prefixspring2026.jl

### A Pluto.jl notebook ###
# v0.20.20

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    #! format: off
    return quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
    #! format: on
end

# ╔═╡ 01ab2b8f-cb97-425b-9c43-e7f2889dfb7d
begin 
	using PlutoUI
	using StatsBase
	using LinearAlgebra
	using Gadfly
	using Compose
end

# ╔═╡ b5fb46e8-ff75-4eb1-ba63-92e3b4a764fe
n = @bind n Slider(1:10; default=8, show_value=true)

# ╔═╡ 2670e66e-bc2a-11ed-2f25-c554deffd3bf
reduce(+, 1:n), sum(1:n)  # triangular numbers

# ╔═╡ ee52bbc2-1e4a-47e4-8b43-8cb0e730069d
reduce(*, 1:n), prod(1:n) # factorials

# ╔═╡ d3e62a10-69ef-47c3-8ab3-4d1299bbfc82
begin	
	boring(a,b)=a
	reduce(boring, Vector('a':'z')[1:n])
end

# ╔═╡ 82b5c025-9ae1-461e-bf88-c6be0ed093f7
begin
	boring2(a,b)=b
	reduce(boring2, Vector('a':'z')[1:n])
end

# ╔═╡ ab0a8204-65ee-459d-98be-6087187b4096
M=[1 1; 1 0]

# ╔═╡ 6a673be9-c6c0-4dce-b970-74f3e68fc990
N = @bind N Slider(1:100; default=8, show_value=true)

# ╔═╡ a2642146-a331-48a2-9a6c-850b3debe241
prod(fill(big.(M),N))

# ╔═╡ 0e1005c7-47b7-493f-99e6-be069e944c8b
begin
	fib(j)=reduce(*, fill(M,j))
	fib.([4,7])
end

# ╔═╡ 8839cc30-2d1d-4c43-bccf-82944ee2a166
begin
	A=[1 2;3 4]
	B=[10 100; 1 -10]
	⊗(A,B)=kron(A,B)
	
	let M=[ 1 1;1 -1]
		H=⊗(⊗(⊗(M,M),M),M)
		H*H'
	end
end

# ╔═╡ e5e40188-08df-4b82-a776-86e645f925ff
begin
	Hadamard(n)=reduce(⊗, fill([1 1;1 -1],n))
	H = Hadamard(3)
end

# ╔═╡ 56d09724-e48f-4003-a168-0ef2e000e2ec
H'H

# ╔═╡ 5c285f77-31e6-4636-8702-7e99ed173491
H*H' #This is a legitimate Hadamard matrix

# ╔═╡ b07ded7a-21fb-4bcf-a4a7-96d6c73a0ee8
begin 
	rolldice() = rand(1:6)+rand(1:6)
	rolls(n)=fit(Histogram,[rolldice() for i=1:n ],2:12,closed=:left)
end

# ╔═╡ 3a93cb87-6e17-4fe4-a87c-db3ceb1a9fb8
rolls(1000)

# ╔═╡ 7d3b7168-ceea-484a-831b-dbf53dd3ac10
reduce(merge,[rolls(100) for i=1:10])

# ╔═╡ 61e583f4-b9f9-4727-bd0c-07360382cbf6
let 
M=[rand(-3:3,2,2) for i=1:4]


(M[4]*M[3]*M[2]*M[1], 
 reduce((A,B)->B*A, M), #backward multiply
 reduce(*, M)
)
end

# ╔═╡ 861d9a43-24c9-4cec-b82f-94a1b547c5b8
(sin ∘ cos)(1), sin(cos(1))

# ╔═╡ 7ba55f3d-509f-41a8-af58-95f3293ebe96
((-) ∘ sin)(1), -sin(1)

# ╔═╡ e7a16f12-4d98-4caf-9161-31035d971ec3
begin
	h=reduce(∘, [sin cos tan])
	[h(π) sin(cos(tan(π)))]
end

# ╔═╡ e6fffc89-420d-40af-b9ab-d23a56a72bf3
begin
	cumsum(1:8)  # It is useful to know that cumsum is a linear operator
	# You can use power method! Below is the underlying matrix
	Ac=tril(ones(Int,8,8)) 
end

# ╔═╡ 3738164e-73d3-4b13-8386-26d1dfca3983
(Ac*(1:8),cumsum(1:8))

# ╔═╡ a9aa8aec-c265-4bf9-9b15-1c810cbb0b7e
mutable struct DummyArray
    length :: Int
    read :: Vector
    history :: Vector{Any}
    DummyArray(length, read=[], history=[])=new(length, read, history)
end

# ╔═╡ 087e4786-a933-47df-8526-60d20df63271
begin
	Base.length(A::DummyArray)=A.length

	function Base.getindex(A::DummyArray, i)
	    push!(A.read, i)
	    missing
	end
	
	function Base.setindex!(A::DummyArray, x, i)
	    push!(A.history, (A.read, [i]))
	    A.read = []
	end

	
end

# ╔═╡ 531b3af5-07e1-4823-818d-ab540865494b
md"### Reduce"

# ╔═╡ 5e3f0741-fe66-4ccb-822f-63cda1b2c89b
md"""
Sum Reduce\
Example: Triangular Numbers
"""

# ╔═╡ 4a77c84d-6a7b-4dd3-8701-ab405adb364b
md"Example: Factorials"

# ╔═╡ 206df737-3eb2-4275-861a-a0ba958c23d1
md"Examples: Boring"

# ╔═╡ 36a0f36a-1d3f-40c8-8514-7a94ad7ea5c9
md"""
Example:Fibonacci

``\begin{pmatrix} f_2 \\ f_1 \end{pmatrix} = \begin{pmatrix} f_1 + f_0 \\ f_1 \end{pmatrix}``

``\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} f_{n+1} & f_n\\  f_n & f_{n-1} \\  \end{pmatrix}``


"""

# ╔═╡ 829b05e4-df24-46ed-981a-2bee358ae0aa
md"""
You can solve recurrences of any complexity using `reduce`. For example, `reduce` can compute a Hadamard matrix from its definition in terms of its submatrices:

`` H_2 = \begin{pmatrix} H_1 & H_1 \\ H_1 & -H_1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \otimes H_1 ``

and so on. 

(Note: this is just using reduce to compute a matrix power.\n",
    "One can think of alternative ways for sure.
"""

# ╔═╡ d0ba43bc-18e4-4ccc-aa9d-d83285ea59b2
md"""
```
# [A B]
# If A is m x n
# If B is p x q
# then kron(A,B) is mp x nq and has all the elements of A 
```
"""

# ╔═╡ 47c9d5a3-274b-4612-a835-db3fadd3373e
md"""
In the following example we apply `reduce` to function composition:
"""

# ╔═╡ bd0267c5-2047-43e4-851e-dadfc25d1ed9
md"""
## `prefix`

Having discussed `reduce`, we are now ready for the idea behind prefix sum. Prefix or scan is long considered an important parallel primitive as well.

Suppose you wanted to compute the partial sums of a vector, i.e. given `y[1:n]`, we want to overwrite the vector y with the vector of partial sums.

```julia
new_y[1] = y[1]
new_y[2] = y[1] + y[2]
new_y[3] = y[1] + y[2] + y[3]
...
```

At first blush, it seems impossible to parallelize this, since

```julia
new_y[1] = y[1]
new_y[2] = new_y[1] + y[2]
new_y[3] = new_y[2] + y[3]
...
```

which appears to be an intrinsically serial process.
"""

# ╔═╡ 8322a1cf-0236-49b3-a169-f8ae7b4eedb7
function prefix_serial!(y, ⊕)
    for i=2:length(y)
        y[i] = y[i-1] ⊕ y[i]
        end
    y
end

# ╔═╡ 434211b7-1055-47d2-aa44-b4b2d9f97203
prefix_serial!([1:8;],*)

# ╔═╡ be349102-60fc-4d27-aa5a-f1f74a96109b
prefix_serial!([rand(1:5,2,2) for i=1:4],*)

# ╔═╡ 05d88bce-6bf7-415c-bee1-a45f8eeb0b34
md"""
However, it turns out that because addition `(+)` is associative, we can regroup the order of how these sums are carried out. (This of course extends to other associative operations such as multiplication.) Another ordering of 8 associative operations is provided by `prefix8!`:
"""

# ╔═╡ 96b1616b-9ea6-4a11-b4b2-9a4464c36ab9
# Eight only for pedagogy
function prefix8!(y, ⋅)
    length(y)==8 || error("length 8 only")
    for i in [2,4,6,8]; y[i] = y[i-1] ⋅ y[i]; end
    for i in [  4,  8]; y[i] = y[i-2] ⋅ y[i]; end
    for i in [      8]; y[i] = y[i-4] ⋅ y[i]; end
    for i in [    6  ]; y[i] = y[i-2] ⋅ y[i]; end
    for i in [ 3,5,7 ]; y[i] = y[i-1] ⋅ y[i]; end
    y
end

# ╔═╡ f2d04cda-df53-4867-ac56-f597267cc807
# Generalization to any n
function prefix!(y, ⋅)
    l=length(y)
    k=ceil(Int, log2(l))
    @inbounds for j=1:k, i=2^j:2^j:min(l, 2^k)              #"reduce"
        y[i] = y[i-2^(j-1)] ⋅ y[i]
    end
    @inbounds for j=(k-1):-1:1, i=3*2^(j-1):2^j:min(l, 2^k) #"broadcast"
        y[i] = y[i-2^(j-1)] ⋅ y[i]
    end
    y
end

# ╔═╡ be1208cb-1621-469a-b41a-d7411f66e91d
prefix!([1:12;],*)

# ╔═╡ 947d4c14-dd06-41ba-9edf-e09b9c55aacb
md"""
## Polymorphism for visualization

We can visualize the operations with a little bit of trickery. In Julia, arrays are simply types that expose the array protocol. In particular, they need to implement methods for the generic functions `length`, `getindex` and `setindex!`. The last two are used in indexing operations, since statements

```julia
y = A[1]
A[3] = y
```

get desugared to

```julia
y = getindex(A, 1)
setindex!(A, y, 3)
```

respectively.

We can trace through the iterable by introduce a dummy array type, `DummyArray`, which stores no useful information but records every access to `getindex` and `setindex!`.

Specifically:

- `length(A::DummyArray)` returns an integer that is stored internally in the `A.length `field
- `getindex(A::DummyArray, i)` records read access to the index `i` in the `A.read` field and always returns `missing`.
- `setindex!(A::DummyArray, x, i)` records write access to the index `i`. The `A.history` field is appended with a new tuple consisting of the current `A.read` field and the index `i`.

The way `DummyArray` works, it assumes an association between a single `setindex!` call and and all the preceding `getindex` calls since the previous `setindex!` call, which is sufficient for the purposes of tracing through prefix calls.
"""

# ╔═╡ 54f9bed6-37f0-4b9e-901c-f4c5fb9ca2ba
let
	M = DummyArray(4)
	M[7] = M[3] + M[2]
	M
end

# ╔═╡ bdf9341c-ec26-423e-985d-834903c2eac0
DummyA = prefix8!(DummyArray(8),+)

# ╔═╡ af31d1d7-ae8e-4400-aeb4-cb42ce0c4ec7
DummyA.history

# ╔═╡ 8eec5baf-02a8-4414-a761-37612bbccae9
md"""
Now let's visualize this! Each entry in A.history is rendered by a gate object:
"""

# ╔═╡ 104438b2-1ad4-4ec8-89eb-e5c0b3225525
struct Gate
	ins :: Vector
    outs:: Vector
end

# ╔═╡ ec968447-7496-42c4-b5b7-5fda3e899e29
function Gadfly.render(G::Gate, x₁, y₁, y₀; rᵢ=0.1, rₒ=0.25)
    ipoints = [(i, y₀+rᵢ) for i in G.ins]
    opoints = [(i, y₀+0.5) for i in G.outs]
    igates  = [Compose.circle(i..., rᵢ) for i in ipoints]
    ogates  = [Compose.circle(i..., rₒ) for i in opoints]
    lines = [line([i, j]) for i in ipoints, j in opoints]
    compose(context(units=UnitBox(0.5,0,x₁,y₁+1)),
    compose(context(), stroke(colorant"black"), fill(colorant"white"),
            igates..., ogates...),
    compose(context(), linewidth(0.3mm), stroke(colorant"black"),
            lines...))
end

# ╔═╡ 4bf453fe-97a2-4f28-ad46-f10eb253cc8b
function Gadfly.render(A::DummyArray)
    #Scan to find maximum depth
    olast = depth = 0
    for y in A.history
        (any(y[1] .≤ olast)) && (depth += 1)
        olast = maximum(y[2])
    end
    maxdepth = depth
    
    olast = depth = 0
    C = []
    for y in A.history
        (any(y[1] .≤ olast)) && (depth += 1)
        push!(C, render(Gate(y...), A.length, maxdepth, depth))
        olast = maximum(y[2])
    end
    
    push!(C, compose(context(units=UnitBox(0.5,0,A.length,1)),
      [line([(i,0), (i,1)]) for i=1:A.length]...,
    linewidth(0.1mm), stroke(colorant"grey")))
    compose(context(), C...)
end

# ╔═╡ 8c9cfbd2-96e1-4d76-9b09-a5c35e5a5e17
render(Gate([1,2],[2]),2,0,0)

# ╔═╡ d9b1b5b9-3415-4fcf-96c7-e22628e51e4c
render(prefix!(DummyArray(8), +))

# ╔═╡ 0e4afe65-a002-48b0-9206-8f9e36954dff
md"""
Now we can see that prefix! rearranges the operations to form two spanning trees:
"""

# ╔═╡ 56dfc572-3a75-4846-bb41-811c4ddb42f8
render(prefix!(DummyArray(120),+))

# ╔═╡ fdb173ff-2aea-443d-a99b-ed5f95279edb
render(prefix!(DummyArray(9),+))

# ╔═╡ 9f7e6763-e2ef-424e-8dee-0aebf3f74816
render(prefix_serial!(DummyArray(8),+))

# ╔═╡ d40c697d-89d2-4194-974e-ef778baa5d7b
npp = @bind npp Slider(1:180; default=20, show_value=true)

# ╔═╡ 7b48451a-028c-48a4-ba9e-5975cbd656d4
render(prefix!(DummyArray(npp),+))

# ╔═╡ cf94afa3-779d-4c4e-bd24-2e86d923d337
render(prefix_serial!(DummyArray(npp),+))

# ╔═╡ c5ccea9d-1a5f-4a9c-8424-5904532b35af
# Generalization to any n
function prefix_kogge_stone!(y, ⋅)
    l=length(y)
    k=ceil(Int, log2(l))

     for j=1:k
		  i=2^(j-1)
		    y[i+1:l] = y[1:l-i] .⋅ y[i+1:l]	  
		  println(y)
	 end
	 y
end

# ╔═╡ 01f7a94b-1b4d-4731-8185-7f894263ce1f
prefix_kogge_stone!([1,2,3,4,5,6,7,8],+)

# ╔═╡ 9804906f-8489-4185-abad-0064bd1c610d
function prefix_kogge_stone2!(y, ⋅)
    l=length(y)
    k=ceil(Int, log2(l))

     for j=1:k
		  i=2^(j-1)
		  for m = l:-1:(i+1)
		    y[m] = y[m-i] .⋅ y[m]	 
		  end
		  #println(y)
	 end
	 y
end

# ╔═╡ 2ed31d97-6e9d-4bf6-8bdb-2afb2bb4a776
prefix_kogge_stone2!([1,2,3,4,5,6,7,8],+)

# ╔═╡ 295f2954-498b-4df3-8ef6-f7b3f427f08b
npp2 = @bind npp2 Slider(1:180; default=20, show_value=true)

# ╔═╡ ba0bb57a-6a7b-47dd-844c-204ed0e0aec7
render(prefix_kogge_stone2!(DummyArray(npp2),+))

# ╔═╡ 00000000-0000-0000-0000-000000000001
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StatsBase = "~0.33.21"
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deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"
version = "1.11.0"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"
version = "1.11.0"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "64d974c2e6fdf07f8155b5b2ca2ffa9069b608d9"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.2.2"

[[deps.SparseArrays]]
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
version = "1.12.0"

[[deps.SpecialFunctions]]
deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "2700b235561b0335d5bef7097a111dc513b8655e"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.7.2"
weakdeps = ["ChainRulesCore"]

    [deps.SpecialFunctions.extensions]
    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"

[[deps.StaticArrays]]
deps = ["LinearAlgebra", "PrecompileTools", "Random", "StaticArraysCore"]
git-tree-sha1 = "246a8bb2e6667f832eea063c3a56aef96429a3db"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.9.18"
weakdeps = ["ChainRulesCore", "Statistics"]

    [deps.StaticArrays.extensions]
    StaticArraysChainRulesCoreExt = "ChainRulesCore"
    StaticArraysStatisticsExt = "Statistics"

[[deps.StaticArraysCore]]
git-tree-sha1 = "6ab403037779dae8c514bad259f32a447262455a"
uuid = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
version = "1.4.4"

[[deps.Statistics]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "ae3bb1eb3bba077cd276bc5cfc337cc65c3075c0"
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.11.1"
weakdeps = ["SparseArrays"]

    [deps.Statistics.extensions]
    SparseArraysExt = ["SparseArrays"]

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "178ed29fd5b2a2cfc3bd31c13375ae925623ff36"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.8.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.21"

[[deps.StatsFuns]]
deps = ["HypergeometricFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "91f091a8716a6bb38417a6e6f274602a19aaa685"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "1.5.2"

    [deps.StatsFuns.extensions]
    StatsFunsChainRulesCoreExt = "ChainRulesCore"
    StatsFunsInverseFunctionsExt = "InverseFunctions"

    [deps.StatsFuns.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.StyledStrings]]
uuid = "f489334b-da3d-4c2e-b8f0-e476e12c162b"
version = "1.11.0"

[[deps.SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "7.8.3+2"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
version = "1.11.0"

[[deps.Tricks]]
git-tree-sha1 = "311349fd1c93a31f783f977a71e8b062a57d4101"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.13"

[[deps.URIs]]
git-tree-sha1 = "bef26fb046d031353ef97a82e3fdb6afe7f21b1a"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.6.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
version = "1.11.0"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
version = "1.11.0"

[[deps.WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "248a7031b3da79a127f14e5dc5f417e26f9f6db7"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "1.1.0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.3.1+2"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.15.0+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.64.0+1"

[[deps.oneTBB_jll]]
deps = ["Artifacts", "JLLWrappers", "LazyArtifacts", "Libdl"]
git-tree-sha1 = "1350188a69a6e46f799d3945beef36435ed7262f"
uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
version = "2022.0.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.5.0+2"
"""

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