider using `factorial(big(", n, "))` instead")))	13✔
20	n == 0 && return one(n)	13✔
21	@inbounds f = table[n]	13✔
22	return oftype(n, f)	13✔
23	end
24
25	factorial(n::Int128) = factorial_lookup(n, _fact_table128, 33)	×
26	factorial(n::UInt128) = factorial_lookup(n, _fact_table128, 34)	×
27	factorial(n::Union{Int64,UInt64}) = factorial_lookup(n, _fact_table64, 20)	25✔
28
29	if Int === Int32
30	factorial(n::Union{Int8,UInt8,Int16,UInt16}) = factorial(Int32(n))	×
31	factorial(n::Union{Int32,UInt32}) = factorial_lookup(n, _fact_table64, 12)	×
32	else
33	factorial(n::Union{Int8,UInt8,Int16,UInt16,Int32,UInt32}) = factorial(Int64(n))	2✔
34	end
35
36
37	# Basic functions for working with permutations
38
39	@inline function _foldoneto(op, acc, ::Val{N}) where N	2,352✔
40	@assert N::Integer > 0	2,054✔
41	if @generated	×
42	quote	253✔
43	acc_0 = acc	2,054✔
44	Base.Cartesian.@nexprs $N i -> acc_{i} = op(acc_{i-1}, i)	2,352✔
45	return $(Symbol(:acc_, N))	2,054✔
46	end
47	else
48	for i in 1:N
49	acc = op(acc, i)
50	end
51	return acc
52	end
53	end
54
55	"""
56	isperm(v) -> Bool
57
58	Return `true` if `v` is a valid permutation.
59
60	# Examples
61	```jldoctest
62	julia> isperm([1; 2])
63	true
64
65	julia> isperm([1; 3])
66	false
67	```
68	"""
69	isperm(A) = _isperm(A)	498✔
70
71	function _isperm(A)	498✔
72	n = length(A)	498✔
73	used = falses(n)	498✔
74	for a in A	518✔
75	(0 < a <= n) && (used[a] ⊻= true) \|\| return false	50,232✔
76	end	50,710✔
77	true	498✔
78	end
79
80	isperm(p::Tuple{}) = true	×
81	isperm(p::Tuple{Int}) = p[1] == 1	×
82	isperm(p::Tuple{Int,Int}) = ((p[1] == 1) & (p[2] == 2)) \| ((p[1] == 2) & (p[2] == 1))	528✔
83
84	function isperm(P::Tuple)	398✔
85	valn = Val(length(P))	398✔
86	_foldoneto(true, valn) do b,i	472✔
87	s = _foldoneto(false, valn) do s, j	1,460✔
88	s \|\| P[j]==i	4,568✔
89	end
90	b&s	1,242✔
91	end
92	end
93
94	isperm(P::Any32) = _isperm(P)	×
95
96	# swap columns i and j of a, in-place
97	function swapcols!(a::AbstractMatrix, i, j)	839✔
98	i == j && return	839✔
99	cols = axes(a,2)	839✔
100	@boundscheck i in cols \|\| throw(BoundsError(a, (:,i)))	839✔
101	@boundscheck j in cols \|\| throw(BoundsError(a, (:,j)))	839✔
102	for k in axes(a,1)	1,678✔
103	@inbounds a[k,i],a[k,j] = a[k,j],a[k,i]	44,615✔
104	end	44,615✔
105	end
106
107	# swap rows i and j of a, in-place
108	function swaprows!(a::AbstractMatrix, i, j)	3✔
109	i == j && return	3✔
110	rows = axes(a,1)	3✔
111	@boundscheck i in rows \|\| throw(BoundsError(a, (:,i)))	3✔
112	@boundscheck j in rows \|\| throw(BoundsError(a, (:,j)))	3✔
113	for k in axes(a,2)	6✔
114	@inbounds a[i,k],a[j,k] = a[j,k],a[i,k]	18✔
115	end	18✔
116	end
117
118	# like permute!! applied to each row of a, in-place in a (overwriting p).
119	function permutecols!!(a::AbstractMatrix, p::AbstractVector{<:Integer})	138✔
120	require_one_based_indexing(a, p)	138✔
121	count = 0	138✔
122	start = 0	138✔
123	while count < length(p)	451✔
124	ptr = start = findnext(!iszero, p, start+1)::Int	830✔
125	next = p[start]	313✔
126	count += 1	313✔
127	while next != start	1,148✔
128	swapcols!(a, ptr, next)	835✔
129	p[ptr] = 0	835✔
130	ptr = next	835✔
131	next = p[next]	835✔
132	count += 1	835✔
133	end	835✔
134	p[ptr] = 0	313✔
135	end	313✔
136	a	138✔
137	end
138
139	function permute!!(a, p::AbstractVector{<:Integer})	×
140	require_one_based_indexing(a, p)	×
141	count = 0	×
142	start = 0	×
143	while count < length(a)	×
144	ptr = start = findnext(!iszero, p, start+1)::Int	×
145	temp = a[start]	×
146	next = p[start]	×
147	count += 1	×
148	while next != start	×
149	a[ptr] = a[next]	×
150	p[ptr] = 0	×
151	ptr = next	×
152	next = p[next]	×
153	count += 1	×
154	end	×
155	a[ptr] = temp	×
156	p[ptr] = 0	×
157	end	×
158	a	×
159	end
160
161	"""
162	permute!(v, p)
163
164	Permute vector `v` in-place, according to permutation `p`. No checking is done
165	to verify that `p` is a permutation.
166
167	To return a new permutation, use `v[p]`. This is generally faster than `permute!(v, p)`;
168	it is even faster to write into a pre-allocated output array with `u .= @view v[p]`.
169	(Even though `permute!` overwrites `v` in-place, it internally requires some allocation
170	to keep track of which elements have been moved.)
171
172	See also [`invpermute!`](@ref).
173
174	# Examples
175	```jldoctest
176	julia> A = [1, 1, 3, 4];
177
178	julia> perm = [2, 4, 3, 1];
179
180	julia> permute!(A, perm);
181
182	julia> A
183	4-element Vector{Int64}:
184	1
185	4
186	3
187	1
188	```
189	"""
190	permute!(v, p::AbstractVector) = (v .= v[p])	1,177✔
191
192	function invpermute!!(a, p::AbstractVector{<:Integer})	×
193	require_one_based_indexing(a, p)	×
194	count = 0	×
195	start = 0	×
196	while count < length(a)	×
197	start = findnext(!iszero, p, start+1)::Int	×
198	temp = a[start]	×
199	next = p[start]	×
200	count += 1	×
201	while next != start	×
202	temp_next = a[next]	×
203	a[next] = temp	×
204	temp = temp_next	×
205	ptr = p[next]	×
206	p[next] = 0	×
207	next = ptr	×
208	count += 1	×
209	end	×
210	a[next] = temp	×
211	p[next] = 0	×
212	end	×
213	a	×
214	end
215
216	"""
217	invpermute!(v, p)
218
219	Like [`permute!`](@ref), but the inverse of the given permutation is applied.
220
221	Note that if you have a pre-allocated output array (e.g. `u = similar(v)`),
222	it is quicker to instead employ `u[p] = v`. (`invpermute!` internally
223	allocates a copy of the data.)
224
225	# Examples
226	```jldoctest
227	julia> A = [1, 1, 3, 4];
228
229	julia> perm = [2, 4, 3, 1];
230
231	julia> invpermute!(A, perm);
232
233	julia> A
234	4-element Vector{Int64}:
235	4
236	1
237	3
238	1
239	```
240	"""
241	invpermute!(v, p::AbstractVector) = (v[p] = v; v)	2,293✔
242
243	"""
244	invperm(v)
245
246	Return the inverse permutation of `v`.
247	If `B = A[v]`, then `A == B[invperm(v)]`.
248
249	See also [`sortperm`](@ref), [`invpermute!`](@ref), [`isperm`](@ref), [`permutedims`](@ref).
250
251	# Examples
252	```jldoctest
253	julia> p = (2, 3, 1);
254
255	julia> invperm(p)
256	(3, 1, 2)
257
258	julia> v = [2; 4; 3; 1];
259
260	julia> invperm(v)
261	4-element Vector{Int64}:
262	4
263	1
264	3
265	2
266
267	julia> A = ['a','b','c','d'];
268
269	julia> B = A[v]
270	4-element Vector{Char}:
271	'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
272	'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
273	'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
274	'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
275
276	julia> B[invperm(v)]
277	4-element Vector{Char}:
278	'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
279	'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
280	'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
281	'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
282	```
283	"""
284	function invperm(a::AbstractVector)	743✔
285	require_one_based_indexing(a)	743✔
286	b = zero(a) # similar vector of zeros	5,807✔
287	n = length(a)	743✔
288	@inbounds for (i, j) in enumerate(a)	1,483✔
289	((1 <= j <= n) && b[j] == 0) \|\|	5,804✔
290	throw(ArgumentError("argument is not a permutation"))
291	b[j] = i	5,804✔
292	end	10,868✔
293	b	743✔
294	end
295
296	function invperm(p::Union{Tuple{},Tuple{Int},Tuple{Int,Int}})	183✔
297	isperm(p) \|\| throw(ArgumentError("argument is not a permutation"))	183✔
298	p # in dimensions 0-2, every permutation is its own inverse	183✔
299	end
300
301	function invperm(P::Tuple)	128✔
302	valn = Val(length(P))	128✔
303	ntuple(valn) do i	128✔
304	s = _foldoneto(nothing, valn) do s, j	384✔
305	s !== nothing && return s	1,146✔
306	P[j]==i && return j	764✔
307	nothing	384✔
308	end
309	s === nothing && throw(ArgumentError("argument is not a permutation"))	382✔
310	s	380✔
311	end
312	end
313
314	invperm(P::Any32) = Tuple(invperm(collect(P)))	×
315
316	#XXX This function should be moved to Combinatorics.jl but is currently used by Base.DSP.
317	"""
318	nextprod(factors::Union{Tuple,AbstractVector}, n)
319
320	Next integer greater than or equal to `n` that can be written as ``\\prod k_i^{p_i}`` for integers
321	``p_1``, ``p_2``, etcetera, for factors ``k_i`` in `factors`.
322
323	# Examples
324	```jldoctest
325	julia> nextprod((2, 3), 105)
326	108
327
328	julia> 2^2 * 3^3
329	108
330	```
331
332	!!! compat "Julia 1.6"
333	The method that accepts a tuple requires Julia 1.6 or later.
334	"""
335	function nextprod(a::Union{Tuple{Vararg{Integer}},AbstractVector{<:Integer}}, x::Real)	6✔
336	if x > typemax(Int)	6✔
337	throw(ArgumentError("unsafe for x > typemax(Int), got $x"))	1✔
338	end
339	k = length(a)	5✔
340	v = fill(1, k) # current value of each counter	13✔
341	mx = map(a -> nextpow(a,x), a) # maximum value of each counter	18✔
342	v[1] = mx[1] # start at first case that is >= x	5✔
343	p::widen(Int) = mx[1] # initial value of product in this case	5✔
344	best = p	5✔
345	icarry = 1	5✔
346
347	while v[end] < mx[end]	63✔
348	if p >= x	58✔
349	best = p < best ? p : best # keep the best found yet	36✔
350	carrytest = true	36✔
351	while carrytest	72✔
352	p = div(p, v[icarry])	36✔
353	v[icarry] = 1	36✔
354	icarry += 1	36✔
355	p *= a[icarry]	36✔
356	v[icarry] *= a[icarry]	36✔
357	carrytest = v[icarry] > mx[icarry] && icarry < k	36✔
358	end	36✔
359	if p < x	36✔
360	icarry = 1	22✔
361	end
362	else
363	while p < x	65✔
364	p *= a[1]	43✔
365	v[1] *= a[1]	43✔
366	end	43✔
367	end
368	end	58✔
369	# might overflow, but want predictable return type
370	return mx[end] < best ? Int(mx[end]) : Int(best)	5✔
371	end

JuliaLang / julia / #37527

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