Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015

This talk Focus on single-source shortest paths (SSSP) 3 Settings: Distributed, Decremental, Streaming The three settings seem to share some common features: All we can do is essentially BFS Better guess for the right solution by looking at these settings at the same time 2 * * There are exceptions

This talk: Organization 3 Graph structures Unweighted, Undirected UnWeighted, Undirected UnWeighted, UnDirected Tools A pproximation R andomness (Things are pretty much the same for unweighted directed graphs)

4 ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) ???? Semi-Stream (# passes) ???? Decremental (total update time) ????

Preliminaries 5 Part 0

Notations n = number of nodes m = number of edges W = (max weight) / (min weight) SSSP = single-source shortest paths problem APSP = all-pairs shortest paths problem 6

7 Remarks polylog n and polylog W are mostly hidden Some great results may not be mentioned (sorry!) If I seem to miss something, please let me know (thank you!)

Introduction 8 Part 1

Distributed Setting (CONGEST) 9 Part 1.1

Network represented by a weighted graph G with n nodes and diameter D. n=6 D=2

Nodes know only local information

Time complexity “number of days” 12

Days: Exchange O(log n) bit 1 Day

Nights: Perform local computation Day 1 Night Assume: Any calculation finishes in one night

1 Night Days: Exchange O(log n) bit 2 Day

2 Day 2 Night Nights: Perform local computation

Finish in t days  Time complexity = t 17

Example s-t distance 18

s t Goal: Node t knows distance from s Distance from s = ?

s t Distance from s = 4 Goal: Node t knows distance from s

s t Distance from s = approximate solution

Computing s-t distance can be done in O(D) time by using the Breadth-First Search (BFS) algorithm. 22 Unweighted Case

s t 0 Source node sends its distance to neighbors 1 Day 23

Each node updates its distance 1 Day Night s t

Nodes tell new knowledge to neighbors 2 Day s t

Each node updates its distance 1 Day Night s t

This algorithm takes  (D) time 27

(Multi-pass) Streaming Setting 28 Part 1.2

29 Small RAM Huge Harddisk 3 rd pass (1, 2) (2, 4) (1, 3) (2, 3)  (n 2 ) space O(n) space

30 Small RAM Huge Harddisk (1, 2) (2, 4) (1, 3) (2, 3)

31 Small RAM Huge Harddisk (1, 2) (2, 4) (1, 3) (2, 3)

32 Small RAM Huge Harddisk (1, 2) (2, 4) (1, 3) (2, 3)

33 Small RAM Huge Harddisk (1, 2) (2, 4) (1, 3) (2, 3)

34 Small RAM Huge Harddisk 2 nd pass (1, 2) (2, 4) (1, 3) (2, 3)

35 Small RAM Huge Harddisk 3 nd pass (1, 2) (2, 4) (1, 3) (2, 3)

Complexity = # of passes Ideally: (polylog n) passes Limitation: (n polylog n) space 36

Example s-t distance 37

38 Huge Harddisk 3 rd pass s 2 3 t (1, 2) (2, 4) (1, 3) (2, 3) Small RAM Initially 0

Small RAM 39 Huge Harddisk 3 rd pass s 2 3 t (1, 2) (2, 4) (1, 3) (2, 3) 1 st pass 0 1 1

Small RAM 40 Huge Harddisk 3 rd pass s 2 3 t (1, 2) (2, 4) (1, 3) (2, 3) 2 st pass

This algorithm takes  (D)=O(n) passes 41

Decremental Setting 42 Part 1.3

43 We start with a graph with of n nodes and m edges.

44 Edges are gradually deleted

45 Edges are gradually deleted

Goal Maintain some graph property under edge deletions 46

Total Update Time = Total time to maintain graph property after all m deletions 47

Example s-t distance 48

49 Goal Maintain the distance between s and t after every deletions

Naive algorithm Compute Breadth-First Search Tree (BFS) after every deletion 50 Total update time = O(m 2 )

Better Solution Dynamic BFS Tree (Even-Shiloach Tree [JACM 1981] ) O(m 2 )  O(mn) 51

Algorithm description as nodes talking to each other 52

53 s e b c f d Single-Source Shortest Paths from s

54 s e b c f d Every node v maintains its level in the BFS level=1 level=2

s e b c f d Delete (s,b)  b connects to a new parent level=1 level=2 55 s e b c f d level=1 level=2

b announces its level change 56 s e b c f d level=1 level=2 s e b c f d level=1 level=2 level(b)=2

f connects to a new parent. e changes level. 57 s e b c f d level=1 level=2 s e b c f d level=3 level=1 level(b)=2

Again, e announces level change 58 level(e)=3 s e b c f d level=2 level=3 level=1

Again, e announces level change 59 s e b c f d level=2 level=3 level=1 This is what we obtain after deleting (s,b)

Even-Shiloach tree can be implemented in such a way that total update time = number of messages 60

s e b c f d level=1 level=2 level(b)=2 61 Takes 3 time steps Even-Shiloach tree can be implemented in such a way that total update time = number of messages

Exercise Number of messages (thus time complexity) is O(mD) = O(mn) 62 Hint Node v sends degree(v) messages every time level(v) increases.

Unweighted, Undirected Graphs 63 Part 2

64 Unweighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS] Stream (# passes) O(D) [BFS] Decremental (total update time) O(mD) [BFS]

65 Unweighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS] Stream (# passes) O(D) O(n) [BFS] Decremental (total update time) O(mD) O(mn) [BFS]

66 Unweighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS]  (D) (even for approx) [Folklore] see weighted case Stream (# passes) O(D) O(n) [BFS]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13] Decremental (total update time) O(mD) O(mn) [BFS]  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15]

67 Unweighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS]  (D) (even for approx) [Folklore] see weighted case Stream (# passes) O(D) O(n) [BFS]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13] Decremental (total update time) O(mD) O(mn) [BFS]  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15]

Lower bounds for streaming SSSP Feigenbaum et al. SODA’05: computing the set of vertices at distance p from source s in ≤ p/2 passes requires n 1+Ω(1/p) space. – Guruswami, Onak CCC’13: Same space lower bound holds even for (p−1) passes Guruswami, Onak, CCC’13: A p passes algorithm requires n 1+  (1/p) /p O(1) space to check if dist(s, t) ≤ 2(p + 1) – Superlinear space when p is small 68

69 Unweighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS]  (D) (even for approx) [Folklore] see weighted case Stream (# passes) O(D) O(n) [BFS]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13] Decremental (total update time) O(mD) O(mn) [BFS]  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15]  (n)?

Hardness for Decremental SSSP Roditty, Zwick, ESA’04: – Assume: no combinatorial O(n 3-  )-time algorithm for Boolean Matrix Multiplication – Then: no combinatorial exact decremental SSSP algorithm with O(mn 1-  ) total update time Henzinger et al. STOC’15: – Assume: no combinatorial O(n 3-  )-time algorithm for Online Boolean Matrix-Vector Multiplication – Then: no combinatorial exact decremental SSSP algorithm with O(mn 1-  ) total update time 70

Online Boolean Matrix-Vector Multiplication Given an (n x n)-matrix M. Given an n-vector v 1. Must answer Mv 1. … Given an n-vector v n. Must answer Mv n. Conjecture: No O(n 3-  )-time algorithm Current best: O(n 3 /log 2 n) [Williams, SODA’07] 71

72 Unweighted, Undirected SSSP -- Conclusion ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS]  (D) (even for approx) [Folklore] see weighted case Stream (# passes) O(D) O(n) [BFS]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13] Decremental (total update time) O(mD) O(mn) [BFS]  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15]  (n)?

UnWeighted, Undirected Graphs 73 Part 3

74 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford] Stream (# passes) O(n) [Bellman-Ford] Decremental (total update time) O(m 2 ) [trivial]

75 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx [Das Sarma et al STOC’11] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[ Decremental (total update time) O(m 2 ) [trivial]  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15]

 (n 1/2 +D) lower bound for distributed weighted SSSP  (D) is from the unweighted case. Das Sarma et al. STOC’11: There exists a family of O(log n)-diameter graphs s.t. poly(n)-approximating dist(s, t) requires  (n 1/2 ) time (Klauck et al. PODC’14: Also hold for quantum distributed algorithms) 76

77 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx [Das Sarma et al STOC’11] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[ Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] ? ? ?

Related open problems Bernstein STOC’13: Exists O(mn) time for decremental exact APSP on undirected graphs? – Exists: O(mn) time (1+  ) approximation on weighted directed graphs – Interesting even for unweighted undirected case – Weighted case: O(mn 2 ) total update time via fully- dynamic algorithm [Demetrescu, Italiano, STOC’03] – Unweighted case: O(n 3 ) total update time [Demetrescu, Italiano FOCS’01] [Baswana et al., STOC’02] 78

Related open problems Bernstein STOC’13: Exists O(mn) time for decremental exact APSP on undirected graphs? One more here: Getting O(mn) for exact weighted SSSP? Also: distributed APSP in O(n) time – Known: O(n)-time (1+  )-approximation Also from Bernstein: Can we remove log W? 79

80 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx [Henzinger et al. ‘15] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[ O(n o(1) ) (1+  )-approx O(n 1+o(1) ) space [Henzinger et al.’15] Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(m 1+o(1) ) (1+  )-approx [Henzinger et al. FOCS’14]

(1+  )-approximation for weighted undirected case Henzinger et al. FOCS’14: (1+  )-approximation decremental SSSP in O(m 1+o(1) ) total update time – Hidden in o(1): O(1/  log 1/2 n) – Heavily rely on randomization Henzinger et al.’15: (1+  )-approximation SSSP in – Streaming: n o(1) passes and n 1+o(1) space – Distributed: n 1/2+o(1) time 81

Key subroutine: BFS Algorithms 82 Hop set Thorup-Zwick Clusters previously used for distance oracles and spanners Bounded-depth BFS trees from every node with special stopping rules

Note: 1-pass streaming algorithm Feigenbaum et al. [ICALP’04]: A (2k-1)- spanner can be constructed in one pass, O(kn 1/k ) space – Implies, e.g., O(log n)-approximation 1-pass O(n)- space algorithm for SSSP 83

84 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx [Henzinger et al. ‘15] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[ O(n o(1) ) (1+  )-approx O(n 1+o(1) ) space [Henzinger et al.’15] Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(m 1+o(1) ) (1+  )-approx [Henzinger et al. FOCS’14] ? ? ?

Open: Eliminate n o(1) terms E.g. (1+  )-approx O(n polylog n)-space (polylog n)-pass streaming algorihtm for SSSP? Exists an (polylog n,  )-hop set of size polylog n? – Known: (n o(1),  )-hop set of size n 1+o(1) 85

Hop Set 86 Skip

a d e f c b a d e f c b Spanner (Sparsify graph) a d e f c b Hopset (Densify graph) a d e f c b Two orthogonal approaches

Hopset [Cohen, JACM’00] 88 (h,  )-hopset of a network G = (V,E) is a set E* of new weighted edges such that h-edge paths in H=(V, E ∪ E*) give (1+ε) approximation to distances in G.

Example (1) Add shortcuts between every pair Input graph 89 Picture from Cohen [JACM’00] 4 a 2 5 6

Example (1) Add shortcuts between every pair Input graph 90 Picture from Cohen [JACM’00] 4 a

Example (1) Input graph Picture from Cohen [JACM’00] 4 a a 6 b 91 (1, 0)-hopset one edge is enough to get distance no error

Example (2) Input graph with (5, 0) -hopset Input graph 92 Picture from Cohen [JACM’00] 11

Hopset constructions 93 References (h,  ) SizeNote Cohen [JACM’00] (polylog n,  n 1+o(1) PRAM alg Bernstein [FOCS’09] (n o(1),  ) n 1+o(1) Use Thorup-Zwick Clusters Static O(m) time alg Henzinger et al. [FOCS’14] ”” Decremental O(m 1+o(1) )-time alg

94 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx [Henzinger et al. ‘15] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[ O(n o(1) ) (1+  )-approx O(n 1+o(1) ) space [Henzinger et al.’15] Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(m 1+o(1) ) (1+  )-approx [Henzinger et al. FOCS’14]

95 UnWeighted, Undirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx   (n 1/2+o(1) +D ) (1+  )-approx [Henzinger et al. ‘15] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[  O(n o(1) ) (1+  )-approx O(n 1+o(1) ) space [Henzinger et al.’15] Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(m 1+o(1) ) (1+  )-approx [Henzinger et al. FOCS’14] ? Derandomization Ideas from [Roditty et al., ICALP’05], [Lenzen, Patt- Shamir’15], [Goldberg et al., STOC’87]

Related question: Deterministic weighted APSP Deterministic decremental (1+  )- approximation O(mn)-time algorithm for weighted APSP Known for unweighted APSP [Henzinger et al., FOCS’13] – Derandomized [Roditty, Zwick, FOCS’04] – Tight [Dor et al, FOCS’96], [Henzinger et al, STOC’15] Randomized decremental (1+  )-approximation O(mn)-time algorithm for weighted directed APSP [Bernstein, STOC’13] 96

97 UnWeighted, Undirected SSSP -- Conclusion ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx   (n 1/2+o(1) +D) (1+  )-approx [Henzinger et al. ‘15] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[  O(n o(1) ) (1+  )-approx O(n 1+o(1) ) space [Henzinger et al.’15] Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(m 1+o(1) ) (1+  )-approx [Henzinger et al. FOCS’14] ? ? ?? ? ?

UnWeighted, UnDirected Graphs 98 Part 4

99 UnWeighted, UnDirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(n) [Bellman-Ford]  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx  (n 1/2 D 1/2 +D) (1+  )-approx Stream (# passes) O(n) [Bellman-Ford]  (log n) any approx (reachability) [Guruswami,Onak CCC’13[ r-pass n 2 /r space [trivial] Decremental (total update time) O(mn) (1+  )-approx O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(mn 1+o(1) ) (1+  )-approx O(mn 0.9 ) (1+  )-approx

Lower bounds for streaming directed SSSP Guruswami, Onak, CCC’13: A p passes algorithm for s-t reachability requires n 1+  (1/p) /p O(1) space (Superlinear space when p is small) 100

101 UnWeighted, UnDirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(n) [Bellman-Ford]  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx  (n 1/2 D 1/2 +D) (1+  )-approx Stream (# passes) O(n) [Bellman-Ford]  (log n) any approx (reachability) [Guruswami,Onak CCC’13[ r-pass n 2 /r space [trivial] Decremental (total update time) O(mn) (1+  )-approx O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(mn 1+o(1) ) (1+  )-approx O(mn 0.9 ) (1+  )-approx ? ? ?

102 UnWeighted, UnDirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(n) [Bellman-Ford]  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx  (n 1/2 D 1/2 +D) (1+  )-approx Stream (# passes) O(n) [Bellman-Ford]  (log n) any approx (reachability) [Guruswami,Onak CCC’13[ r-pass n 2 /r space [trivial] Decremental (total update time) O(mn) (1+  )-approx O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(mn 1+o(1) ) (1+  )-approx O(mn 0.9 ) (1+  )-approx

103 UnWeighted, UnDirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(n) [Bellman-Ford]  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx  (n 1/2 D 1/2 +D) (1+  )-approx Stream (# passes) O(n) [Bellman-Ford]  (log n) any approx (reachability) [Guruswami,Onak CCC’13[ r-pass n 2 /r space [trivial] Decremental (total update time) O(mn) (1+  )-approx O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(mn 1+o(1) ) (1+  )-approx O(mn 0.9 ) (1+  )-approx

Upper Bounds for Directed SSSP Nanongkai STOC’14 (implicit): (1+  )- approximation  (n 1/2 D 1/2 +D)-time distributed algorithm Henzinger et al. STOC’14: (1+  )-approximation decremental algorithm with  (mn 0.99 ) total update time – Recently improve to  (mn 0.9 ) total update time 104

105 UnWeighted, UnDirected SSSP ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(n) [Bellman-Ford]  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx  (n 1/2 D 1/2 +D) (1+  )-approx Stream (# passes) O(n) [Bellman-Ford]  (log n) any approx (reachability) [Guruswami,Onak CCC’13[ r-pass n 2 /r space [trivial] Decremental (total update time) O(mn) (1+  )-approx O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(mn 1+o(1) ) (1+  )-approx O(mn 0.9 ) (1+  )-approx ? ? ?

Hop Set for Directed Graphs? k-Transitive-Closure Spanner [Thorup WG’92]: – Has the same transitive closure as in the original graph – Diameter at most k There is a n 1/2 -TC-spanner of size O(n). How efficient can we compute it in various settings? 106

Conjecture Two parties each gets part of the directed graph. Conjecture: There exists no communication protocol that takes r rounds and o(n 2 /r) communication that can solve s-t shortest path on n-node directed graphs. Might be true even for reachability Will imply a tight lower bound in the streaming setting Will imply a non-trivial (perhaps tight) lower bound in the distributed setting 107

Conclusion 108

109 Unweighted, Undirected SSSP -- Conclusion ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) [BFS]  (D) (even for approx) [Folklore] see weighted case Stream (# passes) O(D) O(n) [BFS]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13] Decremental (total update time) O(mD) O(mn) [BFS]  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15]  (n)?

110 UnWeighted, Undirected SSSP -- Conclusion ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(D) O(n) [Bellman-Ford]  (D)  (n 1/2 +D) even for approx   (n 1/2+o(1) +D) (1+  )-approx [Henzinger et al. ‘15] Stream (# passes) O(n) [Bellman-Ford]  (R) for distance R=O(log n), exact only [Feigenbaum et al SODA’05] [Guruswami,Onak CCC’13[  O(n o(1) ) (1+  )-approx O(n 1+o(1) ) space [Henzinger et al.’15] Decremental (total update time) O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(m 1+o(1) ) (1+  )-approx [Henzinger et al. FOCS’14] ? ? ?? ? ?

111 UnWeighted, UnDirected SSSP -- Conclusion ModelExact alg.Lower boundApprox + Rand Distributed (# rounds) O(n) [Bellman-Ford]  (n 1/2 +D) even for approx  (n 1/2+o(1) +D) (1+  )-approx  (n 1/2 D 1/2 +D) (1+  )-approx Stream (# passes) O(n) [Bellman-Ford]  (log n) any approx (reachability) [Guruswami,Onak CCC’13[ r-pass n 2 /r space [trivial] Decremental (total update time) O(mn) (1+  )-approx O(m 2 )  (mn) conditional, exact only [Roditty, Zwick, ESA’04] [Henzinger et al. STOC’15] O(mn 1+o(1) ) (1+  )-approx O(mn 0.9 ) (1+  )-approx ? ? ? ? ? ?

Thank you 112