Simpler and faster implementation of Floyd's F2#1277
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dhardy merged 1 commit intorust-random:masterfrom Jan 1, 2023
Merged
Simpler and faster implementation of Floyd's F2#1277dhardy merged 1 commit intorust-random:masterfrom
dhardy merged 1 commit intorust-random:masterfrom
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On my machine - an M1 MacBook I get the following benchmarks: before: After: |
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The 8% slower performance on |
dhardy
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Dec 31, 2022
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Good simplification. The standard proof-by-induction of Floyd's combination algorithm can be extended to prove that the output is fully shuffled. It's a little tedious. A copy of my notes for those interested.
Proof by induction
for j in (len - amount) .. len {
let t = rng.gen_range(0..=j);
if X(t) {
choose j
} else {
choose t
}
}
Where:
m = amount
X(i) = event that i is chosen
Expect:
P(X(i)) = m / len
Standard proof by induction, using notation:
Xm(i) = event that i is chosen by (no later than) step m
Xm1(i) = event i chosen at step m - 1
Rlen = a random value from 0..=len
Case m=1 is trivial.
Assume m > 1 and correct for previous step:
Assumption: for i in 0..(len-1) { P(Xm1(i)) = (amount-1) / (len-1) }
Case: i in 0..(len-1)
Xm(i) = Xm1(i) ⊻ {Rlen=i}
P(Xm(i)) = P(Xm1(i)) + P(!Xm(i))*(1/len)
P(Xm(i)) = (m-1)/(len-1) + (len-1-m+1)/(len-1)*(1/len)
P(Xm(i)) = len*(m-1)/(len*(len-1)) + (len-1-m+1)/(len*(len-1))
P(Xm(i)) = (len*m-m)/(len*(len-1))
P(Xm(i)) = m/len
Case: i=len-1
Xm(i) = {Rlen=i} ⊻ {Rlen=k where Xm1(k)}
P(Xm(i)) = 1/len + Sum(k in 0..len){1/len * P(Xm1(k))}
P(Xm(i)) = 1/len + (len-1)/len * (m-1)/(len-1)
P(Xm(i)) = 1/len + 1/len * (m-1)
P(Xm(i)) = m/len
Ok.
Fully shuffled variant:
for j in (len - amount) .. len {
let t = rng.gen_range(0..=j);
if X(t) {
choose j
swap elts x, j
} else {
choose t
}
}
Proof by induction:
X(i,j) = event that i is chosen and appears in output at position j
Expect:
P(X(i,j)) = P(X(i)) / m = 1 / len
Assume for step m-1:
Assumption: for i in 0..(len-1), j in 0..(m-1) { P(Xm1(i,j)) = 1/(len-1) }
Case: i in 0..(len-1), j in 0..(m-1)
Xm(i,j) = if Xm1(Rlen,j) then 0 else Xm1(i,j)
Xm(i,j) = !Xm1(Rlen,j) AND Xm1(i,j)
P(Xm(i,j)) = (1 - Sum(k in 0..(len-1)){1/len * 1/(len-1)}) / (len-1)
P(Xm(i,j)) = (1 - 1/len) / (len-1)
P(Xm(i,j)) = (len - 1) / len / (len-1)
P(Xm(i,j)) = 1/len
Case: i in 0..(len-1), j=m-1: P(Xm1(i,j)) = 0
Xm(i,j) = Rlen=i
P(Xm(i,j)) = 1/len
Case: i=len-1, j=m-1:
Xm(i,j) = Rlen=i
P(Xm(i,j)) = 1/len
Case: i=len-1, j in 0..(m-1):
Xm(i,j) = Xm1(Rlen, j)
P(Xm(i,j)) = Sum(k in 0..(len-1)){1/len / (len-1) = 1/len
Ok.
The one thing that needs changing is the CHANGELOG: this is a value-breaking change. Affected API must be noted.
The previous implementation used either `Vec::insert` or a second F-Y shuffling phase to achieve fair random order. Instead, use the random numbers already drawn to achieve a fair shuffle.
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Have added some notes on this, let me know if this is worded the way you need - thanks! |
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The previous implementation used either
Vec::insertor a second F-Y shuffling phase to achieve fair random order. Instead, use the random numbers already drawn to achieve a fair shuffle.