1. Answering distance queries in directed graphs using fast matrix multiplication
Raphael Yuster
Haifa University
Uri Zwick
Tel Aviv University

2. Answering distance queries
Generalizes both SSSP and APSP
Preprocessing:
data structure
u,v
δ(u,v)
Query answering:

3. New result: Answering distance queries
Directed graphs. Edge weights in {−M,…,0,…M}
Preprocessing time
Query time
Authors
Mn2.38 n
[Yuster-Zwick ’05]
In particular, any Mn1.38 distances can be computed in Mn2.38 time.
For dense enough graphs with small enough edge weights, this improves on Goldberg’s SSSP algorithm. Mn vs. mn0.5log M

4. Single-source Shortest Paths in directed graphs with real edge weights
Non-negative edge weights
Running time
Authors
m + n log n
[Dijkstra ’59] [Fredman-Tarjan ’86]
Positive and negative edge weights
Running time
Authors mn
[Bellman ’58] [Ford ’58]

5. Single-source Shortest Paths in directed graphs with integer edge weights
Positive and negative edge weights greater than –N
Running time
Authors
mn1/2log(nN)
[Gabow-Tarjan ’89]
mn1/2logN
[Goldberg ’95]
For dense graphs, the running time is O(n2.5) !!!

6. All-Pairs Shortest Paths in directed graphs with real edge weights
Running time
Authors
mn + n2 log n
[Dijkstra ’59], [Johnson ’77] [Fredman-Tarjan ’86]
mn + n2 log log n
([Thorup ’99] [Hagerup ’00]) [Pettie ’02]
For dense graphs, the running time is O(n3) !!!

7. All-Pairs Shortest Paths in graphs with small integer weights
Undirected graphs. Edge weights in {0,1,…M}
Running time
Authors
Mn2.38
[Shoshan-Zwick ’99]
Improves results of [Alon-Galil-Margalit ’91] [Seidel ’95]

8. All-Pairs Shortest Paths in graphs with small integer weights
Directed graphs. Edge weights in {−M,…,0,…M}
Running time
Authors
M0.68 n2.58
[Zwick ’98]
Improves results of [Alon-Galil-Margalit ’91] [Takaoka ’98]

9. New result: Answering distance queries
Directed graphs. Edge weights in {−M,…,0,…M}
Preprocessing time
Query time
Authors
Mn2.38 n
[Yuster-Zwick ’05]
In particular, any Mn1.38 distances can be computed in Mn2.38 time.
For dense enough graphs with small enough edge weights, this improves on Goldberg’s SSSP algorithm. Mn vs. mn0.5log M

10. An interesting special case of the APSP problem20 17 30 2 23 10 5 20
Min-Plus product

11. Solving the APSP problem by repeated squaring
If W is an n by n matrix containing the edge weights of a graph. Then Wn is the distance matrix.
D  W
for i 1 to log2n do D  D*D
Thus: APSP(n)  MPP(n) log n
Actually: APSP(n) = O(MPP(n))

12. O(n2.38) [Strassen ’69] … [Coppersmith- Winograd ’90]
Min-Plus Product
Algebraic Product
O(n2.38) [Strassen ’69] … [Coppersmith- Winograd ’90]
The fast algebraic algorithms cannot be used, as the min operation has no inverse

13. Using matrix multiplication to compute min-plus products

14. Using matrix multiplication to compute min-plus products
Assume: 0 ≤ aij , bij ≤ M
M operations per polynomial product
Mn2.38 operations per max-plus product
n2.38 polynomial products  =

15. Trying to implement the repeated squaring algorithm
Consider an easy case: all weights are 1.  *
Iteration
Max size of elements
Cost 1
n2.38 2
2·n2.38 i 2i
2i·n2.38
log n n
n·n2.38

16. Sampled Repeated Squaring (Z ’98)
D  W
for i 1 to log3/2n do {
s  (3/2)i+1
B  rand( V , (9n ln n)/s )
D  min{ D , D[V,B]*D[B,V] } }
Choose a subset of V of size (9n ln n)/s
Select the columns of D whose indices are in B
Select the rows of D whose indices are in B
With high probability, all distances are correct!
The is also a slightly more complicated deterministic algorithm

17. Sampled Distance Products (Z ’98)
In the i-th iteration, the set B is of size n ln n / s, where s = (3/2)i n
The matrices get smaller and smaller but the elements get larger and larger n
|B|

18. Sampled Repeated Squaring - Correctness
Invariant: After the i-th iteration, distances that are attained using at most (3/2)i edges are correct.
D  W
for i 1 to log3/2n do {
s  (3/2)i+1
B  rand(V,(9 ln n)/s)
D  min{ D , D[V,B]*D[B,V] } }
Consider a shortest path that uses at most (3/2)i+1 edges
at most
Failure probability :
Let s = (3/2)i+1

19. Rectangular Matrix multiplication = n p
Naïve complexity: n2p
[Coppersmith ’97]: n1.85p0.54+n2+o(1)
For p ≤ n0.29, complexity = n2+o(1) !!!

20. Complexity of APSP algorithm
The i-th iteration:  n
n ln n / s
s=(3/2)i+1
The elements are of absolute value at most Ms

21. The new preprocessing algorithm
D  W ; B V
for i 1 to log3/2n do {
s  (3/2)i+1
B  rand(B,(9n ln n)/s)
D[V,B]  min{D[V,B] , D[V,B]*D[B,B] }
D[B,V]  min{D[B,V] , D[B,B]*D[B,V] } }

22. The old algorithm D  W for i 1 to log3/2n do { s  (3/2)i+1
B  rand(V,(9nln n)/s) }
D  min{ D , D[V,B]*D[B,V] }

23. Twice Sampled Distance Products
|B|
|B| n

24. The query answering algorithm
δ(u,v)  D[{u},V]*D[V,{v}] u v
Query time: O(n)

25. The preprocessing algorithm: Correctness
Let Bi be the i-th sample. B1 B2  B3  …
Invariant: After the i-th iteration, if u Bi or vBi and there is a shortest path from u to v that uses at most (3/2)i edges, then D(u,v)=δ(u,v).
Consider a shortest path that uses at most (3/2)i+1 edges
at most

26. The query answering algorithm: Correctness
Suppose that the shortest path from u to v uses between (3/2)i and (3/2)i+1 edges
at most u v

27. Open problems
An O(n3-ε) algorithm for the APSP problem with edge weights in {1,2,…,n}?
More generally, an O(n3-ε(log M)c) algorithm for the APSP problem?
An O(n2.5-ε) algorithm for the SSSP problem with edge weights in {0,±1, ±2,…, ±n}?
More generally, an O(n2.5-ε(log M)c) algorithm for the SSSP problem?