The document discusses practical and worst-case efficient apportionment algorithms developed by Raphael Reitzig and Sebastian Wild, focusing on divisor methods for allocating seats based on votes from multiple parties. It outlines various apportionment wishes like pairwise vote monotonicity and house monotonicity, as well as details computational methods for implementing these algorithms. Additionally, it includes runtime analysis and references to related literature on the topic.

1. Practical and Worst-Case Efficient Apportionment
Raphael Reitzig @ Theorietage 2015
worked with Sebastian Wild
FACHBEREICH
INFORMATIK
FACHBEREICH
INFORMATIK
&Algorithmen
Komplexität

2. Apportionment
Votes
19777721
12843458
3585178
3180299 →
?
311
193
63
64
Seats

3. Apportionment
Wishes:
Pairwise Vote
Monotonicity
House
Monotonicity
Quota
Rule

4. Apportionment
Wishful thinking:
Pairwise Vote
Monotonicity
House
Monotonicity
Quota
Rule

5. Apportionment
Wishful thinking:
Pairwise Vote
Monotonicity
House
Monotonicity
Quota
Rule
⇐=

6. Apportionment
Wishful thinking:
Pairwise Vote
Monotonicity
House
Monotonicity
Quota
Rule
⇐=
Divisor Methods
⇐⇒

7. Divisor Methods
Input
Votes v of n parties for k seats, k n.

8. Divisor Methods
Input
Votes v of n parties for k seats, k n.
Method
DivisorMethod(v, k) :
1. s ← 0n.
2. For k, . . . , 1:
2.1 I ← arg maxn
i=1 vi /(si + 1).
2.2 sI ← sI + 1.
3. Return s.

9. Divisor Methods
Input
Votes v of n parties for k seats, k n.
Increasing divisor sequence d = (dj )j∈N0 .
Method
DivisorMethodd (v, k) :
1. s ← 0n.
2. For k, . . . , 1:
2.1 I ← arg maxn
i=1 vi /dsi
.
2.2 sI ← sI + 1.
3. Return s.

10. Divisor Methods
Input
Votes v of n parties for k seats, k n.
Increasing divisor sequence d = (dj )j∈N0 .
Method
DivisorMethodd (v, k) :
1. s ← 0n.
2. For k, . . . , 1:
2.1 I ← arg maxn
i=1 vi /dsi
.
2.2 sI ← sI + 1.
3. Return s. Θ
(k
logn)
using
priority
queues

11. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72

12. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66

13. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36

14. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36
2 44

15. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36
2 44
29

16. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36
2 44
29
3 33

17. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36
2 44
29
3 33
20

18. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36
2 44
29
3 33
20
24
=: a

19. Divisor Methods
dj = j + 1, k = 7
j A B C D
0 58 132 40 72
1 66 36
2 44
29
3 33
20
24
=: a
A := {132, 72, 66, 58, 44, 40, 36, 33, . . . }
≤ a > a

20. Reduction to Rank Selection
Observations
Knowing the value a selected last is sufficient.
We select the kth largest value last.

21. Reduction to Rank Selection
Observations
Knowing the value a selected last is sufficient.
We select the kth smallest value last.
Idea
Use a selection algorithm on multiset
A = ai,j =
dj
vi
i ∈ [1..n], j ∈ N0
and obtain a = A(k) “directly”.

22. Reduction to Rank Selection
Observations
Knowing the value a selected last is sufficient.
We select the kth smallest value last.
Idea
Use a selection algorithm on a finite subset of multiset
A = ai,j =
dj
vi
i ∈ [1..n], j ∈ N0
and obtain a = A(k) “directly”.

23. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
A :

24. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
A :

25. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
k
A :

26. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
k
A :
Θ(kn) algorithm.

27. Finding Good Cutoffs
Given (dj )j≥−1 with [technical details], we assume δ : R≥0 → R≥d0
with [technical details] so that
ai,j =
δ(j)
vi
and
δ−1
(y) = max{j ∈ Z≥−1 | dj ≤ y}
is the (zero-based) rank function of d.

28. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
A :

29. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
A :
δ−1(vi a ) = si − 1

30. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
[
[
[
[
[
]
]
]
]
]
]
]
]
A :
δ−1(vi a ) = si − 1
δ−1(vi a)
δ−1(vi a)
a ≤ a ≤ a

31. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
ˆA :
δ−1(vi a ) = si − 1
δ−1(vi a)
δ−1(vi a)
a ≤ a ≤ a

32. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
ˆA :
δ−1(vi a ) = si − 1
δ−1(vi a) =: ji
δ−1(vi a) =: ji
a ≤ a ≤ a

33. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
ˆA :
δ−1(vi a ) = si − 1
δ−1(vi a) =: ji
δ−1(vi a) =: ji
a = A(k) = ˆA(ˆk) with ˆk = k − i ji
.

34. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
ˆA :
δ−1(vi a ) = si − 1
δ−1(vi a) =: ji
δ−1(vi a) =: ji
i ji
< k ≤ i ji

35. Finding Good Cutoffs
i
j
1 2 3 4 5 6 7 8
0
1
2
3
4
5
...
...
...
...
...
...
...
...
...
ˆA :
δ−1(vi a ) = si − 1
δ−1(vi a) =: ji
δ−1(vi a) =: ji
i ji
< k ≤ i ji
Make tight!

36. Finding Good Cutoffs
In the Article
Observation:
k ≤ rank(a , A) ≤ k + n.
Ansatz:
rank(a, A) < k and k + n ≤ rank(a, A).
Express rank in terms of bounds on δ−1.
Make tight!

37. Good Bounds On a
If
αx + β ≤ δ(x) ≤ αx + β
with β ≤ α and [technical details], then
a := max 0,
αk − (α − β) · n)
n
i=1 vi
and
a :=
αk + βn
n
i=1 vi
work.

38. Good Bounds On a
If
αx + β ≤ δ(x) ≤ αx + β
with β ≤ α and [technical details], then
a := max 0,
αk − (α − β) · n)
n
i=1 vi
and
a :=
αk + βn
n
i=1 vi
work.
Furthermore,
| ˆA| ≤ 2(1 + (β − β)/α) · n ∈ Θ(n) .

39. An Aside: Applicability
Does anybody use d with suitable δ?

40. An Aside: Applicability
Does anybody use d with suitable δ?
Method Divisor Sequence δ(x) Sandwich
Smallest divisors 0, 1, 2, 3, . . . x —
Greatest divisors 1, 2, 3, 4, . . . x + 1 —
Sainte-Lagu¨e 1, 3, 5, 7, . . . 2x + 1 —
Modified S-L 1.4, 3, 5, 7, . . . 2x+1
1.6x+1.4
x≥1
x<1 2x + 6
5 ± 1
5
Equal Proportions 0,
√
2,
√
6,
√
12, . . . x(x + 1) x + 1
4 ± 1
4
Harmonic Mean 0, 4
3, 12
5 , 24
7 , . . . 2x(x+1)
2x+1 x + 1
4 ± 1
4
Imperiali 2, 3, 4, 5, . . . x + 2 —
Danish 1, 4, 7, 10, . . . 3x + 1 —
Yes, they all do.

41. The Algorithm
RW15d (v, k) :
1. Compute a and a.
2. Initialize ˆA := ∅ and ˆk := k.
3. For each i ∈ [1..n] do:
3.1 Compute ji
and ji .
3.2 Add all dj/vi to ˆA for which ji
≤ j ≤ ji .
3.3 Update ˆk ← ˆk − ji
.
4. Select and return ˆA(ˆk).

42. The Algorithm
RW15d (v, k) :
1. Compute a and a.
2. Initialize ˆA := ∅ and ˆk := k.
3. For each i ∈ [1..n] do:
3.1 Compute ji
and ji .
3.2 Add all dj/vi to ˆA for which ji
≤ j ≤ ji .
3.3 Update ˆk ← ˆk − ji
.
4. Select and return ˆA(ˆk).
Θ
(n)
runtim
e!

43. In Context
Puk CE14 RW15
WC O(n)

44. In Context
Puk CE14 RW15
WC O(n)
Implementable

45. Advantage: Code Complexity
CE14: ≈ 250 loc PukPQ: ≈ 80 loc RW15: ≈ 50 loc DMPQ: ≈ 25 loc
plus library and shared code

46. In Context
Puk CE14 RW15
WC O(n)
Implementable
Simple

47. Advantage: Runtime Cost
2 4 6 8 10
0
1
2
3
4
n
Averagerunningtimeinµs/n
(α, β) = (2, 1) and k = 100n
0 50 100 150 200
0
0.2
0.4
0.6
n
(α, β) = (1, 3/4) and k = 10n
PukPQ CE14 RW15 DMPQ

48. In Context
Puk CE14 RW15
WC O(n)
Implementable
Simple
Fast

49. Details
Literature
Michel L. Balinski und H. Peyton Young. Fair Representation. Meeting the
Ideal of One Man, One Vote. 2nd. Brookings Institution Press, 2001. isbn:
978-0-8157-0111-8
Friedrich Pukelsheim. Proportional Representation. Apportionment Methods
and Their Applications. 1. Aufl. Springer, 2014. isbn: 978-3-319-03855-1. doi:
10.1007/978-3-319-03856-8
Zhanpeng Cheng und David Eppstein. “Linear-time Algorithms for Proportional
Apportionment”. In: International Symposium on Algorithms and Computation
(ISAAC) 2014. Springer, 2014. doi: 10.1007/978-3-319-13075-0_46
Raphael Reitzig und Sebastian Wild. A Practical and Worst-Case Efficient
Algorithm for Divisor Methods of Apportionment. 2015. arXiv: 1504.06475
Code
github.com/reitzig/2015 apportionment