Fast Monotone Summation over Disjoint Sets Petteri Kaski 1 , 2 - - PowerPoint PPT Presentation
Fast Monotone Summation over Disjoint Sets Petteri Kaski 1 , 2 Mikko Koivisto 1 , 3 Janne H. Korhonen 1 , 3 1 Helsinki Institute for Information Technology HIIT 2 Department of Information and Computer Science, Aalto University 3 Department of
Petteri Kaski1,2 Mikko Koivisto1,3 Janne H. Korhonen1,3
1Helsinki Institute for Information Technology HIIT 2Department of Information and Computer Science, Aalto University 3Department of Computer Science, University of Helsinki
Input: graph G = (V, E) Output: number of 2k-paths with middle vertex v
Input: graph G = (V, E) Output: number of 2k-paths with middle vertex v
◮ Trivial algorithm nO(1) n 2k
k
1 2
f (Y) · f (X)
◮ Inclusion-exclusion in time nO(1)n k
e(Y) =
f (X) e(Y) =
(−1)|X|
Z⊇X
f (Z)
◮ Recover the result as
f (Y) · f (X) =
f (Y) · e(Y)
Let [n] = {1, 2, . . . , n}. Input: f (X) for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of [n] of size k.
Let [n] = {1, 2, . . . , n}. Input: f (X) for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of [n] of size k. f e n k
k
Outputs
Let [n] = {1, 2, . . . , n}. Input: f (X) for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of [n] of size k.
◮ Trivial algorithm O
n
k
n−k
k
◮ Inclusion-exclusion O
n
k
e(Y) =
(−1)|X|
Z⊇X
f (Z)
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of X of size k.
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of X of size k.
Input: f (X) ∈ Q for all X, Output: e(Y) = max
X : X∩Y=∅ f (X)
for all Y.
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of X of size k.
◮ Trivial algorithm O
n
k
n−k
k
◮ Inclusion-exclusion?
e(Y) =
(−1)|X|
Z⊇X
f (Z)
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of X of size k.
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of X of size k.
Monotone k-disjoint summation can be solved in time O(nk+2 polylog(n)).
Input: graph G = (V, E) with edge weights Output: number of 2k-paths with middle vertex v and weight w∗, where w∗ the maximum weight of a 2k-path.
1 2 3 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 3 5 3 3 3 3 3 3 3 1 4 4 4 4 3 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 5 3 3 3 3 3 1 1 1 1 1 5 2 3 3
Input: graph G = (V, E) with edge weights Output: number of 2k-paths with middle vertex v and weight w∗, where w∗ the maximum weight of a 2k-path.
2 1 3 3 1 1 2 2 3 3 1 5 3 1
Input: graph G = (V, E) with edge weights Output: number of 2k-paths with middle vertex v and weight w∗, where w∗ the maximum weight of a 2k-path.
◮ If weights are integers at most W, then problem can be
solved with counting in halves and polynomial encoding in time W · nO(1)n
k
Input: graph G = (V, E) with edge weights Output: number of 2k-paths with middle vertex v and weight w∗, where w∗ the maximum weight of a 2k-path.
◮ If weights are integers at most W, then problem can be
solved with counting in halves and polynomial encoding in time W · nO(1)n
k
monotone disjoint summation nk+O(1)
The Cartesian product N × (R ∪ {−∞}) is a commutative semiring when equipped with operations (c, w) ⊕ (d, v) =
(c, w) if w > v, (d, v) if w < v, (c + d, w) if w = v and (c, w) ⊙ (d, v) = (cd, w + v) .
2 1 1 3 3 2 3 3 4 5 3
f (X) = (3, 18)
◮ Let
f (X) = (c, w), where
◮ w is the maximum weight of k-path on X ∪ {v} ending
at v, and
◮ c is the number of such paths with weight w.
2 1 1 3 3 2 3 3 4 5 3 2 1 1 1 2 3 1 1 5
(3, 18) ⊗ (2, 15) = (6, 33)
◮ Compute
(c, w∗) =
f (Y) ⊗ f (X)
◮ Then the maximum weight of a path is w∗ and the
number of maximum weight paths is 1
2c
◮ k × n matrix permanent in time nk/2+O(1) over
noncommmutative semirings: Input: A ∈ Sk×n Output: perm(A) =
a1σ(1)a2σ(2) · · · akσ(k)
◮ Maximisation with restrictions, e.g. in machine
learning, nk+O(1) vs. nO(1)n
k
2:
Input: f (X) ∈ Q for all X Output: e(Y) = max
X : X∩Y=∅ f (X)
for all Y
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: fi ∈ S for i = 1, 2, . . . , n Output: ej =
fi for j = 1, 2, . . . , n
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: fi ∈ S for i = 1, 2, . . . , n Output: ej =
fi for j = 1, 2, . . . , n
There is an S-arithmetic circuit of size O(n) for monotone 1-disjoint summation.
f₁ f₂ f₃ f₄ f₅ f₆ f₇ f₈ e₁ e₂ e₃ e₄ e₅ e₆ e₇ e₈
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of [n] of size at most k.
Let [n] = {1, 2, . . . , n} and let (S, ⊕) be a commutative semigroup. Input: f (X) ∈ S for all X, Output: e(Y) =
f (X) for all Y, where X and Y range over subsets of [n] of size at most k.
There is an S-arithmetic circuit of size O(nk log n) for extended monotone k-disjoint summation. Furthermore, the circuit can be constructed in time O(k2nk polylog n).
[n]
[n] 1 2 3 4
1 2 3 4 X
1 2 3 4 X X|2
1 2 3 4 X X|2 X|1
1 2 3 4 W
1 2 3 4 W Y
1 2 3 4 W Y
1 2 3 4 {r} Y
◮ Working with semigroups and -rings
◮ Working with semigroups and -rings ◮ Open question:
O(nk log n) vs. O
n
k
◮ Working with semigroups and -rings ◮ Open question:
O(nk log n) vs. O
n
k
◮ [Sergeev 2012]:
O
n
k
α = 3 + √ 5 2
◮ Working with semigroups and -rings ◮ Open question:
O(nk log n) vs. O
n
k
◮ [Sergeev 2012]:
O
n
k
α = 3 + √ 5 2