1. Dynamic Graph Algorithms
III
Surender Baswana
Visiting Professor till 31st May 2020
Heinz Nixdorf Institute & Department of Computer Science (Algorithms and Complexity Group)

2. Dynamic DFS tree problem

3. Revisiting DFS traversal
Part I
Revisiting DFS traversal

4. DFS traversal A recursive way to traverse the graph
DFS(v)
{ Visited(v) ← true;
For each neighbor w of v {
{ DFS(w) ; }
DFS-traversal(G)
{ For each vertex vϵ V
Visited(v)← false;
For each vertex v ϵ V
If (Visited(v ) = false) DFS(v); v z y c d b f g h u w r s
if (Visited(w) = false)
……..;
……..;
A DFS tree rooted at v

5. DFS traversal Applications: Connected components of a graph.
a milestone in the area of graph algorithms
Applications:
Connected components of a graph.
Finding articulation points in a graph.
Finding bridges in a graph.
Planarity testing of a graph
Strongly connected components of a directed graph.
O(𝑚+𝑛) time

6. How will an edge appear in DFS traversal ?

7. How will an edge appear in DFS traversal ?
It can never happen u x y

8. DFS traversal from a non-root vertex?y f b h v w d g x y z s c

9. Flexibility versus restrictionh v w d g x y z s c

10. Problem: Dynamic DFS tree
Part II
Problem: Dynamic DFS tree

11. Dynamic DFS tree
Problem: Preprocess a given graph 𝐺=(𝑉,𝐸) to build a data structure s.t. for any online sequence of edge insertion/deletion a DFS tree can be maintained efficiently.

12. Partially dynamic algorithms
Directed Acyclic Graph (DAG):
Undirected Graphs:
Problem
Total Update time
Reference
Incremental
𝑂(𝑚𝑛)
Franciosa et al. (ESA 1996)
Decremental
𝑂 𝑚𝑛 log 𝑛 (expected)
B and Choudhary (MFCS 2015)
Problem
Total Update time
Reference
Incremental
𝑂(𝑚𝑛)
Franciosa et al. (1997)
Emphasize on the slide Notice the following: Firstly the only known results are partially dynamic
Problem
Total Update time
Reference
Incremental
𝑂( 𝑛 2 )
B and Khan (ICALP 2014)

13. Fully dynamic algorithms
Undirected Graphs:
Problem
Time per Update
Reference
Fully Dynamic*
𝑶( 𝑚𝑛 log 2.5 𝑛 )
B, Chaudhury, Choudhary, Khan
(SODA 2016)
Simple enough to be taught at Undergrad course on algorithms.
Recent, unpublished, joint work with
Shiv Gupta and Ayush Tulsyan at IITK  
𝑶( 𝑚𝑛 log 𝑛 )
* Supports both edge and vertex updates.

14. Familiarizing with the dynamic DFS tree problem
Part III
Familiarizing with the dynamic DFS tree problem

15. Insertion of an edge u y x
Reroot this subtree at y

16. Deletion of an edge x y v
Reroot this subtree at v

17. A novel decomposition of a tree
Part IV
A novel decomposition of a tree

18. Heavy path decomposition of tree

19. Heavy path decomposition of tree

20. Heavy path decomposition of tree

21. Heavy path decomposition of tree

22. Heavy path decomposition of tree

23. Shallow Tree 𝑝 1 𝑝 2 𝑝 3 𝑝 4 𝑝 6 𝑝 5 A tree of arbitrary height
Heavy path decomposition
Shallow Tree
𝑝 1
𝑝 2
𝑝 3
𝑝 4
𝑝 6
𝑝 5

24. Algorithm for Rerooting a DFS tree
Part V
Algorithm for Rerooting a DFS tree

25. 𝑝 T

26. 𝑝

27. Exploit
Flexibility of DFS 𝑝

28. How to ensure fewer edges in a reduced adjacency list ?
Time to respect
the Restriction of DFS
Idea: 𝑣
Compute a reduced adjacency list
for each vertex 𝑣 in a lazy manner 𝑝

29. More insight into DFS traversal
Part VI
More insight into DFS traversal

30. u
Observation 1
Redundant v w x
Connected components

31. Observation 2
Redundant v
Redundant u w x
Connected components

32. Resuming the algorithm for Rerooting a DFS tree
Part VII
Resuming the algorithm for Rerooting a DFS tree

33. For each 𝑢 on 𝒑,
For each ancestor path of 𝒑
Add only one edge to Adj(𝑢)
No. of edges added to Adj(𝑢):
O(log 𝑛)
Observation 2 𝑣 𝑝

34. But before that, can you figure out the entire algorithm now ?
For each descendant 𝑥 of 𝒑,
edge incident on 𝒑
Add (𝑥,𝑤) to Adj(𝑤)
No. of edges added by 𝑥 to its ancestors ?
(𝑥,𝑤) 
nearest to 𝑣.
Seems many 
Observation 1
Data Structure for each vertex 𝑥 :
But before that, can you figure out the entire algorithm now ?
Edges to ancestors in sorted order 𝑣 𝑤 𝑝 𝑥

35. Begin with the heavy path decomposition of the original DFS tree.
Traverse the longer portion of the path Populate reduced adj. list of traversed path Do DFS starting from most recently visited vertex You might enter another path Do DFS traversal from the most recently visited vertex Do DFS traversal from the most … 
𝑝 1
(say 𝑝 2 )
𝑝 7
Begin with the heavy path decomposition of the original DFS tree.
𝑝 6
(say 𝑝 7 )
𝑝 1
𝑝 3
𝑝 2
New root
𝑝 5
𝑝 4

36. Algorithm can be seen as a walk on the shallow tree
𝑝 7
𝑝 6
𝑝 1
𝑝 2
𝑝 3
𝑝 4
𝑝 5

37. For each descendant 𝑥 of 𝒑,
edge incident on 𝒑
Add (𝑥,𝑤) to Adj(𝑤)
No. of edges added by 𝑥 to its ancestors ?
(𝑥,𝑤) 
nearest to 𝑣.
Seems many 
O( log 2 𝑛)
Observation 1 𝑣 𝑤 𝑝 𝑥

38. Theorem: A DFS tree 𝑇 of a given graph 𝐺=(𝑉,𝐸) can be preprocessed in O(𝑚) time to build a data structure of O(𝑚) size s.t. for any vertex 𝑣∈𝑉, a DFS tree rooted at 𝑣 can be reported in O(𝑛 log 2 𝑛) time. Thefault tolerant DFS tree problem and fully dynamic DFS tree problem can be solved by “just epsilon” modification to the above algorithm for rerooting . The entire paper is available at Arxiv: Surender Baswana, Shiv Kumar Gupta, Ayush Tulsyan: Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient. CoRR abs/ (2018)