Dynamic Graph Algorithms Giuseppe F. (Pino) Italiano University of - - PowerPoint PPT Presentation

Dynamic Graph Algorithms

Giuseppe F. (Pino) Italiano

University of Rome Tor Vergata giuseppe.italiano@uniroma2.it

http://people.uniroma2.it/giuseppe.italiano/

Main Goals

Understand what’s going on in a rich area (35+ years, still very active) Master the main algorithmic ideas, tools and techniques introduced Get involved and contribute to solving some exciting open problems!

Prerequisites

Basic data structures:

• Binary search trees, linking and cutting trees,

union-find, etc… Basic (graph) algorithms:

• DFS, BFS, MST, shortest paths, etc…

Analysis of algorithms:

• worst-case, average, amortized, randomized,

etc…

Outline

Dynamic Graph Problems – Quick Intro Topic 1. (Undirected Graphs) Dynamic Connectivity & MST Topic 2. (Undirected/Directed Graphs) Dynamic Shortest Paths Topic 3. (Non-dynamic?) 2-Connectivity in Directed Graphs

Theory is when you know something, but it doesn't work.

Theory

Practice is when something works, but you don't know why.

The real world out there…

Theory is when you know something, but it doesn't work. Practice is when something works, but you don't know why.

Combining Theory and Practice?

Theory is when you know something, but it doesn't work. Practice is when something works, but you don't know why. Big challenge: combine theory and practice… …i.e., nothing works and you don't know why.

Combining Theory and Practice?

Outline

Dynamic Graph Problems – Quick Intro Topic 1. (Undirected Graphs) Dynamic Connectivity & MST Topic 2. (Undirected/Directed Graphs) Dynamic Shortest Paths Topic 3. (Non-dynamic?) 2-Connectivity in Directed Graphs

Dynamic Graphs

Graphs subject to update operations

Insert(u,v) Delete(u,v) ChangeWeight(u,v,w)

Typical updates:

A graph

Initialize Insert Delete Query

Dynamic Graphs

Dynamic Graph Algorithms

The goal of a dynamic graph algorithm is to support query and update operations as fast as possible (usually much faster than recomputing from scratch). We will use also amortized analysis:

Total worst-case time over sequence of ops # operations

Notation:

G = (V,E) n = |V| m = |E|

Dynamic Graphs

Partially Dynamic Problems

Graphs subject to insertions only (incremental),

• r deletions only (decremental), but not both.

Fully Dynamic Problems

Graphs subject to intermixed sequences

• f insertions and deletions.

Usually much more difficult problems.

Dynamic Graph Problems

Support queries about properties on a dynamic graph

Dynamic Connectivity (undirected graph G) 


Connected(): Connected(x,y):

Is G connected? Are x and y connected in G? Dynamic Minimum Spanning Tree (undirected graph G) 
 Any property on a MST of G

Dynamic Graph Problems

Dynamic All Pairs Shortest Paths


Distance(x,y):

What is the distance from x to y in G?

ShortestPath(x,y):

What is the shortest path from x to y in G? Dynamic Transitive Closure (directed graph G) 


Reachable(x,y):

Is y reachable from x in G?

Dynamic Graph Problems

Dynamic Min Cut

MinCut(): Cut(x,y):

Min cut? Are x and y on the same side of a min cut of G? Dynamic Planarity Testing

planar():

Is G planar? Dynamic k-connectivity

k-connected(): k-connected(x,y):

Is G k-connected? Are x and y k-connected?

Dynamic Graph Problems

Dynamic (Approximate) Maximum Matching

Matching():

Maximum Matching?

ApproximateMatching():

Approximate Maximum Matching? ValueofMatching(): Dynamic (Approximate) Minimum Vertex Cover VertexCover(): Approximate Minimum Vertex Cover?

Outline

Dynamic Graph Problems – Quick Intro Topic 1. (Undirected Graphs) Dynamic Connectivity & MST Topic 2. (Undirected/Directed Graphs) Dynamic Shortest Paths Topic 3. (Non-dynamic?) 2-Connectivity in Directed Graphs

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Fully Dynamic Graph Connectivity

Maintain an undirected graph G under an intermixed sequence of

• perations of the following type:
• insert(u,v) : Add a new edge (u,v)
• delete(u,v) : Remove edge (u,v) from G (assumes (u,v) in G)
• connected(u,v) : Return yes if there is a path between u and v;

return no otherwise Subproblem (basic ingredient) in many other problems Minimum spanning trees, 2-connectivity, … Simple problem but lots of interesting ideas!

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connected(v,w)

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connected(v,w)

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connected(v,w)

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insert(v,w)

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insert(v,w)

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delete(v,w)

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delete(v,w)

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delete(v,w)

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delete(v,w)

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Incremental connectivity: Without deletions, union-find data structures would be just enough

Observation

Disjoint Set-Union:

• makeset(x): {x}
• find(x): returns set containing x
• union(x,y): replaces find(x) and find(y) by

their union

Incremental Connectivity

• insert(x,y): if find(x) ≠ find(y) then union(x,y)
• connected(x,y): return (find(x) == find(y))

(true iff x and y are connected) Amortized cost per operation is O(α(m,n)) (inverse Ackermann)

Incremental Connectivity

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Incremental connectivity: Without deletions, union-find data structures would be just enough

Observation

Incremental MST: Without deletions, linking and cutting trees [Sleator & Tarjan 1983] would be just enough. Why?

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Back to Fully Dynamic Connectivity

But simple case first: Graph is a forest

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Operations we need to do on the forest

link(v,w) : Join two trees in the forest by inserting edge (v,w) (assume v and w are in different trees) cut(v,w) : Split a tree by deleting edge (v,w) (assume v and w are adjacent in a tree) findtree(v) : Return the tree containing vertex v in the forest

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Operations we need to do on the forest

link(v,w) : Join two trees in the forest by inserting edge (v,w) (assume v and w are in different trees) cut(v,w) : Split a tree by deleting edge (v,w) (assume v and w are adjacent in a tree) findtree(v) : Return the tree containing vertex v in the forest Can do this in O(log n) per operation in the worst case with several data structures, e.g.:

• Sleator & Tarjan dynamic trees [1983],
• ET-trees (Euler Tour trees) [Tarjan & Vishkin 1984]

We refer to those as dynamic tree data structures

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ET-trees

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A B D E R G C F

ET-trees

Euler tour of a tree

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ET-trees

Euler tour of a tree

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ET-trees

ET-tree is a balanced binary tree over the Euler tour of a tree. Can perform link, cut and findtree in O(log n)

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d b-c-d-c-b-a- e-f-e-g-e-h-e

• a-b

b d b e e c c a f e g h a e b

How to transform a tree into a one dimensional data structure ?

Euler tour tree :

A data structure for dynamic trees

: minimum value of all

nodes in the subtree.

a b c e f g h d b-c-d-c-b-a- e-f-e-g-e-h-e

• a-b

b d b e e c c a f e g h a e b a f b c d e g h

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d b-c-d-c-b-a- e-f-e-g-e-h-e -a-b

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d b-c-d-c-b-a- e-f-e-g-e-h-e

• a-b

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d a-b b-c-d-c-b-a e-f-e-g-e-h-e T1 T2 T1

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d a-b b-c-d-c-b-a T1 T2 T1 T1 e-f-e-g-e-h-e

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d a-b b-c-d-c-b-a e-f-e-g-e-h-e T1 T2 T1

T1

Euler tour tree :

A data structure for dynamic trees

a b c e f g h d a-b b-c-d-c-b-a e-f-e-g-e-h-e T1 T2 T1 T1

Euler tour tree :

A data structure for dynamic trees

• Split(T,(e,a))
• Merge(T1,T2,(u,v))
• Change-origin(T,x) :

change the origin of Euler tour to vertex x. Each operation can be implemented in O(log n) worst-case time

a b c e f g h d b-c-d-c-b-a-b e-f-e-g-e-h-e T1 T2 T1 T1 T2 T2

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Fully Dynamic Connectivity

Get to the real thing: Graph is a graph

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Maintain a spanning forest of the graph We will have to link trees, cut trees, and determine whether two vertices are in the same tree in this forest Reduce the problem to a problem on trees (i.e., maintain a certificate for the property)

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But dynamic tree data structures are not enough: we still have a problem with deleting a tree edge

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How do we find

• ut whether

there is a “replacement” edge for the forest or it really got disconnected ? For dynamic MSF it is not enough to find a repacement edge, we need to find the best replacement edge

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Recap (so far)

Tree Edge Non-Tree Edge Insert Link Easy Delete Cut, Replacement? Easy

(Main) History of the Problem

Update Query Reference

O(log3 n) [Henzinger, King JACM’99] [Holm, de Lichtenberg & log n log log n O( )

Type

O(m1/2 ) O(1) [Frederickson SICOMP’85] det/w-c O(n1/2 ) O(1) [Eppstein, Galil, I. & det/w-c rand/amort O(log2 n) [Henzinger, Thorup log n log log n O( ) rand/amort O(log2 n) log n log log n O( ) det/amort O(log n (log log n)) log n log log log n O( ) rand/amort [Thorup STOC’00] [Wulff-Nilsen SODA’13] log n log log n O( ) log2 n log log n O( ) det/amort O(log5 n) [Kapron, King & O( ) rand/w-c log n log log n Nissenzweig JACM’97]

• Rand. Struct. & Algs. ’97]

Thorup JACM’01] Mountjoy SODA’13]

Very Recent Results

Update Query Reference Type

n log n O( ( )½ log log n ) O(log4 n) [Gibb, Kapron, King & O( ) rand/w-c log n log log n Thorn arXiv’15] O(1) [Keilberg-Rasmussen, det/w-c Thorup, arXiv’15] Kopelowitz, Pettie &

Very Recent Results

Update Query Reference Type

n log n O( ( )½ log log n ) O(log4 n) [Gibb, Kapron, King & O( ) rand/w-c log n log log n Thorn arXiv’15] O(1) [Keilberg-Rasmussen, det/w-c Thorup, arXiv’15] Kopelowitz, Pettie &

Questions (Open): For worst-case per operation, can we bridge the gap between randomized (polylog) and deterministic (polynomial)? Or at least get a Las Vegas polylog bound? (I.e., no errors)

Will see

Update Query Reference

O(log3 n) [Henzinger, King JACM’99] [Holm, de Lichtenberg & log n log log n O( )

Type

O(m1/2 ) O(1) [Frederickson SICOMP’85] det/w-c O(n1/2 ) O(1) [Eppstein, Galil, I. & det/w-c rand/amort O(log2 n) [Henzinger, Thorup log n log log n O( ) rand/amort O(log2 n) log n log log n O( ) det/amort O(log n (log log n)) log n log log log n O( ) rand/amort [Thorup STOC’00] [Wulff-Nilsen SODA’13] log n log log n O( ) log2 n log log n O( ) det/amort O(log5 n) [Kapron, King & O( ) rand/w-c log n log log n Nissenzweig JACM’97]

• Rand. Struct. & Algs. ’97]

Thorup JACM’01] Mountjoy SODA’13]

Will actually see

Update Query Reference

O(log3 n) [Henzinger, King JACM’99] [Holm, de Lichtenberg & log n log log n O( )

Type

O(m1/2 ) O(1) Simpler version det/w-c O(n1/2 ) O(1) [Eppstein, Galil, I. & det/w-c rand/amort O(log2 n) [Henzinger, Thorup log n log log n O( ) rand/amort O(log2 n) O(log n) det/amort O(log n (log log n)) log n log log log n O( ) rand/amort [Thorup STOC’00] [Wulff-Nilsen SODA’13] log n log log n O( ) log2 n log log n O( ) det/amort O(log5 n) [Kapron, King & O( ) rand/w-c log n log log n Nissenzweig JACM’97]

• Rand. Struct. & Algs. ’97]

Thorup JACM’01] Mountjoy SODA’13]