Param etric an d Kin etic Min im um Span n in g Trees Pan kaj K. - - PDF document

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

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λ < −2 5 + 2λ −2 < λ < −1 3 + λ −1 < λ < 1 2 1 < λ < 2 3 − λ 2 < λ 5 − 2λ

Param etric Min im um Span n in g Tree:

Fin d MST for each possible value of λ Given graph, edges labeled by lin ear fun ction s

2 3 + λ − λ λ 3 − λ

Geom etric In terpretation :

Poin t (-B,A) for tree w/ weight A+ Bλ Then MST(λ)= tan gen t to lin e w/ slope λ so param etric MST = lower con vex hull

5 2 3+λ 3−λ λ=1.5 5+2λ 5−2λ 6+λ 6−λ

Application s

For any quasicon cave fun ction f (A, B)

• ptim 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 − B2 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

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Previous Results on Param etric MST

Num ber of breakpoin ts:

• O(m n1/3) [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]

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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(log4 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

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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

• f 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

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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

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New Results

Gen eral graphs:

• O(m 2/3 log4/3 m ) per output chan ge
• O(n2/3 log4/3 n) tim es worst case # chan ges

Min or-closed graph fam ilies (in cludin g plan ar graphs):

• O(n1/2 log3/2 n) per output chan ge

Min or-closed fam ilies with on ly fun ction al updates:

• O(n3/2) preprocessin g (n on plan ar graphs on ly)
• O(n1/4 log3/2 n) tim es worst case # chan ges

Som e ran dom ized im provem en ts to polylogs

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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

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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

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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

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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)

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An alysis of Clusterin g

Gen eral graphs: Total tim e ˜ O(m /k + k2) Optim al k = ˜ O(m 1/3) ˜ O(m 2/3) per MST chan ge Sparse (m in or-closed) graph fam ilies: Total tim e ˜ O(n/k + k) Optim al k = ˜ O(n1/2) ˜ O(n1/2) per MST chan ge

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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 = G1 ∪ G2 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 (T1 ∪ T2)

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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(n2/3) tim es worst-case # chan ges

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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 ) Total tim e: ˜ O(n1/4) per worst-case chan ge

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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(n19/12)) 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

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