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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

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
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
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
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
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 optimal fully dynamic connectivity
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 bounds connectivity
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 bounds incremental connectivity
Lower BoundsIncrementalConnectivity
  • Boolean Union-Find Incremental Connectivity
  • Same reduction algorithm
    • When , then

Kaplan et. al. (STOC'02)

lower bounds 1d rmq
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 minima open problems
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 trees with dynamic weights
Static Treeswith Dynamic Weights

Path Minima on

Transformation: add O(m) edges

make it rooted

degree

static trees with dynamic weights1
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 t rees with static weights
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 minima open problems1
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)