Filter-Kruskal MST Algorithm Vitaly Osipov, Peter Sanders, Johannes Singler. Universität Karlsruhe (TH)

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1 Vitaly Osipov, Peter Sanders, Johannes Singler Universität Karlsruhe (TH)

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

3 Motivation What is the best (practical) algorithm for Minimum Spanning Trees? Kruskal? for very sparse graphs

4 Motivation What is the best (practical) algorithm for Minimum Spanning Trees? Kruskal? Jarník Prim? for very sparse graphs

5 Motivation What is the best (practical) algorithm for Minimum Spanning Trees? Kruskal? Jarník Prim? for very sparse graphs else

6 Motivation What is the best (practical) algorithm for Minimum Spanning Trees? Kruskal? for very sparse graphs Jarník Prim? else Karger Klein Tarjan, Chazelle, Pettie Ramachandran,...?

7 Motivation What is the best (practical) algorithm for Minimum Spanning Trees? Kruskal? Jarník Prim? for very sparse graphs else Karger Klein Tarjan, Chazelle, Pettie Ramachandran,...? too complicated

8 Kruskal s Algorithm Procedure kruskal(e, T : Sequence of Edge, P : UnionFind) sort E by increasing edge weight foreach {u, v} E do if u and v are in different components of P then add edge {u, v} to T join the partitions of u and v in P

9 Kruskal s Algorithm Procedure kruskal(e, T : Sequence of Edge, P : UnionFind) sort E by increasing edge weight foreach {u, v} E do if u and v are in different components of P then add edge {u, v} to T join the partitions of u and v in P Time O((n+m) log m)

10 Quick-Kruskal Procedure qkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p qkruskal(e, T, P) qkruskal(e >, T, P) p

11 Quick-Kruskal Procedure qkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p qkruskal(e, T, P) qkruskal(e >, T, P) p

12 Quick-Kruskal Procedure qkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p qkruskal(e, T, P) qkruskal(e >, T, P) ( ) O m + n log 2 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

13 Filter-Kruskal Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p filterkruskal(e, T, P) E > := filter(e >, P) filterkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P p

14 Filter-Kruskal Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p filterkruskal(e, T, P) E > := filter(e >, P) filterkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P p

15 Filter-Kruskal Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p filterkruskal(e, T, P) E > := filter(e >, P) filterkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P p

16 Filter-Kruskal Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) else pick a pivot p E E := e E : e p ; E > := e E : e > p filterkruskal(e, T, P) E > := filter(e >, P) filterkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P For random edge weights Time - O(m+n log n log m n )

17 Filter-Kruskal, Parallelization Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) parallel else pick a pivot p E E := e E : e p ; E > := e E : e > p parallel filterkruskal(e, T, P) E > := filter(e >, P) parallel filterkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P For random edge weights Time - O(m+n log n log m n )

18 Filter-Kruskal, Parallelization Use STL algorithms from libstdc++ parallel mode Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, E, T ) then kruskal(e, T, P) sort else pick a pivot p E E := e E : e p ; E > := e E : e > p partition filterkruskal(e, T, P) E > := filter(e >, P) remove if filterkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P For random edge weights Time - O(m+n log n log m n )

19 Getting rid of the log m/n? E.g., random graphs 4.5 comparisons / edges m = n log(n) filterkruskal log(log(x)) number of nodes n

20 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

21 Linear Time? E.g., random graphs 20 m=n m=2n m=4n number of comparisons / n number of nodes n

22 Running Time: Random graph with 2 16 nodes 1000 Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp time / m [ns]

23 Conversion time, random graphs 1000 time / m [ns] 100 n=2 16, Edges -> Adj. Array n=2 22, Edges -> Adj. Array n=2 16, Adj. Array -> Edges n=2 22, Adj. Array -> Edges number of edges m / number of nodes n

24 Conclusions filterkruskal improves on Kruskal and Jarník 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 ( n 0.6)? more efficient implementation

25 Thank you!

26 Random graph with 2 10 nodes time / m [ns] Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp

27 Random graph with 2 22 nodes 1000 time / m [ns] 100 Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp number of edges m / number of nodes n

28 Random graph, n = 2 16, anti-prim weights time / m [ns] Kruskal qkruskal Kruskal8 qjp pjp filterkruskal number of edges m / number of nodes n

29 Random geometric graph with 2 16 nodes 100 time / m [ns] 10 Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp number of edges m / number of nodes n

30 Lollipop graph with 2 10 nodes 1000 Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp time / m [ns] number of edges m / number of nodes n

31 Lollipop graph with 2 16 nodes time / m [ns] Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp number of edges m / number of nodes n

32 Lollipop graph with 2 22 nodes 1000 time / m [ns] 100 Kruskal qkruskal Kruskal8 filterkruskal+ filterkruskal filterkruskal8 qjp pjp number of edges m / number of nodes n

33 Image Segmentation Application time / m [ns] m / n Kruskal qkruskal Kruskal8 filterkruskal m / n filterkruskal filterkruskal8 qjp pjp

Improving Kruskal s Algorithm Vitaly Osipov, Peter Sanders, Johannes Singler. Universität Karlsruhe (TH)

Improving Kruskal s Algorithm Vitaly Osipov, Peter Sanders, Johannes Singler. Universität Karlsruhe (TH) Vitaly Osipov, Peter Sanders, Johannes Singler Universität Karlsruhe (TH) Motivation What is the best (practical) algorithm for Minimum Spanning Trees? Kruskal? Jarník Prim? for very sparse graphs else