PRESENTATION
Liang Zhuo
Nondecreasing Paths
■ given an edge weighted graph 𝐺 = (𝑉, 𝐸, 𝑤) and a source 𝑠 ∈
𝑉 , report for every 𝑡 ∈ 𝑉 , the minimum last edge weight on a
nondecreasing path from 𝑠 to 𝑡.
■ single source minimum nondecreasing paths problem (SSNP)
■ planning a train
■ tripall pairs nondecreasing paths problem (APNP)
■ Related Problems: Shortest Paths, Bottleneck Paths
Previous Results
SSNP
𝑂(𝑚 loglog 𝑛)
Vassilevska, 08
APBP
𝑂(𝑛(3+𝜔)/2 )
Duan & Pettie, 09
APNP
𝑂(𝑛(15+𝜔)/6 )
Vassilevska, 08
could be improved
SSNP
■ Easy 𝑂(𝑛 + 𝑚 log 𝑛) by Dijikstra’s
■ To achieve 𝑂(𝑚 loglog 𝑛), combine 2 algorithms:
– Alg. 1: remember d[.]; state (v, last); DFS with initial state
(s, -inf); for one node, from small to large edges. <- need
order for each node.
– Alg. 2: Dijkstra’s. <- need to decrease heap size
■ Divide nodes to 2 classes: low deg. & high deg. (>O(log 𝑛))
– Low: use alg. 1. 𝑂 𝑚 ⋅ 𝑂(log log 𝑛) (if using Q-Heaps:
𝑂 𝑚 ⋅ 𝑂(1))
– High: use alg. 2. 𝑂
𝑚
log 𝑛
⋅ 𝑂(log 𝑛)
APNP
■ Use matrix.
■ Define (min, ≤) product C = 𝐴 ∘ 𝐵 where 𝐶 𝑖, 𝑗 =
min{𝐵 𝑘, 𝑗 : 𝐴 𝑖, 𝑘 ≤ 𝐵[𝑘, 𝑗]}
𝑘
■ We just need to do (min, ≤) product.
■ Borrow dominance and max-min product in APBP
– Dominance product C = 𝐴 ⊙ 𝐵 where 𝐶 𝑖, 𝑗 =
| 𝑘: 𝐴 𝑖, 𝑘 ≤ 𝐵 𝑘, 𝑗 |
– Max-min product C = 𝐴 ⋄ 𝐵 where 𝐶 𝑖, 𝑗 =
max min{𝐴 𝑖, 𝑘 , 𝐵[𝑘, 𝑗]}
𝑘
Dominance Product
■ The dominance product of two 𝑛 × 𝑛 matrices 𝐴 and 𝐵 with
entries from a totally ordered set is computable in
𝑂(𝑛(3+𝜔)/2 ) time. (Matousek)
■ Let 𝐴 and 𝐵 be two 𝑛 × 𝑛 matrices where the number of non(∞) values in 𝐴 is 𝑚1 and the number of non-(−∞) values in 𝐵
𝑚 𝑚
is 𝑚2 . Then 𝐴 ⊙ 𝐵 can be computed in time 𝑂( 1 2 + 𝑛𝜔 ) .
𝑛
(Duan & Pettie)
■ Let 𝐴 and 𝐵 be two 𝑛 × 𝑛 matrices where the number of non(∞) values in 𝐴 is 𝑚1 and the number of non-(−∞) values in 𝐵
is 𝑚2 . 𝑚1 𝑚2 ≥ 𝑛1+𝜔 . Then 𝐴 ⊙ 𝐵 can be computed in time
𝑂( 𝑚1 𝑚2 𝑛(𝜔−1)/2 ) . (Duan & Pettie)
Max-min Product
■ Given two real 𝑛 × 𝑛 matrices 𝐴 and 𝐵, 𝐴 ⋄ 𝐵 can be computed in
𝑂(𝑛(3+𝜔)/2 ). (Duan & Pettie)
■ How it is proved?
■ It computes 𝐶 𝑖, 𝑗 = max{𝐴 𝑖, 𝑘 : 𝐴 𝑖, 𝑘 ≤ 𝐵[𝑘, 𝑗]} and 𝐶 ′ 𝑖,𝑗 =
𝑘
max 𝐵 𝑘, 𝑗 : 𝐴 𝑖, 𝑘 ≥ 𝐵[𝑘, 𝑗]}
𝑘
■ It let 𝐴 ⋄ 𝐵 𝑖, 𝑗 = max 𝐶 𝑖, 𝑗 , 𝐶 ′ 𝑖, 𝑗
■ 𝐶 𝑖, 𝑗 = max{𝐴 𝑖, 𝑘 : 𝐴 𝑖, 𝑘 ≤ 𝐵[𝑘, 𝑗]} could also be computed in
𝑘
𝑂(𝑛(3+𝜔)/2 ).
■ 𝐶 𝑖, 𝑗 = min{−𝐴 𝑖, 𝑘 : −𝐵[𝑘, 𝑗] ≤ −𝐴 𝑖, 𝑘 }. Thus (min, ≤)
𝑘
product in 𝑂(𝑛(3+𝜔)/2 ).
Sampling
■ Given a collection of 𝑁 subsets of {1, . . . , 𝑛}, each of size 𝑙,
one can find in 𝑂(𝑁𝑙) time a set of 𝑛 log 𝑛/𝑙 elements of
{1, . . . , 𝑛} hitting every one of the subsets. (Galil & Margalit)
Algorithm
■ Short Paths, Long Paths
■ Alg 1: length ≤ 𝑙. (min, ≤) product for 𝑙 times: 𝐴𝑙
■ Alg 2: length > 𝑙. Get a set of size 𝑛 log 𝑛/𝑙 hitting all paths of
length 𝑙. Use the alg. for SSNP for each node 𝑣 in the set, and
run it reversely to get the APSP through 𝑣.
■ Running time:
– 𝑂(𝑛(3+𝜔)/2 𝑙)
𝑛 log 𝑛
𝑛2 log 𝑛)
𝑙
𝑂(𝑛(3−𝜔)/4 ), running time:
– 𝑂(𝑛2 𝑙 +
■ Let 𝑙 =
𝑂(𝑛(9+𝜔)/4 ).
Discussion
■ Multiple edge? Increase #node => complexity increases.
■ Short Paths, Long Paths technique makes the complexity
greater than APBP. 𝑂(𝑛1.5 𝑀(𝑛)),where 𝑀(𝑛) is the
complexity of matrix multiplication.
■ Dense Dominance Product could be done better?
( 𝑚1 𝑚2 𝑛(𝜔−1)/2 ) is worse when, say, 𝑚1 , 𝑚2 = 𝑂(𝑛1.5 ).
■ Other way to solve this problem?
Discussion
■ Sort all edges, and divide into 𝑃 parts, each contains
𝑚
𝑃
edges
■ For each part generate a bool matrix, then multiply them:
𝑂(𝑛𝜔 𝑃)
■ Inside each part, select 𝑡 key edges. Paths between key
𝑚
edges/point and other edges: 𝑂(𝑡 ). APNP through key
𝑃
2
edges/point: 𝑂(𝑛 𝑡).
■ We need to compute APNP that do not pass through our key
edges: ?.
Reference
■
Vassilevska, Virginia, Ryan Williams, and Raphael Yuster. "All-pairs
bottleneck paths for general graphs in truly sub-cubic time." Proceedings
of the thirty-ninth annual ACM symposium on Theory of computing. ACM,
2007.
■
Vassilevska, Virginia. "Nondecreasing paths in a weighted graph or: How
to optimally read a train schedule." Proceedings of the nineteenth
annual ACM-SIAM symposium on Discrete algorithms. Society for
Industrial and Applied Mathematics, 2008.
■
Duan, Ran, and Seth Pettie. "Fast algorithms for (max, min)-matrix
multiplication and bottleneck shortest paths." Proceedings of the
twentieth Annual ACM-SIAM Symposium on Discrete Algorithms. Society
for Industrial and Applied Mathematics, 2009.
■
Kowaluk, Miroslaw, and Andrzej Lingas. "LCA queries in directed acyclic
graphs." Automata, Languages and Programming. Springer Berlin
Heidelberg, 2005. 241-248.
Thank you!