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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
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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