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Path Minima on Dynamic Weighted Trees. Joint work with Gerth Stølting Brodal and S. Srinivasa Rao. Pooya Davoodi Aarhus University. Aarhus University, November 17, 2010. Path Minima Problem Definition. Applications:
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Path Minima on Dynamic Weighted Trees Joint work with Gerth Stølting Brodal and S. SrinivasaRao Pooya Davoodi Aarhus University Aarhus University, November 17, 2010
Path Minima Problem Definition Applications: Network Flows, Minimum Spanning Trees, Transportation Problem, Network Optimization Algorithms • Forest of unrooted trees • Operations:make-tree, path-minima,weight-update, link, cut h 12 a 10 i c e 1 1 15 4 2 g b 6 make-tree(i) f link(g,b,2) d path-minima(d,f) (g,b) cut(e,g) path-minima: bottleneck edge query (beq) weight-update(b,c,1)
Computational Models • Unit-cost RAM with word size bits • Operations on the edge-weights: • semigroup operations • the weights are from a semigroup • a straight line program (no comparisons) • should work for any semigroup operation (e.g., +, *, min) • comparisons • standard RAM operations
Outline • Path Minima Problem • make-tree, beq, update, link, cut • Dynamic Trees ofSleator and Tarjan (STOC’81) • Dynamic Trees is Optimal Patrascu and Demaine (STOC’04) • Lower Bounds • The Problem is Open • Variants • make-tree, beq, update, link, cut • Previous Works • Lower Bounds • Static Trees withDynamic Weights • Leaf-Link-Cut Trees withStatic Weights Reductions New Reductions New
Dynamic Trees (Link-Cut Trees)Sleator and Tarjan (STOC’81) • Arbitrary roots with operation evert(more operations: parent, root, LCA) • Vertex-disjoint path decomposition • Each path represented by a biased search tree or a splay tree • Operations in O(log n) • Model: Semigroup by J. Erickson, C. Osborn
Dynamic Trees is OptimalFully Dynamic Connectivity • Reduced to Sleator and Tarjan’s • connectivity: root or evert • insert: link • delete: cut • Patrascu and Demaine (STOC’04) • Reduction from Dynamic Partial Sums (Cell Probe) • They are optimal(logarithmic bounds) • What If we do not exploit root and evert? • Even in Comparison and RAM models? v u
Lower BoundsConnectivity (Cell Probe) • Reduction from Fully Dynamic Connectivity • connectivity(u,v): beq(u,v) • insert(u,v,w): cut (beq(u,r)) + link(u,v,w) • delete(u,v): (2*beq) + (4*link) + (4*cut) • , and • when , then • When , then • If , then Patrascu and Demaine (STOC’04) r w u v
Lower BoundsIncrementalConnectivity • Boolean Union-Find Incremental Connectivity • Same reduction algorithm • When , then Kaplan et. al. (STOC'02)
Lower Bounds1D-RMQ • Just a Path with no link & cut • Brodal et. al.(SWAT'96) • reduction from Insert-Delete-FindMinin (Comparison) • Alstrup et. al.(FOCS'98): • reduction from Priority Search Trees (Cell Probe) • Patrascu and Demaine (SODA'04): • reduction from Dynamic Partial Sums (Semigroup)
Path MinimaOpen Problems • When , improve to • For polylog , lower bound of ? • Touch the curve:when , then • When , then • When , then (RAM model) Conjecture of Patrascu and Thorup (STOC’06) (Comparison and RAM models)
Static Treeswith Dynamic Weights Path Minima on Transformation: add O(m) edges make it rooted degree
Static Treeswith Dynamic Weights Path Minima on cont. • Heavy-path decomposition • path-minima: Tabulating in small subtrees, , • update: Using Q-heap, v u
Leaf-Link-Cut Trees with Static Weights Path Minima on Topological Partitioning Recursion link: Split & Update cut: Global Rebuilding make it rooted Preprocessing: Path Minima: Leaf-link and Leaf-cut:
Path MinimaOpen Problems • When , improve to • For polylog , lower bound of ? • Touch the curve:when , then • When , then • When , then (RAM model) Conjecture of Patrascu and Thorup (STOC’06) (Comparison and RAM models)
