Challenges in Distributed Shortest Paths Algorithms Danupon - PowerPoint PPT Presentation

Challenges in Distributed Shortest Paths Algorithms Danupon Nanongkai KTH Royal Institute of Technology, Sweden 1 SIROCCO 2016

About this talk Main focus s-t Distance Known (1+ e )-approx. in Q (n 1/2 +D) time Open problem Technical challenge 1. Exact O(n 1- e +D) time Avoid bounded-hop distances! 2. Directed O(n 1/2 +D) time Avoid sparse spanner, etc.! 2

Note polylog terms will be hidden most of the time 3

Plan 1. Problem & Known Results 2. Open Problems 3. Technical Challenges 4

Part 1.1 CONGEST Model 5

1 4 4 1 3 1 3 1 1 n=6 2 4 D=2 4 7 6 1 5 Network represented by a weighted graph G with n nodes and hop-diameter D . 6

1 4 4 1 3 1 4 3 4 1 1 1 1 2 1 3 4 6 4 7 6 1 5 Nodes know only local information 7

Time complexity “number of days” 8

Days: Exchange one bit Day 1 1 4 3 2 6 5 9

Nights: Perform local computation Night Day 1 1 1 4 3 2 6 5 Assume: Any calculation finishes in one night 10

Days: Exchange one bit Night Day 1 1 2 4 3 2 6 5 11

Nights: Perform local computation Night Day 1 2 2 4 3 2 6 5 12

Finish in t days  Time complexity = t 13

Part 1.2 Unweighted s-t distance 14

s 4 3 2 t 5 Distance from s = ? Goal: Node t knows distance from s 15

s 4 3 2 t 5 Distance from s = 2 16

Unweighted Case O(D) time using Breadth-First Search (BFS) algorithm. There is an W (D) lower bound. 17

s Day 4 0 1 3 2 t 5 Source node sends its distance to neighbors 18

s Night Day Day Day 4 0 1 1 1 1 1 3 1 2 1 t 5 Each node updates its distance 19

s Day 4 0 2 1 3 1 2 1 t 5 Nodes tell new knowledge to neighbors 20

s Night Day Day Day 4 0 1 1 2 1 1 3 1 2 1 t 5 2 2 Each node updates its distance 21

This algorithm takes Q (D) time 22

Part 1.3 How about weighted graphs? 23

1 4 4 1 3 1 4 3 4 1 1 1 1 2 1 3 4 6 4 7 6 1 5 Input: weighted network Remark : Weights do not affect the communication, edge weights ≤ O( polylog n) 24

s 4 4 1 3 1 4 3 1 1 2 4 7 4 7 t 1 5 4 s-t distance 25

s 4 4 1 3 1 4 3 1 1 2 4 7 4 7 t 1 5 8 4 2 -approximation 26

A naïve solution Aggregate everything into one node. Then solve the problem on that node. Time = O(# of edges) (using “pipelining” technique) 27

Can we do better? 28

Why distributed s-t distance? • Among active research on distributed algorithms for basic graph problems – MST, Connectivity, Matching, etc. • Connection to other distributed algorithmic problems – Routing, APSP, Diameter, Eccentricity, Radius, etc. • Provide distributed algorithmic viewpoint – Complement with data streams, dynamic algorithms, parallel algorithms, etc. 29

Part 1.4 Known Results for s-t distance (All results also hold for sing-source distance.) 30

Reference Time Approximation W (D) Folklore any 31

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact 32

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact W ((n/ a ) 1/2 + D) any a Elkin [STOC 2006] - polylog(n/ e ) factors are hidden 33

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact W ((n/ a ) 1/2 + D) any a Elkin [STOC 2006] W (n 1/2 + D) any a Das Sarma et al [STOC 2011] Elkin et al. [PODC 2014] also quantum - polylog(n/ e ) factors are hidden 34

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact W ((n/ a ) 1/2 + D) any a Elkin [STOC 2006] W (n 1/2 + D) any a Das Sarma et al [STOC 2011] Elkin et al. [PODC 2014] also quantum O(n 1/2+ e + D) O(1/ e ) Lenzen,Patt-Shamir [STOC 2013] - polylog(n/ e ) factors are hidden 35 - Lenzen&Patt-Shamir actually achieve more than computing distances

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact W ((n/ a ) 1/2 + D) any a Elkin [STOC 2006] W (n 1/2 + D) any a Das Sarma et al [STOC 2011] Elkin et al. [PODC 2014] also quantum O(n 1/2+ e + D) O(1/ e ) Lenzen,Patt-Shamir [STOC 2013] O (n 1/2 D 1/4 + D) 1+ e N [STOC 2014] - polylog(n/ e ) factors are hidden 36 - Lenzen&Patt-Shamir actually achieve more than computing distances

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact W ((n/ a ) 1/2 + D) any a Elkin [STOC 2006] W (n 1/2 + D) any a Das Sarma et al [STOC 2011] Elkin et al. [PODC 2014] also quantum O(n 1/2+ e + D) O(1/ e ) Lenzen,Patt-Shamir [STOC 2013] O (n 1/2 D 1/4 + D) 1+ e N [STOC 2014] O (n 1/2+o(1) + D 1+o(1) ) 1+ e Henzinger,Krinninger,N [STOC 2016] (Deterministic) - polylog(n/ e ) factors are hidden 37 - Lenzen&Patt-Shamir actually achieve more than computing distances

Reference Time Approximation W (D) Folklore any Bellman&Ford [1950s] O(n) exact W ((n/ a ) 1/2 + D) any a Elkin [STOC 2006] W (n 1/2 + D) any a Das Sarma et al [STOC 2011] * Elkin et al. [PODC 2014] also quantum O(n 1/2+ e + D) O(1/ e ) Lenzen,Patt-Shamir [STOC 2013] O (n 1/2 D 1/4 + D) 1+ e N [STOC 2014] O (n 1/2+o(1) + D 1+o(1) ) 1+ e Henzinger,Krinninger,N [STOC 2016] (Deterministic) O (n 1/2 + D) 1+ e Becker, Karrenbauer, * Krinninger, Lenzen [2016] (Deterministic) 38 - All previous results except Becker et al. can compute shortest-paths tree

Summary of Part 1 Main focus s-t Distance Known (1+ e )-approx. in Q (n 1/2 +D) time Distributed approximate s-t distance are essentially solved . 39

Plan 1. Problem & Known Results 2. Open Problems 3. Technical Challenges 40

Part 2.1 Exact algorithms 41

Is there a sublinear-time exact algorithm for s-t distance? • Current lower bound: W (n 1/2 + D) • (1+ e )-approx. algorithms need O (n 1/2 + D) time • Exact algorithm: no O (n 1- e + D) known 42

Exact case also open for many other graph problems. 43

Is there a linear-time exact algorithm for all-pairs distances ? • Current lower bound: W (n). • We have linear-time (1+ e )-approx. algorithm. 44

1 4 source 4 1 3 1 ? 3 1 1 ? 2 4 4 ? 7 6 ? 5 1 Distance from 1, 2, …, 5 = ? ? All-Pairs Shortest Paths 45

Part 2.2 Directed Case 46

1 4 4 1 3 1 3 1 1 2 4 dist(6, 3)=1 dist(3, 6)=2 4 1 7 6 1 5 Directed case Note: Two-way communication, not affected by weights. 47

Directed s-t & single-source distances Reference Time Approximation 1+ e N [STOC’14] O(n 1/2 D 1/2 +D) Ghaffari, Udwani [PODC’15] O(n 1/2 D 1/4 +D) Reachability Open O(n 1/2 +D)-time (any)-approximation algorithm. 48

Part 2.3 Congested Cliques 49

2 1 3 2 1 3 6 4 6 4 5 5 Congested Clique: The underlying network is fully connected 50

s-t distance, congested clique Reference Time Approximation N [STOC’14] O(n 1/2 ) exact O (n 1/3 ) Censor-Hillel et al. exact [PODC’15] * O (n 0.15715 ) 1+e O (n o(1) ) 1+ e Henzinger,Krinninger,N [STOC’16] 1+ e Becker, Karrenbauer, polylog(n) Krinninger, Lenzen [2016] Open: Better exact algorithm? Lower bound? 51 * Censor- Hillel et al.’s result works for APSP

All-pairs distances Reference Time Approximation 2+ e O(n 1/2 ) N [STOC’14] O (n 1/3 ) Censor-Hillel, exact O (n 0.15715 ) 1+e Kaski, Korhonen, Lenzen, Paz, Connection to matrix multiplication Suomela [PODC’15] • Additional algebraic tools (e.g. determinant) Le Gall [DISC’16] • Applications to, e.g., maximum matching Open: 1. Better exact and approximation algorithm. 2. Explore the power of algebraic techniques on congested cliques. 3. Lower bounds? 52

Lower Bounds on Congested Clique? Drucker et al [PODC’14]: • Not so easy. • Will imply something big in circuit complexity . 53

Part 2.4 Other Related Problems 54

1 4 ? 3 ? 2 ? 6 ? 5 Network Diameter = ? ? Diameter 55

Diameter (unweighted) Algorithm Time Approximation BFS D 2 W (n) 3/2- e Holzer et al. [PODC’12] for small D [Censor- Hillel et al. DISC’16 ]: Result holds for any D and even for sparse graph Holzer et al. [PODC’12] O(n) exact Peleg et al. [ICALP’12] W ((n/D) 1/2 +D) 3/2- e Frischknecht et al. [SODA’12] O (n 1/2 +D) Lenzen-Peleg [PODC’13] 3/2 O ((n/D) 1/2 +D) 3/2+ e Holzer et al. [DISC’14] W( n/D+D ) 1+e Open (By Holzer) Also: Eccentricity, radius, etc. 56

Diameter (weighted) Algorithm Time Approximation W (n) 2- e Holzer et al. [PODC’12] O (n 1/2 + D) 2+ e Becker et al. [2016] Open sublinear 2 Intermediate problem to exact SSSP (Getting a sublinear-time exact algorithm for SSSP will resolve this) 57

1 4 T 1 3 T 4 2 T 3 6 T 2 5 small “routing table” T 6 T 5 Routing 58

Some open problems Thanks: Christoph Lenzen • Eliminate n o(1) term as in the case of s-t shortest path. – Techniques from s-t SP usually transfer to the routing problem. – Exception: Becker et al [2016] • Lower bounds on the construction time for stateful routing. • Further read: Elkin, Neiman [PODC’16] & Lenzen, Patt- Shamir [STOC’13] 59