5.3 dyn algo-i

Dynamic Graph Algorithms - I
Surender Baswana
Department of CSE, IIT Kanpur.

About this talk
• Prerequisite: a course on data structure and algorithms
• Survey of the results: (not the main objective of the talk)
• Main contents:
-- A novel data structure
-- A fully dynamic algorithm for a fundamental graph problem
AIM:
To give an exposure to the beautiful area of dynamic algorithmsanimation

A (static) Graph Algorithm
AlgorithmG=(V,E) Solution

A dynamic graph algorithm
• For problems involving queries (connectivity, distance,…):
initial graph G=(V,E) followed by a sequence
q,u,u,q,u,u,u,q,u,q,q,u, …
q: query
u: insertion/deletion of edge
Each query has to be answered in an online manner.
• For problems that aim to maintain some structure (matching, spanner, min-cut,…)
initial graph G=(V,E) followed by a sequence
u,u,u,u,u,u,u, …
The structure has to be maintained in an online manner.
No assumption about the updates.

Dynamic graph algorithm
Aim: Maintain a data structure which can
• answer each query efficiently (or maintain the structure), and
• process each update efficiently (much faster than the static algo)
Types of dynamic graph algorithms:
• Incremental (only insertion of edges)
• Decremental (only deletion of edges)
• Fully dynamic (both insertion and deletion of edges)

A motivating example :
Undirected Connectivity
u
v
a aaabb b b b bc c c cccd d d d
Static
solution:

• Incremental Algorithm: O(log* n) update time
(Disjoint Set Union Algorithm [Tarjan 1975])
• Decremental Algorithm:
– O(n) update time [Even and Shiloach, 1981]
– O(log n) update time [Thorup 1997]

Fully Dynamic Algorithms:
1. O() update time [Frederickson, 1982]
2. O() update time [Eppstein, Galil, Italiano, Nissenzweig 1991]
3. O(polylog n) expected update time, O(log n) query time
[King and Henzinger 1995]
4. O(polylog n) update time, O(log n) query time
[Holm, Litchenberg, Thorup 1998]
•

Outline of the talk
• Dynamic graph algorithms for some important problems
• Data structure for dynamic trees
• Fully dynamic connectivity with polylog n update time
• Open problems

Examples of fully dynamic
algorithms

Fully dynamic algorithms for undirected
graphs
1. Connectivity
2. 2-edge connectivity
3. Bi-connectivity
4. Bipartiteness
5. Min. spanning tree
O(polylog n) update time [Holm, Litchenberg, Thorup 1998]

Fully dynamic algorithms for undirected
graphs
• Min-cut
– Best static algorithm: O(m polylog n) (Randomized) [Karger, 1996]
– Fully dynamic algorithm: O() update time [Thorup, 2001]
• Graph spanner
Definition:
A subgraph which is sparse and yet preserves all-pairs distances approximately.
– Best static algorithm: O(m) [Halperin, Zwick, 1996]
– Fully dynamic algorithm: O(polylog n) update time
[Baswana, Khurana, and Sarkar, 2008]
•

Dynamic algorithms for directed graphs
Maintaining BFS tree under deletion of edges :
O(n) time per edge deletion [Even & Shiloach, 1981]
• Unbeaten till date.
• Used in many dynamic algorithms for directed graphs.
Not as good bounds as
undirected graphs

Dynamic algorithms for directed graphs
Transitive Closure
• Incremental algorithm: O(n) update time [Italiano, 1986]
• Decremental algorithm: O(n) update time
– Randomized [Roditty and Zwick, 2002]
– Deterministic [Lacki, 2011 ]
• Fully Dynamic algorithm: O() update time [Roditty, 2003]
All-pairs Shortest paths
• Fully dynamic algorithms:
– Amortized O() update time [Demetrescu and Italiano, 2003]
– Worst case O() update time [Thorup, 2005]
•

Data structure for dynamic trees

Data structures
• Stacks
• Queues
• Binary heap
• Binary search tree
...
• Fibinaacci heap
• Too elementary
• Limited applications
• Too complex
• Hardly any application !

Power of Data structures:
An inspirational example
Maintain n records r(1),…,r(n) under the following operations
• Add(i,j,x) : Add x to each record from r(i),…,r(j).
• All-swap(i,j) : r(i)↔r(j), r(i+1) ↔ r(j-1), r(i+2) ↔ r(j-2).
• Report(i) : report record r(i).
• Report-min(i,j) : report the smallest record from r(i),…,r(j).
Each operation in O(log n)
worst case time.

Balanced Binary Tree : a very powerful data structure
: Additional information

Dynamic Trees
ef
a
u
c
gj
d
b
w v u

Dynamic Trees
Aim : Maintain a forest of trees on n vertices under the following
operation.
• Link(u,v) : Add an edge between u and v
• Cut(u,v) : Delete an edge between u and v
• Update() : Update information associated with nodes/edges
• Query() :
– Topological
– information associated with a tree, or a path

Data Structures for Dynamic Trees
• ST Tree [Sleator & Tarjan, 1983]
Operations and queries on edges of paths
• ET tree [Henzinger and King, 1995]
Operations and queries on nodes of a tree
• Top tree [Alstrup et al., TALG 2005]
(generalization of Topology Tree [Frederickson, 1982])
Topological properties (diameter, center)

Dynamic Trees
query and updates on trees
Operations :
• Link(u,v)
• Cut(u,v)
• Update-weight-node(v,a):
weight(v)  a
• Add-weight-tree(v,x):
add x to weight of each node of tree of
v
• ReportMin(u):
report min weight in the entire tree
containing u
a
u
c
g
ef
j
d
b
w v u
5
12
-3
741
172
32
3
44 67 15

Dynamic Trees
ReportMin(v) = 12
Operations :
• Link(u,v)
• Cut(u,v)
weight(v)  a
v
• ReportMin(u):
containing u
a
u
c
g
ef
j
d
b
w v u
5
12
-3
741
172
32
3
44 67 15

Dynamic Trees
ReportMin(v) changes …
Operations :
• Link(u,v)
• Cut(u,v)
weight(v)  a
v
• ReportMin(u):
containing u
a
u
c
g
ef
j
d
b
w v u
5
12
-3
741
172
32
3
44 67 1

Euler tour tree :
A data structure for
dynamic trees a b ce
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 :
dynamic trees
: minimum value of all nodes
in the subtree.
a b ce
f
g
h d
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 :
f
g
h d
b-c-d-c-b-a- e-f-e-g-e-h-e -a-b

Euler tour tree :
f
g
h d

Euler tour tree :
f
g
h d
a-bb-c-d-c-b-a
e-f-e-g-e-h-e
T1 T2
T1

Euler tour tree :
f
g
h d
a-bb-c-d-c-b-a
T1 T2
T1
T1
e-f-e-g-e-h-e

Euler tour tree :
f
g
h d
a-bb-c-d-c-b-a
e-f-e-g-e-h-e
T1 T2
T1
T1

Euler tour tree :
dynamic trees
• Split(T,(e,a))
• Merge(T1,T2,(u,v))
• Change-origin(T,x) :
change the origin of Euler tour
to vertex x.
a b ce
f
g
h d
b-c-d-c-b-a-be-f-e-g-e-h-e
T1 T2
T1
T1
T2
T2

Fully dynamic algorithm for
connectivity

Fully dynamic randomized algorithm for connectivity with
polylogarithmic update time

Maintain ET tree for each tree in
the spanning forest

A Hierarchical algorithm
1
2
3
2 log
n

Fully dynamic randomized algorithm for
connectivity with O() update time
1
2
A 2-level algorithm

Decremental O() update time algorithm for
connectivity
Key tools in addition to ET tree data structure:
• Trivial algorithm (for handling deletion of a tree edge) :
Let (u,v) be a tree edge in the spanning forest.
Let its deletion creates trees T1 and T2.
Let µ be the number of non-tree edges incident on T2.
Replacement edge can be found in time O(µ log n) time.
• Random sampling

The role of random sampling
• Exercise: If none of 2k log n balls is blue, then with probability
, the fraction of blue balls is less than 1/k.
•
Uniform random sampling
with replacement

Handling the deletion of a tree edge
T2T1 T
T2
How to augment ET-tree
to sample an edge ?
Few samplings
needed if the
fraction of blue
edges is large
What if fraction of
blue edges is small ?

2-Level approach
• A partition of E into two levels : (E1, E2)
In the beginning, E1 = E and E1 = Ø
• F1 : spanning forest of E1
• F2 : spanning forest of E, F1 is subset of F2
Level 1
Level 2

Level 1
Level 2

Level 1
Level 2
Trivial algorithm at level 2
Random sampling at level 1

Algorithm for handling deletion of a tree
edge
If (e ϵ F2 F1) scan non-tree edges at level 2 to find replacement edge.
Else Let T be the tree to which e belongs;
(T1,T2)  Split(T,e);
Repeat k log n times
{ (u,v)  Sample-edge(T2);
If (u,v) is a cut-edge
{ add (u,v) to F1;
Merge(T1,T2, (u,v)); return;
}
}
Scan all non-tree edges incident on T2;
If less than 1/k fraction are cut-edges
move all edges of cut(T1,T2) to Level 2 and add one of them to F2.
Else add an edge of cut(T1,T2) to F1
O(µ2) time
O(k logn) time
O(µ1(T2)) time
probability

Bounding µ2 (number of non-tree edges at level 2)
Upon splitting T into T1 and T2, how many edges are passed to level 2 ?
≤
charge to each Tϵ 2
•
Level 1
Level 2
T2T1

Analysis
• A vertex v is processed only O(log n) times.
Whenever v is processed
The processing cost of v is O(deg(v))
The number of edges that move to level 2 is less than
Hence µ2 , the number of edges at level 2 is O(log n)
• Processing cost per update :
log n + k + O(m)
= O()
•

Transforming to fully dynamic environment
• Add every newly inserted edge to level 2.
• Periodically rebuild the data structure after every ()
insertions.
Expected amortized time per update : O()
•

• A partition of E into 2log n levels : (E1, E2, …)
In the beginning, E1 = E and Ei = Ø for all i>1
• F1 : spanning forest of E1
• Fi : spanning forest of
• Fi-1 is subset of Fi for all i>1
•

A Hierarchical algorithm
1
2
2 log
n
c
2

Open problems
• Amortized cost versus worst case bounds
• Specific problems : Min-cut, s-t min cut, max-flow, …
• Specific graph family : Planar graphs
• Better lower bounds ?

5.3 dyn algo-i

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5.3 dyn algo-i