Improving Kruskal’s Algorithm · Kruskal’s Algorithm Procedure kruskal(E,T : Sequence of Edge,P...

Improving Kruskal’s AlgorithmVitaly Osipov, Peter Sanders, Johannes Singler

Universitat Karlsruhe (TH)

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Motivation

What is the best (practical) algorithm for Minimum Spanning Trees?

Kruskal? for very sparse graphs Jarnık–Prim? else Karger–Klein–Tarjan, Chazelle, Pettie–Ramachandran,. . . ?

too complicated

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Kruskal’s Algorithm

Procedure kruskal(E , T : Sequence of Edge, P : UnionFind)sort E by increasing edge weightforeach u, v ∈ E do

if u and v are in different components of P thenadd edge u, v to Tjoin the partitions of u and v in P

Time O ((n + m) log m)

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Quick-Kruskal (Moret–Shapiro, Paredes–Navarro)

Procedure qKruskal(E , T : Sequence of Edge, P : UnionFind)if m ≤ kruskalThreshold(n, m, |T |) then kruskal(E , T , P)else

pick a pivot p ∈ EE≤:= 〈e ∈ E : e ≤ p〉E>:= 〈e ∈ E : e > p〉qKruskal(E≤, T , P)if |T | = n − 1 then exitqKruskal(E>, T , P)

p

Time: O

(

m + n log2 n)

– random graphs, random edge weights

Θ ((n + m) log m) if there is any heavy MST edge,e.g., Lollipop graph with random edge weights

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Filter-Kruskal

Procedure filterKruskal(E , T : Sequence of Edge, P : UnionFind)if m ≤ kruskalThreshold(n, m, |T |) then kruskal(E , T , P)else

pick a pivot p ∈ EE≤:= 〈e ∈ E : e ≤ p〉E>:= 〈e ∈ E : e > p〉qKruskal(E≤, T , P)if |T | = n − 1 then exitE>:= filter(E>, P)qKruskal(E>, T , P)

Function filter(E)return 〈u, v ∈ E : u, v are in different components of P〉

p

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Analysis – Arbitrary Graph, Random Weights

Lemma: It suffices to count edge comparisons

Chan’s sampling lemma:

r lightest edges processed ⇒ P [e ∈ E> survives filtering] ≤nr

Optimistic Analysis:Assume edge with rank i is filtered when r = i.

E[#survivors] ≤ n +∑

i>n

ni

= Θ(

n logmn

)

Partitioning m edges and sorting n log mn edges:

Ω(m + n log n logmn

)

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Analysis – Upper Bound (Outline)

Idea: Generalize textbook analysis of quicksort.0–1-RV X ij := 1 iff edges with ranks i and j are compared.

E [#comparisons] =∑

i≤m

∑

i<j≤m

P[

Xij = 1]

.

strengthen bound on P[

Xij = 1]

using Chan’s sampling lemma.. . .

. . .O(m + n log n log

mn

) expected comparisons

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Getting rid of the log m/n? E.g., random graphs

2

2.5

3

3.5

4

4.5

32 64 27 28 29 210211212213214215216217218219220221222

com

paris

ons

/ edg

es m

= n

log(

n)

number of nodes n

filterKruskallog(log(x))

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Filter-Kruskal+

Function filter+(E , T , P)E ′:= 〈u, v ∈ E : u, v are in different components of P〉T ′:= 〈u, v ∈ E ′ : u or v have degree one in comp. graph 〉T := T ∪ T ′

return E ′ \ T ′

p

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Linear Time? E.g., random graphs

5

10

15

20

25

210 211 212 213 214 215 216 217 218 219 220 221 222 223 224

num

ber

of c

ompa

rison

s / n

number of nodes n

m=nm=2nm=4n

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Parallelization

Procedure filterKruskal(E , T : Sequence of Edge, P : UnionFind)if m ≤ kruskalThreshold(n, m, |T |) then

kruskal(E , T , P) // parallel sortelse

pick a pivot p ∈ EE≤:= 〈e ∈ E : e ≤ p〉 // parallelE>:= 〈e ∈ E : e > p〉 // partitioningqKruskal(E≤, T , P)if |T | = n − 1 then exitE>:= filter(E>, P) // parallel removeIfqKruskal(E>, T , P)

Easy: available in the Multi-Core Parallel STL (e.g. g++)

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Running Time: Random graph with 216 nodes

10

100

1 4 16 64 256 1024

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8

filterKruskal+filterKruskal

filterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Graph Formatting: Random graph with 222 nodes

10

100

1000

10000

1 2 4 8 16 32

time

/ m [n

s]

number of edges m / number of nodes n

List of edges -> Adjacency ArrayAdjacency Array -> List of Edges

filterKruskalpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Conclusions

filterKruskal improves on Kruskal and Jarnık–Prim very simple often faster parallelizable needs only edge sequence

Todo: More real world inputs, tight analysis,. . .

Open Problem: What about filterKruskal+ provably linear time? scalable parallelization? Sequential component O

(

n0.6)

? more efficient implementation

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Thank you!

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Random graph with 210 nodes

10

100

1000

1 2 4 8 16 32 64 128 256

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8

filterKruskal+filterKruskal

filterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Random graph with 222 nodes

100

1000

1 2 4 8 16

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8filterKruskal+filterKruskalfilterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Random graph, n = 216, anti-Prim weights

10

100

1000

10000

100000

1 2 4 8 16 32 64 128 256

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8

qJPpJP

filterKruskal

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Random geometric graph with 216 nodes

10

100

4 8 16 32 64 128 256

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8filterKruskal+filterKruskalfilterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Lollipop graph with 210 nodes

10

100

1000

1 2 4 8 16 32 64 128

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8

filterKruskal+filterKruskal

filterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Lollipop graph with 216 nodes

10

100

1 4 16 64 256 1024

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8filterKruskal+filterKruskalfilterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Lollipop graph with 222 nodes

32

64

128

256

512

1024

1 2 4 8 16

time

/ m [n

s]

number of edges m / number of nodes n

KruskalqKruskalKruskal8filterKruskal+filterKruskalfilterKruskal8qJPpJP

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm

Image Segmentation Application

10

100

4 8 16

time

/ m [n

s]

m / n

KruskalqKruskalKruskal8

filterKruskal+filterKruskal

filterKruskal8qJPpJP

4 8 16

m / n

Vitaly Osipov, Johannes Singler, Peter Sanders Improving Kruskal’s Algorithm