Param etric an d Kin etic Min im um Span n in g Trees Pan kaj K. Agarwal David Eppstein Leon idas J. Guibas Mon ika R. Hen zin ger 1
Param etric Min im um Span n in g Tree: Given graph, edges labeled by lin ear fun ction s λ − λ 2 3 − λ 3 + λ Fin d MST for each possible value of λ λ < −2 −2 < λ < −1 −1 < λ < 1 1 < λ < 2 2 < λ 5 + 2λ 3 + λ 2 3 − λ 5 − 2λ
Geom etric In terpretation : Poin t (-B,A) for tree w/ weight A+ B λ 6+λ 6−λ 5+2λ 5 5−2λ λ=1.5 3+λ 3−λ 2 Then MST( λ )= tan gen t to lin e w/ slope λ so param etric MST = lower con vex hull
Application s For any quasicon cave fun ction f ( A , B ) optim um tree m ust be a con vex hull vertex Tree w/ m in im um cost-reliability ratio ( A = cost, B = − log probability all edges exist): f ( A , B ) = A exp ( B ) Tree w/ m in im um varian ce in total weight (if edge weights in depen den t ran dom variables): f ( A , B ) = A − B 2 Tree with high probability of low total weight (if edge weights in depen den t Gaussian variables): √ f ( A , B ) = A + B So each of these optim a can be foun d from param etric MST solution 2
Previous Results on Param etric MST Num ber of breakpoin ts: • O ( m n 1 / 3 ) [Dey 1997] • Ω ( m α ( n )) [Eppstein 1995] Tim e to com pute all trees: • O ( m n log n ) [Fern ´ an dez-Baca, Slutzki, Eppstein 1996] 3
Dyn am ic Min im um Span n in g Tree An altern ate form of tim e-varyin g data: Weighted graph subject to discrete updates (like param etric w/ piecewise con stan t fun ction s) Many algorithm s kn own [Sleator, Tarjan 1983] [Frederickson 1985] [Eppstein 1991] [Eppstein , Galil, Italian o, Nissen zweig 1992] [Eppstein , Galil, Italian o, Spen cer 1993] [Hen zin ger, Kin g 1997] [Holm , de Lichten berg, Thorup 1998] Curren t best tim e: O ( log 4 n ) per update better for restricted updates or plan ar graphs Idea: apply dyn am ic graph algorithm techn iques to param etric MST problem 4
How to com bin e param etric an d dyn am ic? Kin etic Algorithm s! In terpret λ as tim e param eter start with param etric problem , sm all λ in crease λ an d perform updates m ain tain in g correct MST at each poin t in process Idea: m odel short-term predictability an d lon g-term un predictability of real-world application s Two kin ds of updates possible: structural: edge in sertion s an d deletion s functional: relabel existin g edge w/ n ew fun ction 5
Other Kin etic Algorithm s [Basch, Guibas, Hershberger 1997] [Basch, Guibas, Zhan g 1997] [Guibas 1998] [Agarwal, Erickson , Guibas 1998] [Basch, Erickson , Guibas, Hershberger, Zhan g 1999] Basic data structures (Priority queue) Com putation al geom etry (Con vex hull, closest pair, bin ary space partition , polygon in tersection ) Typical tim e boun ds are polylog × worst case n um ber of chan ges to solution 6
New Results Gen eral graphs: • O ( m 2 / 3 log 4 / 3 m ) per output chan ge • O ( n 2 / 3 log 4 / 3 n ) tim es worst case # chan ges Min or-closed graph fam ilies (in cludin g plan ar graphs): • O ( n 1 / 2 log 3 / 2 n ) per output chan ge Min or-closed fam ilies with on ly fun ction al updates: • O ( n 3 / 2 ) preprocessin g (n on plan ar graphs on ly) • O ( n 1 / 4 log 3 / 2 n ) tim es worst case # chan ges Som e ran dom ized im provem en ts to polylogs 7
Idea I: Clusterin g Expan d vertices so graph has degree three, then ... Group MST in to k clusters of O( n/ k ) edges, at m ost two edges crossin g each cluster boun dary [Frederickson 1985] Form bun dles of n on -tree edges, accordin g to the clusters con tain in g their en dpoin ts adjust clusters as MST chan ges
Classification of MST chan ges MST always chan ges by swap: in sert n on -tree edge, delete tree edge Three types of swap: • In tra-cluster swap: tree edge belon gs to cluster con tain in g both n on -tree edge en dpoin ts • Dual-cluster swap: tree edge belon gs to cluster con tain in g both n on -tree edge en dpoin ts • In ter-cluster swap: tree edge an d n on -tree edge are in disjoin t clusters 8
Fin din g In tra-Cluster Swaps Use Megiddo’s param etric search to fin d last value of λ for which the cluster has sam e MST Decision oracle is (static) MST algorithm Tim e: ˜ O ( m / k ) per chan ged cluster Each update chan ges O ( 1 ) clusters so ˜ O ( m / k ) total 9
Fin din g In ter-Cluster Swaps Collapse each bun dle or cluster to superedge Weight of bun dle superedge = m in in bun dle Weight of cluster superedge = m ax in path Han dle weight queries usin g con vex hull of coefficien ts of edge labels in bun dle or cluster Fin d swap by param etric search in collapsed graph Tim e for param etric search: ˜ O ( # bun dles) Tim e to rebuild con vex hulls: ˜ O ( m / k ) per chan ged cluster 10
Fin din g Dual-Cluster Swaps “ Am bivalen t data structure” [Frederickson 1997] For each n on -tree edge en dpoin t, there are two tree paths in side the cluster to the two cluster exits. Non -tree edge stores a can didate swap per exit Foun d by traversin g MST within cluster queryin g dyn am ic con vex hull of path edges Each bun dle stores a can didate swap per exit the best am on g all swaps stored by its edges Best dual-cluster swap foun d by checkin g which can didate is correct for each bun dle, pickin g the best of the correct can didates Tim e to update edge an d bun dle can didates: ˜ O ( m / k ) per chan ged cluster Tim e to fin d best swap: ˜ O ( # bun dles) 11
An alysis of Clusterin g Gen eral graphs: O ( m / k + k 2 ) Total tim e ˜ O ( m 1 / 3 ) Optim al k = ˜ O ( m 2 / 3 ) per MST chan ge ˜ Sparse (m in or-closed) graph fam ilies: Total tim e ˜ O ( n / k + k ) O ( n 1 / 2 ) Optim al k = ˜ O ( n 1 / 2 ) per MST chan ge ˜ 12
Idea II: Sparsification [Eppstein , Galil, Italian o, Nissen zweig 1992] [Fern ´ an dez-Baca, Slutzki, Eppstein 1996] Split edges of graph in to two subsets G = G 1 ∪ G 2 Main tain MST of each subset (two sm aller kin etic problem s) Com bin e to get MST of overall graph (on e sparse structurally kin etic problem ) MST ( G ) = MST ( T 1 ∪ T 2 ) 13
Sparsification An alysis Replaces factors of m by factors of n in any gen eral graph MST algorithm But subproblem s chan ges m ay n ot propagate to global MST so also replaces factors of actual MST chan ges with worst-case # chan ges Therefore: gen eral graph kin etic MST O ( n 2 / 3 ) tim es worst-case # chan ges ˜ 14
Separator Based Sparsification [Eppstein , Galil, Italian o, Spen cer 1993] Given functionally kin etic problem Form separator decom position of sparse graph Solve MST problem s on each side of separator (two sm aller fun ction ally kin etic problem s) Use solution s to form com pact certificate (graph with O ( √ n ) vertices havin g sam e kin etic behavior as origin al subgraph) Com bin e certificates (on e very sm all structurally kin etic problem ) O ( n 1 / 4 ) per worst-case chan ge Total tim e: ˜ 15
Con clusion s an d Open Problem s New kin etic MST algorithm s Som e im provem en t to param etric MST especially in the plan ar case (n ow O ( n 19 / 12 ) ) but for gen eral graphs, still n ot o ( m n ) Plan ar graph algorithm uses clusterin g in sparsified subproblem s — can we in stead use sparsification recursively? Geom etric kin etic MST? Edge weights becom e quadratic in stead of lin ear 16
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