1. Path Minima on Dynamic Weighted Trees Pooya Davoodi Aarhus University Aarhus University, November 17, 2010 Joint work with Gerth Stølting Brodal and S. Srinivasa Rao

2. Path Minima Problem Definition Forest of unrooted trees Operations: make-tree, path-minima, weight-update, link, cut 12 1 6 154 2 f b c e a g d i make-tree(i) link(g,b,2) path-minima(d,f) cut(e,g) (g,b) weight-update(b,c,1) 1 path-minima: bottleneck edge query (beq) h 10 2 Applications: Network Flows, Minimum Spanning Trees, Transportation Problem, Network Optimization Algorithms

3. Computational Models 3

4. Outline 4 Path Minima Problem make-tree, beq, update, link, cut Dynamic Trees of Sleator and Tarjan (STOC’81) Dynamic Trees is Optimal Patrascu and Demaine (STOC’04) Lower Bounds The Problem is Open New Reductions

5. 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 5

6. Dynamic Trees is Optimal Fully Dynamic Connectivity 6 u v

7. Lower Bounds Connectivity 7 uv r w Patrascu and Demaine (STOC’04) (Cell Probe)

8. Lower Bounds Incremental Connectivity 8 Kaplan et. al. (STOC'02)

9. Lower Bounds 1D-RMQ Just a Path with no link & cut Brodal et. al.(SWAT'96) reduction from Insert-Delete-FindMin in (Comparison) Alstrup et. al.(FOCS'98): reduction from Priority Search Trees (Cell Probe) Patrascu and Demaine (SODA'04): reduction from Dynamic Partial Sums (Semigroup) 9

10. Path Minima Open Problems 10 (RAM model) Conjecture of Patrascu and Thorup (STOC’06) (Comparison and RAM models)

11. Variants OperationsPreprocessingPath MinimaUpdatelink & cutComments beq, update & link no results beq & link no results Semigroup & Comparisons -RAM, Kaplan et al. (ESA’08) beq Semigroup & Comparison, Chazelle (FOCS’84) Alon & Shieber (TecRep’87) Pettie (FOCS’02) RAM, Kaplan et al. (ESA’08) beq & update Comparison – New RAM - New beq, leaf-link & leaf-cut Semigroup – New RAM, Kaplan et al. (ESA’08) 11

12. Static Trees with Dynamic Weights 12 Transformation: add O(m) edges make it rooted Path Minima on

13. Static Trees with Dynamic Weights 13 Path Minima on u v cont.

14. Leaf-Link-Cut Trees with Static Weights 14 make it rooted Topological Partitioning Recursion link: Split & Update cut: Global Rebuilding Path Minima on

15. Path Minima Open Problems 15 (RAM model) Conjecture of Patrascu and Thorup (STOC’06) (Comparison and RAM models)

16. Variants OperationsPreprocessingPath MinimaUpdatelink & cutComments beq, update & link no results beq & link no results Semigroup & Comparisons -RAM, Kaplan et al. (ESA’08) beq Semigroup & Comparison, Chazelle (FOCS’84) Alon & Shieber (TecRep’87) Pettie (FOCS’02) RAM, Kaplan et al. (ESA’08) beq & update Comparison – New RAM - New beq, leaf-link & leaf-cut Semigroup – New RAM, Kaplan et al. (ESA’08) 16

17. 17 THANK YOU