Path Minima on Dynamic Weighted Trees

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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

h

12

a

10

i

c

e

1

1

15

4

2

g

b

6

make-tree(i)

f

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

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
• 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

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

Path Minima on

Topological Partitioning

Recursion

cut: Global Rebuilding

make it rooted

Preprocessing:

Path Minima: