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

2. Fully dynamic connectivity
Update time
Query time
Reference
O( 𝒎 )
O(𝟏)
Frederickson, SICOMP 83
O( 𝒏 )
O(𝟏)
Eppstein et al. J. ACM 91
expected
amortized
randomized
O( log 𝟑 𝒏)
O(log 𝒏)
Henzinger and King, J.ACM 99
Prerequisites ?
Elementary knowledge of probability

3. Deterministic Algorithm
Input
Output
The output as well as the running time are functions only of the input.
Algorithm

4. Deterministic Algorithm
Randomized
Random bits
Input
Output
The output or the running time are functions of the input and random bits chosen .
Algorithm

5. Organization of the Talk
𝟏. Random sampling 𝟐. Experiencing the difficulty of the problem 𝟑. A simple but inefficient algorithm 𝟒. Designing the efficient algorithm

6. Part I
Random Sampling
A gentle introduction

7. Elementary Probability
Tossing a coin
𝟏 𝟐 𝒌
𝐏[ No HEAD appears in 𝒌 tosses] = ?

8. Problem: Given an array A[1…𝑛] storing 0/1s. Report any index storing 𝟏. Theorem: For every deterministic algorithm, there exists an input for which 𝛀(𝑛) entries will be scanned to find the first 𝟏.
At least ½ of the entries in A are 𝟏. A 𝟎 𝟎 𝟏 𝟎 𝟎

9. Random Sampling
Problem: Given an array A[1…𝑛] storing 0/1s. Report any index storing 𝟏.
At least ½ of the entries in A are 𝟏.
1 𝑘 A 𝟎 𝟏
Randomized Algorithm
count  𝟎;
Repeat
count  count +𝟏;
𝒊 a random number from [𝟏..𝒏];
Until ((A[𝒊] ≠𝟏) and (count < 𝒄 log 𝒏)) 𝒌
𝟏 𝟐
𝒄log 𝒏
𝟏 𝒏 𝒄
Probability of failure :

10. Experiencing the difficulty of the problem
Part II
Experiencing the difficulty of the problem

11. Fully dynamic connectivity

12. Fully dynamic connectivity
Update time
Query time
Reference
O( 𝒎 )
O(𝟏)
Frederickson, SICOMP 83
O( 𝒏 )
O(𝟏)
Eppstein et al. J. ACM 91
expected
amortized
randomized
O( log 𝟑 𝒏)
O(log 𝒏)
Henzinger and King, J.ACM 99 1 2 1 2 3
2 log 𝒏
O( 𝒎 log 𝟐 𝒏)
O( 𝒌 𝒎 log 𝟐 𝒏)
𝒌 levels

13. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time

14. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
The 2-level algorithm 2 1

15. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
Idea:
Maintain a spanning forest

16. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
Decremental
What is the real challenge ?

17. A simple but inefficient algorithm
Part III
A simple but inefficient algorithm

18. A simple decremental algorithm
Question: Can we find a replacement edge in 𝑶 (| 𝝁(𝑻) |) time ?
Operations
Time
Enumerating all non-tree edges incident on a tree 𝑻
O(𝝁(𝑻)log 𝒏)
Checking if two vertices belong to the same tree
O(log 𝒏)
𝝁(𝑻) : non tree edges 𝑻

19. A simple decremental algorithm
Question: Can we find a replacement edge in 𝑶 (| 𝝁(𝑻) |) time ?
Euler Tour tree
𝝁(𝑻) : non tree edges 𝑻
Homework:
Picking a non-tree edge randomly uniformly
in O(log 𝒏) time ?

20. The efficient algorithm
Part IV
The efficient algorithm

21. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
Decremental
A combination of two simple tools
A simple 𝑶 (| 𝝁(𝑻) |) time algorithm
Random sampling Exercise

22. 1 |cut( 𝑻 𝟏 , 𝑻 𝟐 )| |𝝁( 𝑻 𝟏 )| ≥ 𝟏 𝒌 < 𝟏 𝑘 sparse dense
Data structure
for 𝑻 𝟏
≥ 𝟏 𝒌
< 𝟏 𝑘
sparse
dense 1
Pick non tree edge incident on 𝑻 𝟏 𝑘
log 𝑛
uniformly randomly
𝑻 𝟏
𝑻 𝟐 𝑻
cut( 𝑻 𝟏 , 𝑻 𝟐 )

23. Trivial algorithm at Level 𝟐 What if the cut is sparse ?
|𝝁( 𝑻 𝟏 )|
< 𝟏 𝒌
sparse
Trivial algorithm at Level 𝟐
What if the cut is sparse ?
Level 𝟐
Level 𝟏
𝑻 𝟏
𝑻 𝟐
Random sampling at level 𝟏

24. Handling the deletion of a tree edge
2-Level approach
A partition of 𝑬 into two levels : ( 𝑬 𝟏 , 𝑬 𝟐 )
In the beginning,
𝑭 𝟏 : spanning forest of 𝑬 𝟏
𝑭 𝟐 : spanning forest of 𝑬,
𝑬 𝟏 =𝑬
and 𝑬 𝟐 = Ø
𝑭 𝟏 ⊆ 𝑭 𝟐

25. Algorithm for Deletion of tree edge (𝒖,𝒗)
yes
(𝒖,𝒗) ϵ 𝑭 𝟐 \ 𝑭 𝟏 ?
scan 𝝁 𝟐
Level 𝟐 No
𝑻 𝟏 : smaller tree
Sample(𝝁( 𝑻 𝟏 )) 𝒆
yes
𝒆 ∈ cut( 𝑻 𝟏 , 𝑻 𝟐 ) ?
Join 𝑻 𝟏 and 𝑻 𝟐 at Level 1 No
yes
Level 𝟏
counter++
<𝒄𝒌 log 𝒏 ? No
Compute cut( 𝑻 𝟏 , 𝑻 𝟐 ) No
Probability ≤𝒏 −𝒄
Is cut( 𝑻 𝟏 , 𝑻 𝟐 ) sparse ?
yes
Move cut( 𝑻 𝟏 , 𝑻 𝟐 ) to Level 2
Add an edge from cut( 𝑻 𝟏 , 𝑻 𝟐 ) to 𝑭 𝟐

26. Analysis ≤ |𝝁( 𝑻 𝟏 )| 𝒌 charge 𝒅𝒆𝒈⁡(𝒗) 𝒌 edges to each 𝒗 ∈𝑻 𝟏
Upon splitting 𝑻 into 𝑻 𝟏 and 𝑻 𝟐 , how many edges are passed to level 2 ?
≤ |𝝁( 𝑻 𝟏 )| 𝒌
charge 𝒅𝒆𝒈⁡(𝒗) 𝒌 edges to each 𝒗 ∈𝑻 𝟏
| 𝝁 𝟐 | = O( 𝑚 𝑘 log 𝑛)
Level 𝟐
Level 𝟏 𝒗
𝑻 𝟏
𝑻 𝟐

27. Analysis
Theorem: Decrement connectivity can be maintained in expected amortized 𝐎( 𝒎 log 2 𝒏 ) time per edge deletion.
Cases
Time taken
(𝒖,𝒗) ϵ 𝑭 𝟐 \ 𝑭 𝟏
O( 𝒎 𝒌 log 2 𝒏)
(𝒖,𝒗) ϵ 𝑭 𝟏
O(𝒌 log 2 𝒏) +
O(𝝁( 𝑇 1 ) log 𝒏)
×𝒏 −𝒄

28. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
Add every newly inserted edge to level 𝟐.
Periodically rebuild the data structure after every 𝒎 insertions.
Expected amortized time per update : O( 𝒎 log 𝒏)
Theorem:
Fully dynamic connectivity can be maintained
in expected amortized 𝐎( 𝒎 log 𝒏 ) time per update.

29. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
The 2-level algorithm
Trivial algorithm at Level 𝟐 𝟐 𝟏
Random sampling at level 𝟏
Reduce the sampling rate

30. Fully dynamic connectivity with O( 𝒎 lo g 𝟐 𝒏) update time
The 3-level algorithm
Trivial algorithm at Level 𝟑 𝟑
Trivial algorithm at Level 𝟐
Random sampling at level 𝟐 𝟐 𝟏
Random sampling at level 𝟏
Homework
𝒌 levels
O( 𝒌 𝒎 log 𝟐 𝒏)

31. Minimum spanning tree Single source shortest paths tree
An open problem
Minimum spanning tree Single source shortest paths tree
A tree that minimizes the “average” stretch for all pairs.

32. An open problem
𝑑𝑖𝑠 𝑡 𝑇 (𝑢,𝑣) = distance between 𝑢 and 𝑣 in tree 𝑇. 𝑆𝑡𝑟𝑒𝑡𝑐 ℎ 𝑇 (𝑢,𝑣) = A𝑣𝑒𝑟𝑎𝑔𝑒𝑆𝑡𝑟𝑒𝑡𝑐ℎ(𝑇) = There exists an O(𝑚 polylog 𝑛) time algorithm to compute a spanning tree with average stretch (polylog 𝑛). Can we maintain such a tree efficiently dynamically ?
𝑑𝑖𝑠 𝑡 𝑇 𝑢,𝑣 𝑑𝑖𝑠 𝑡 𝐺 𝑢,𝑣
1 |𝐸| 𝑢,𝑣 ∈𝐸 𝑆𝑡𝑟𝑒𝑡𝑐 ℎ 𝑇 (𝑢,𝑣)