Improving Kruskal s 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? Jarník Prim? for very sparse graphs else Karger Klein Tarjan, Chazelle, Pettie Ramachandran,...? too complicated

3 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)

4 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 E E := e E : e p E > := e E : e > p qkruskal(e, T, P) if T = n 1 then exit qkruskal(e >, T, P) p Time: 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

5 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 E E := e E : e p E > := e E : e > p qkruskal(e, T, P) if T = n 1 then exit E > := filter(e >, P) qkruskal(e >, T, P) Function filter(e) return {u, v} E : u, v are in different components of P p

6 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] n r Optimistic Analysis: Assume edge with rank i is filtered when r = i. E[#survivors] n + i>n Partitioning m edges and sorting n log m n edges: n i ( = Θ n log m ) n Ω(m + n log n log m n )

7 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 [ X ij = 1 ]. strengthen bound on P [ X ij = 1 ] using Chan s sampling lemma O(m + n log n log m ) expected comparisons n

8 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

9 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

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

11 Parallelization Procedure filterkruskal(e, T : Sequence of Edge, P : UnionFind) if m kruskalthreshold(n, m, T ) then kruskal(e, T, P) // parallel sort else pick a pivot p E E := e E : e p // parallel E > := e E : e > p // partitioning qkruskal(e, T, P) if T = n 1 then exit E > := filter(e >, P) // parallel removeif qkruskal(e >, T, P) Easy: available in the Multi-Core Parallel STL (e.g. g++)

12 Running Time: Random graph with 2 16 nodes time / m [ns]

13 Graph Formatting: Random graph with 2 22 nodes List of edges -> Adjacency Array Adjacency Array -> List of Edges filterkruskal pjp time / m [ns] number of edges m / number of nodes n

14 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

15 Thank you!

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

17 Random graph with 2 22 nodes 1000 time / m [ns] number of edges m / number of nodes n

18 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

19 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

20 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

21 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

22 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

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

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

Filter-Kruskal MST 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? Motivation What is the best (practical)