1. all-pairs shortest paths in undirected graphs
Dynamic
approximate
all-pairs shortest paths in undirected graphs
Liam Roditty Uri Zwick Tel Aviv University

2. Dynamic Graph Problems
A graph
Initialize
Insert
Delete
Query
n - number of vertices m - number of edges

3. Shortest Paths Problems
Single-Source Shortest Paths (SSSP)
All-Pairs Shortest Paths (APSP)
“Distance Oracles”

4. Approximate Distance Oracles (TZ’01)
n by n distance matrix
APSP algorithm
Compact data structure
mn1/k time n1+1/k space
O(1) query time stretch 2k-1
Stretch-Space tradeoff is essentially optimal!

5. Dynamic Shortest Paths Algorithms
Problem
Total update time
Amortized* update time
Query time
Authors
Decremental SSSP
O(mn)
O(n)
O(1)
Even-Shiloach ’81
Decremental (1+)-approximate APSP
O*(mn)
O*(n)
O(1)
This talk
Problem
Amortized update time
Query time
Authors
Fully-dynamic APSP
O*(n2)
O(1)
Demetrescu-Italiano ’03 (Thorup ’04)
Fully-dynamic (1+)- approximate APSP
O*(m1/2n)
O(m1/2)
This talk

6. Decremental Approximate Distance Oracles
Compact data structure
mn1/k time n1+1/k space
O(1) query time stretch 2k-1
Static
mn time m+n1+1/k space
Decremental
Decremental (up to distance d)
dmn1/k time m+n1+1/k space

7. Thorup-Zwick ’01 Static approximate distance oracles.
Three ingredients
Even-Shiloach ’81 A decremental algorithm for maintaining a single-source shortest path tree of depth d with a total running time of O(dm).
Random Sampling Let P be a set of vertices of size k. Let S be a random subset of vertices of size (cn ln n)/k. Then with high probability PS  .
Thorup-Zwick ’01 Static approximate distance oracles.

8. Decremental SSSP [Even-Shiloach ’81]
Every edge is only examined once per level!
Total complexity is O(dm).

9. Random Sampling n k Select each element independently with probability
The probability that a given set of k elements is not hit is

10. Let pi(v) be the vertex of Si closest to v
A sequence of Samples
Let Si be a subset obtained by selecting each vertex with probability (c ln n)/(2i)
Let pi(v) be the vertex of Si closest to v
If v is contained in a connected component with at least 2i vertices, then:
δ(v,pi(v)) ≤ 2i
2i
pi(v)

11. Maintaining the closest centers
To compute pi(v) for every v:
Add a new source vertex si.
Connect it to all vertices of Si.
Maintain a SSSP tree from si.

12. Maintaining distances from centers
From each center of Si maintain a SSSP tree of depth 2i+2
Complexity:

13. Answering a distance query
pi(u) u v
Suppose:
Return: ≤ ≤

14. Answering a distance query
pi(u) u v
Return:
where i is smallest such that v is in the tree of pi(u), i.e.,
This value of i can be found using binary search.
Query time: O(log log n)
Stretch: 

15. Reporting long distances in O(1) time
pr(u)
pr(v) u v
Suppose:
Let:
Return:
Maintain a table of all Sr to Sr distances.
Size of table is O(n).

16. Reporting short distances in O(1) time
Suppose:
Develop a decremental version of the static approximate distance oracles of TZ.
dmn1/k time m+n1+1/k space stretch 2k-1
Take d=n1/2 k=2
For distances up to d
The 3-approximation can be refined into a (1+)-approximation in O(1) time.

17. Approximate Distance Oracles [TZ’01] A hierarchy of centers
A0V ; Ak  ; Ai sample(Ai-1,n-1/k) ;

18. A0= A1= A2=
Clusters w

19. Bunches (inverse clusters)

20. Bunches
A0= A1= A2=
p2(v) v
p1(v)

21. Approximate Distance Oracles [TZ’01] The data structure
For every vertex vV:
The centers p1(v), p2(v),…, pk-1(v)
A hash table holding B(v)
For every wV, we can check, in constant time, whether wB(v), and if so, what is (v,w).

22. Lemma: E[|B(v)|] ≤ kn1/k
Proof: |B(v)Ai| is stochastically dominated by a geometric random variable with parameter p=n-1/k.

23. Query answering algorithm
Algorithm distk(u,v)
wu , i0
while wB(v)
{ i i+1
(u,v) (v,u)
w pi(u) }
return (w,u)+ (w,v)

24. Query answering algorithm
w3=p3(v)A3
w2=p2(u)A2
w1=p1(v)A1 u v

25. Analysis
wi=pi(u)Ai
wi-1=pi-1(v)Ai-1 i
(i+1) i
(i-1) u v 

26. Open problems Faster dynamic SSSP algorithm?
Exact dynamic APSP algorithm?
Weighted graphs?
Directed graphs?