Path Minima on Dynamic Weighted Trees

Pooya DavoodiAarhus University

Aarhus University, November 17, 2010

Joint work with Gerth Stølting Brodal and S. Srinivasa Rao

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Path Minima Problem Definition

• Forest of unrooted trees• Operations:

make-tree, path-minima, weight-update, link, cut

12

1

6

15 42f

b

cea

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

Applications: Network Flows, Minimum Spanning Trees,

Transportation Problem, Network Optimization Algorithms

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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

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Outline

Path Minima Problemmake-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

Variantsmake-tree, beq, update, link, cut

• Previous Works

• Lower Bounds• Static Trees with

Dynamic Weights• Leaf-Link-Cut Trees with

Static Weights

New

New

Reductions

Reductions

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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

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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?

u

v

Lower BoundsConnectivity

• 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

7

u v

−∞ −∞r

w

−∞

Patrascu and Demaine (STOC’04)

(Cell Probe)

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Lower BoundsIncremental Connectivity

• Boolean Union-Find Incremental Connectivity

• Same reduction algorithm– When , then

Kaplan et. al. (STOC'02)

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Lower Bounds1D-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)

𝑤1 𝑤2 𝑤3 𝑤4 𝑤5 𝑤6

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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)

VariantsOperations Preprocessing Path Minima Update link & cut Comments

beq, update & link no results

beq & linkno 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)

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𝑘−1

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Static Treeswith Dynamic Weights

degree

Transformation: add O(m) edges

make it rooted

Path Minima on

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Static Treeswith Dynamic Weights

• Heavy-path decomposition• path-minima: Tabulating in small subtrees, ,

• update: Using Q-heap,

Path Minima on

𝜖 log log𝑛

O( log𝑛)

O( log𝜖𝑛)

u

v

cont.

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Leaf-Link-Cut Trees with Static Weights

𝑏1

𝑏2𝑏3 𝑏4

𝑏7𝑏5𝑏6 𝑏1

𝑏2

𝑏3

𝑏4

𝑏5𝑏6

𝑏7

𝑏1

𝑏3

𝑏2

𝑏4 𝑏5𝑏6𝑏7

𝒖

𝒗

𝒖 𝒗

make it rooted

Topological Partitioning

𝑂 (𝛼𝑘−1 (𝑛 ))

Preprocessing: Path Minima: Leaf-link and Leaf-cut:

Recursionlink: Split & Update

cut: Global Rebuilding

Path Minima on

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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)

VariantsOperations Preprocessing Path Minima Update link & cut Comments

beq, update & link no results

beq & linkno 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)

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THANK YOU