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Dynamic Graph Algorithms Christian Wulff-Nilsen University of Copenhagen November 14 , 2019 1 / 21 Graph algorithms and data structures 2 / 21 Graph algorithms and data structures Graphs and graph problems are everywhere: 2 / 21 Graph

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SLIDE 1

1 / 21

Dynamic Graph Algorithms

Christian Wulff-Nilsen University of Copenhagen

November 14, 2019

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SLIDE 2

Graph algorithms and data structures

2 / 21

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SLIDE 3

Graph algorithms and data structures

2 / 21

  • Graphs and graph problems are everywhere:
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SLIDE 4

Graph algorithms and data structures

2 / 21

  • Graphs and graph problems are everywhere:
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SLIDE 5

Graph algorithms and data structures

2 / 21

  • Graphs and graph problems are everywhere:
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SLIDE 6

Graph algorithms and data structures

2 / 21

  • Graphs and graph problems are everywhere:
  • Often, we want to model graphs that change over time
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SLIDE 7

Graph algorithms and data structures

2 / 21

  • Graphs and graph problems are everywhere:
  • Often, we want to model graphs that change over time
  • Dynamic algorithms and data structures are tailor-made to

handle changes in very large graphs

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SLIDE 8

Graphs

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  • A graph G is a pair (V, E) where V is the set of vertices and E

is the set of connections between these vertices called edges

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SLIDE 9

Graphs

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  • A graph G is a pair (V, E) where V is the set of vertices and E

is the set of connections between these vertices called edges

  • Let m = |E| and n = |V | in the following
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SLIDE 10

Graphs

3 / 21

  • A graph G is a pair (V, E) where V is the set of vertices and E

is the set of connections between these vertices called edges

  • Let m = |E| and n = |V | in the following
  • Edges may have weights and may be directed or undirected:

4 4 3 3 3 2 2 3 2 1 3 1 2 1 3 2 1

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SLIDE 11

A classical graph problem: minimum spanning forest

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  • We have near-linear time algorithms (Prim, Kruskal,...) to

compute a minimum spanning forest of a given graph:

3 8 6 2 4 1 4 2 1 5 1 3

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SLIDE 12

A classical graph problem: minimum spanning forest

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  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 2 1 5 1

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SLIDE 13

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 2 1 5 1 3

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SLIDE 14

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 2 1 5 1 3

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SLIDE 15

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 2 1 5 1 3 7

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SLIDE 16

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 2 1 5 1 3 7

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SLIDE 17

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 1 5 1 3 7

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SLIDE 18

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 1 5 1 3 7

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SLIDE 19

A classical graph problem: minimum spanning forest

4 / 21

  • Maintaining a minimum spanning forest in a dynamic graph:

3 8 6 2 4 1 4 1 5 1 3 7

  • Can we handle each update faster than rebuilding from scratch?
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SLIDE 20

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

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SLIDE 21

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
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SLIDE 22

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
  • A query for whether two vertices u and v are connected in the

current graph G

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SLIDE 23

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
  • A query for whether two vertices u and v are connected in the

current graph G

  • Example:
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SLIDE 24

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
  • A query for whether two vertices u and v are connected in the

current graph G

  • Example:
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SLIDE 25

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
  • A query for whether two vertices u and v are connected in the

current graph G

  • Example:
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SLIDE 26

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
  • A query for whether two vertices u and v are connected in the

current graph G

  • Example:
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SLIDE 27

Dynamic connectivity

5 / 21

  • A data structure for dynamic connectivity should support the

following in a dynamic graph G:

  • The insertion/deletion of an edge in G
  • A query for whether two vertices u and v are connected in the

current graph G

  • Example:
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SLIDE 28

Clusters

6 / 21

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SLIDE 29

Cluster forest

7 / 21

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SLIDE 30

Handling an update insert(u, v)

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SLIDE 31

Handling an update insert(u, v)

8 / 21

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SLIDE 32

Handling an update insert(u, v)

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SLIDE 33

Handling an update insert(u, v)

8 / 21

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SLIDE 34

A vertex of Mi is explored by both search procedures

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v u

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SLIDE 35

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 36

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 37

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 38

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 39

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 40

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 41

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 42

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 43

A vertex of Mi is explored by both search procedures

9 / 21

v u

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SLIDE 44

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 45

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 46

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 47

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 48

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 49

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 50

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 51

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 52

A search procedure has no more edges to explore

10 / 21

v u

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SLIDE 53

Traversing a single graph edge

11 / 21

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SLIDE 54

Spanning tree of a graph

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  • A spanning tree of a connected undirected graph G is a tree

contained in G which contains all vertices of G:

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SLIDE 55

Spanning tree of a graph

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  • A spanning tree of a connected undirected graph G is a tree

contained in G which contains all vertices of G:

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SLIDE 56

Spanning tree of a graph

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  • A spanning tree of a connected undirected graph G is a tree

contained in G which contains all vertices of G:

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SLIDE 57

Spanning forest of a graph

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  • A spanning forest of a graph G consists of a spanning tree of

each connected component of G:

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SLIDE 58

Spanning forest of a graph

13 / 21

  • A spanning forest of a graph G consists of a spanning tree of

each connected component of G:

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SLIDE 59

Spanning forest of a graph

13 / 21

  • A spanning forest of a graph G consists of a spanning tree of

each connected component of G:

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SLIDE 60

Spanning forest of a graph

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  • A spanning forest of a graph G consists of a spanning tree of

each connected component of G:

  • If we can efficiently maintain a spanning forest of a dynamic

graph G then we also have an efficient data structure for dynamic connectivity

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SLIDE 61

Dynamic Trees

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  • Suppose we want a data structure maintaining some information

about a dynamic forest where there is no underlying graph:

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SLIDE 62

Dynamic Trees

14 / 21

  • Suppose we want a data structure maintaining some information

about a dynamic forest where there is no underlying graph:

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SLIDE 63

Dynamic Trees

14 / 21

  • Suppose we want a data structure maintaining some information

about a dynamic forest where there is no underlying graph:

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SLIDE 64

Dynamic Trees

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  • The data structure should be able to:
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SLIDE 65

Dynamic Trees

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  • The data structure should be able to:
  • Insert/delete an edge
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SLIDE 66

Dynamic Trees

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  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
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SLIDE 67

Dynamic Trees

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  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

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SLIDE 68

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 101

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SLIDE 69

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 101

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SLIDE 70

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 101

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SLIDE 71

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 101

Change to 100

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SLIDE 72

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 100

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SLIDE 73

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 100

Bitwise AND:

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SLIDE 74

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 100

Bitwise AND: 100

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SLIDE 75

Dynamic Trees

14 / 21

  • The data structure should be able to:
  • Insert/delete an edge
  • Update the key value of a vertex
  • Report the sum or the bitwise AND/OR/XOR of all vertices in

the tree containing a given vertex

000 101 111 110 100 100 010 100

Bitwise AND: 100

  • There is a dynamic tree data structure supporting each such
  • peration in O(log n) time
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SLIDE 76

Fully-dynamic connectivity

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  • Efficiently maintaining a spanning forest of a graph:

1 2 4 5 6 7 3

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SLIDE 77

Fully-dynamic connectivity

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  • Efficiently maintaining a spanning forest of a graph:

000 001 010 100 101 110 111 011

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SLIDE 78

Fully-dynamic connectivity

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  • Efficiently maintaining a spanning forest of a graph:

000 001 010 100 101 110 111 011 000001 000010 011111 100111 010101 101110 110111 001011 011100 011110

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SLIDE 79

Fully-dynamic connectivity

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  • Efficiently maintaining a spanning forest of a graph:

000001 000010 011111 100111 010101 101110 110111 001011 011110 011100

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SLIDE 80

Fully-dynamic connectivity

15 / 21

  • Efficiently maintaining a spanning forest of a graph:

001010 010111 111011 010110 000001 000010 011111 100111 010101 101110 110111 001011 011100 011110 000011 111011 000111 001111

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SLIDE 81

Fully-dynamic connectivity

15 / 21

  • Efficiently maintaining a spanning forest of a graph:

001010 010111 111011 010110 000001 000010 011111 100111 010101 110111 001011 011100 011110 000011 010101 101001 001111

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SLIDE 82

Fully-dynamic connectivity

15 / 21

  • Efficiently maintaining a spanning forest of a graph:

001010 010111 111011 010110 000001 000010 011111 100111 010101 110111 001011 011100 011110 000011 010101 101001 001111

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SLIDE 83

Fully-dynamic connectivity

16 / 21

  • The previous only works if there is at most one edge that can

reconnect the tree.

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SLIDE 84

Fully-dynamic connectivity

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  • The previous only works if there is at most one edge that can

reconnect the tree.

  • Handling multiple reconnecting edges with random sampling:

1 2 4 5 6 7 3

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SLIDE 85

Fully-dynamic connectivity

16 / 21

  • The previous only works if there is at most one edge that can

reconnect the tree.

  • Handling multiple reconnecting edges with random sampling:

1 2 4 5 6 7 3

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SLIDE 86

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 2 1 5 1

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SLIDE 87

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 2 1 5 1 3

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SLIDE 88

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 2 1 5 1 3

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SLIDE 89

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 2 1 5 1 3 7

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SLIDE 90

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 2 1 5 1 3 7

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SLIDE 91

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 1 5 1 3 7

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SLIDE 92

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 1 5 1 3 7

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SLIDE 93

Dynamic minimum spanning forest

17 / 21

  • Suppose that instead of maintaining a spanning forest, we want

to maintain a minimum spanning forest:

3 8 6 2 4 1 4 1 5 1 3 7

  • Does the XOR trick work here?
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SLIDE 94

Types of randomized algorithms

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  • For several graph problems, state-of-the-art algorithms are

randomized

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SLIDE 95

Types of randomized algorithms

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  • For several graph problems, state-of-the-art algorithms are

randomized

  • Some of these algorithms are Las Vegas, some are Monte Carlo.
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SLIDE 96

Types of randomized algorithms

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  • For several graph problems, state-of-the-art algorithms are

randomized

  • Some of these algorithms are Las Vegas, some are Monte Carlo.
  • Las Vegas algorithm: produces the correct solution but

time/space random

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SLIDE 97

Types of randomized algorithms

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  • For several graph problems, state-of-the-art algorithms are

randomized

  • Some of these algorithms are Las Vegas, some are Monte Carlo.
  • Las Vegas algorithm: produces the correct solution but

time/space random

  • Monte Carlo algorithm: may give incorrect solutions but

time/space typically not random

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SLIDE 98

Issues with randomized dynamic graph algorithms

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  • A randomized dynamic graph algorithm either needs to:
  • hide its random bits from the user or
  • assume that the user is not adaptive
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SLIDE 99

Issues with randomized algorithms

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  • Concrete example: Randomized approximate shortest path

algorithm supporting:

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SLIDE 100

Issues with randomized algorithms

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  • Concrete example: Randomized approximate shortest path

algorithm supporting:

  • Updates: insertion/deletion of a single edge
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SLIDE 101

Issues with randomized algorithms

20 / 21

  • Concrete example: Randomized approximate shortest path

algorithm supporting:

  • Updates: insertion/deletion of a single edge
  • Queries: given query vertices u and v, output a value

˜ dG(u, v) such that dG(u, v) ≤ ˜ dG(u, v) ≤ δ · dG(u, v)

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SLIDE 102

Issues with randomized algorithms

20 / 21

  • Concrete example: Randomized approximate shortest path

algorithm supporting:

  • Updates: insertion/deletion of a single edge
  • Queries: given query vertices u and v, output a value

˜ dG(u, v) such that dG(u, v) ≤ ˜ dG(u, v) ≤ δ · dG(u, v)

  • Suppose a user only has access to the values ˜

dG(u, v) obtained

from queries

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SLIDE 103

Issues with randomized algorithms

20 / 21

  • Concrete example: Randomized approximate shortest path

algorithm supporting:

  • Updates: insertion/deletion of a single edge
  • Queries: given query vertices u and v, output a value

˜ dG(u, v) such that dG(u, v) ≤ ˜ dG(u, v) ≤ δ · dG(u, v)

  • Suppose a user only has access to the values ˜

dG(u, v) obtained

from queries

  • Even this may potentially reveal too much information about the

random bits of the algorithm

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SLIDE 104

Oblivious versus adaptive adversary

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  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

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SLIDE 105

Oblivious versus adaptive adversary

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  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

  • Adaptive adversary: the opposite of oblivious adversary
  • is allowed to choose subsequent updates/queries based on

the information revealed by the algorithm so far

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SLIDE 106

Oblivious versus adaptive adversary

21 / 21

  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

  • Adaptive adversary: the opposite of oblivious adversary
  • is allowed to choose subsequent updates/queries based on

the information revealed by the algorithm so far

  • Assuming an oblivious adversary:
  • deals with the issue on the previous slides
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SLIDE 107

Oblivious versus adaptive adversary

21 / 21

  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

  • Adaptive adversary: the opposite of oblivious adversary
  • is allowed to choose subsequent updates/queries based on

the information revealed by the algorithm so far

  • Assuming an oblivious adversary:
  • deals with the issue on the previous slides
  • but significantly limits the range of applications for the

algorithm

slide-108
SLIDE 108

Oblivious versus adaptive adversary

21 / 21

  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

  • Adaptive adversary: the opposite of oblivious adversary
  • is allowed to choose subsequent updates/queries based on

the information revealed by the algorithm so far

  • Assuming an oblivious adversary:
  • deals with the issue on the previous slides
  • but significantly limits the range of applications for the

algorithm

  • Two ways of allowing an adversary to be adaptive:
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SLIDE 109

Oblivious versus adaptive adversary

21 / 21

  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

  • Adaptive adversary: the opposite of oblivious adversary
  • is allowed to choose subsequent updates/queries based on

the information revealed by the algorithm so far

  • Assuming an oblivious adversary:
  • deals with the issue on the previous slides
  • but significantly limits the range of applications for the

algorithm

  • Two ways of allowing an adversary to be adaptive:
  • develop a deterministic algorithm
slide-110
SLIDE 110

Oblivious versus adaptive adversary

21 / 21

  • Oblivious adversary: entire sequence of updates/queries fixed in

advance

  • Adaptive adversary: the opposite of oblivious adversary
  • is allowed to choose subsequent updates/queries based on

the information revealed by the algorithm so far

  • Assuming an oblivious adversary:
  • deals with the issue on the previous slides
  • but significantly limits the range of applications for the

algorithm

  • Two ways of allowing an adversary to be adaptive:
  • develop a deterministic algorithm
  • develop a randomized algorithm and prove that it does not

reveal (enough) information about its random bits