DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Search Fundamentals
Radu Nicolescu Department of Computer Science University of Auckland
12 Aug 2020
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
1 Distributed DFS and BFS
2 ClassicalDFS
3 CidonDFS §1
4 CidonDFS §2
5 Bellman-Ford algorithm
6 Maximal Independent Set
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Distributed DFS
• Despite its sequential appearance, DFS can work faster on distributed networks!
• Classical DFS traverses each edge (tree and frond) twice, thus time complexity = message complexity = 2|E|
• Like other distributed versions, Cidon DFS traverses each tree edge twice (needed), but avoids all or most fronds, thus
time complexity = 2|V | − 2, but at the possible expense of increased message complexity = 3|E|
• Small error in Tel’s text: message complexity is not 4|E|…
• Cidon DFS was designed for async networks, thus also works for sync networks (even better).
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Distributed BFS
• For sync networks, Echo (aka SyncBFS) finds a BFS spanning tree; time complexity = 2D + 1, message complexity = 2|E |
• Despite its parallel appearance, BFS is harder to implement on async networks and does not get all the expected benefits (bad time or bad message complexity).
• In fact, on async networks, BFS looks like a “milder” version of Bellman-Ford: shows similar issues, but less severe
• AsyncBFS [Lynch]: messages = O(D|E|); time−pileups = O(D); time+pileups = O(D|V|)
• LayeredBFS [Lynch]: messages = O (D |V | + |E |); time−pileups = O(D2)
• We do not further follow these issues here…
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS
• Cidon DFS uses two types of tokens:
• One single classical tok token, sent on tree edges and,
occasionally, on fronds
• New vis notification tokens, used to mark possible fronds
• A vis token can be sent: • ahead of tok
• together with tok (can be combined) • alone (on fronds only)
Read more: Tel §6.4, or Cidon’s original paper:
Yet Another Distributed Depth-First-Search Algorithm http://cidon.eew.technion.ac.il/files/var/448324- cidon_dfs_87.pdf
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical vs Cidon DFS – Examples
• In all cases, there is one single tok token, thus the time complexity sums the delays in this token’s delivery
• In classical DFS, frond edge {2, 4} is sequentially twice traversed by the tok token, which adds 2 time intervals
• In Cidon DFS §1, frond edge {2, 4} is twice traversed by vis tokens, in parallel with the progress of the tok token, which now saves 2 time intervals (while keeping messages count)
• In Cidon DFS §2, frond edge {2, 4} is overlappingly traversed
• in one direction, by the vis token, in parallel with the progress
of the tok token,
• in the reverse direction, by a pair of tok+vis tokens (vis is not
strictly necessary)
• this still saves 1 time interval (but increases messages count)
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS
2
3
1
6
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Time Units = 0 Messages = 0
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS
{}
{}
2
3
{2}
16
{}
{}
{}
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Time Units = 1 Messages = 1
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS
{3} 2
{}
3
{2}
1
{}
6
{}
{}
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Time Units = 2 Message = 2
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS
{3}
{5, 6} 3
2
{2, 4} 1
{3, 5} 6
{1}
{6, 4}
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Time Units = 10 Message = 10
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS : 4 → tok → 2
{3}
{5, 6} 3
2
{2, 4}
{3, 5}
1
6
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{1, 2}
{6, 4} Time Units = 11
Message = 11
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS : 2 → tok → 4
{3, 4} 2
{5, 6} 3
{2, 4} 1
{3, 5} 6
{1, 2}
{6, 4}
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Time Units = 12 Message = 12
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Classical DFS
{3, 4, 1} 2
{5, 6, 4, 2} 3
{2, 4}
{3, 5}
1
6
{1, 2, 3, 5}
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{6, 4, 3}
Time Units = 18 = 2M Message = 18 = 2M
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §1
2
3
1
6
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Time Units = 0 Messages = 0
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §1
{1} 2
{}
3
{2}
16
{}
45 {1}
Time Units = 1 Messages = 3
{}
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
CidonDFS§1: 2→vis→4
{1, 3} {2}
2
3
{2}
16
{}
{1, 2}
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Time Units = 2 Messages = 6
{}
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §1
{1, 3}
2
{2, 5, 6}
3
{2}
{3, 5} 16
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{1, 2, 3, 5} {3, 6, 4}
Time Units = 6 Messages = 16
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
CidonDFS§1: 4→vis→2
{1, 3, 4} 2
{2, 5, 6, 4} 3
{2, 4}
{3, 5} 16
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{1, 2, 3, 5} {3, 6, 4}
Time Units = 7 Messages = 20
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §1
{1, 3, 4} 2
{2, 5, 6, 4} 3
{2, 4}
{3, 5} 1 6
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{1, 2, 3, 5} {3, 6, 4}
Time Units = 10 = 2N – 2 Messages = 23 ≤ 3M
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §2
2
3
1
6
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Time Units = 0 Messages = 0
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §2
{1} 2
{}
3
{2}
16
{}
45 {1}
Time Units = 1 Messages = 3
{}
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
CidonDFS§2: 2vis4
{1, 3} {2}
2
3
{2}
16
{}
{1}
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Time Units = 1 + ε Messages = 6
{}
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §2
{1, 3}
2
{2, 5, 6}
3
{2}
{3, 5} 16
45 {1, 3, 5} {3, 6, 4}
Time Units = 1 + 5ε Messages = 16
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
CidonDFS§2: 2vis4,4→tok+vis→2
{1, 3, 4} 2
{2, 5, 6, 4} 3
{2, 4}
{3, 5} 16
45 {1, 3, 5} {3, 6, 4}
Time Units = 1 + 6ε Messages = 20
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
CidonDFS§2: 2→vis→4
{1, 3, 4} {2, 5, 6, 4}
2
3
{2, 4}
{3, 5} 16
45 4
{1, 2, 3, 5} {3, 6, 4}
Time Units = 2 Messages = 20
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Cidon DFS §2
{1, 3, 4} 2
{2, 5, 6, 4} 3
{2, 4}
{3, 5} 1 6
45 {1, 2, 3, 5} {3, 6, 4}
Time Units = 6 ≤ 2N – 2 Messages = 24 ≤ 3M
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Distributed Bellman-Ford algorithm
• Classical Bellman-Ford algorithm finds all shortest paths from a single source – like Dijkstra
• Advantage for Bellman-Ford: can cope with negative weights
• Classical Dijkstra: Time complexity = O ((|E | + |V |) log |V |)
• Classical Bellman-Ford: Time complexity = O(|V||E|)
• Distributed Dijkstra: more difficult distribution…
• Distributed Bellman-Ford ≈ a simple extension of Echo
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Distributed Bellman-Ford algorithm
• Sync Bellman-Ford (“Echo ++”): • Time complexity = O(|V|)
• Message complexity = O(|V||E|)
• Async Bellman-Ford
• Message complexity: O(|V||V|) (terrible worst case, but often
much lower in reality)
• Time complexity = O(|V|) – if any message is delivered in at most 1 time units – no FIFO [Tel]
• Time complexity = O(|V||V|) – if we consider the congestion (pileups) on FIFO channels [Lynch]
•
Are these realistic? …
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Sync Bellman-Ford – Start
Problem: Find all shortest paths from node 3.
• red = active
• blue = unvisited • green = visited
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Sync Bellman-Ford – Round 1
Update formula: new distance = minimum between old distance and all newly (received distance + edge length)
• Nodes 4 and 2: update distances
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Sync Bellman-Ford – Round 2
• Nodes 5, 6, 1: update distances
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Sync Bellman-Ford – Round 3
• Node 1:
update distance (recalculation!)
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Sync Bellman-Ford – Round 4
• Node 2:
no recalculation (but could have been)
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Distributed Bellman-Ford – termination?
All nodes have successfully terminated
We can see this, but the nodes don’t know this yet
How to detect and, optionally, disseminate the termination info?
• For Sync and Async Bellmann-Ford: by convergecast and, optionally, a subsequent broadcast (like Echo)
• For Sync Bellmann-Ford: by attaching a time-to-live (TTL) to the broadcast token
• Initially equal to the number of nodes |V | (if this is known)
• After receiving it, each node decrements this by 1, at each
round (thus sync mode required)
• When TTL = 0, each node knows that the algorithm has terminated (guaranteed)
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Async Bellman-Ford – worst case (sketch, cf. Lynch §15.4)
Sketch for
• k = 3 (can be any number) • N=2k+2=8nodes
• M=3k+1=10edges
Costs: as indicated or 0 (default)
Initiator: left-most node (a)
Green: fast link; Red: slow link; Black: slow & critical if FIFO
bdf
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aceyz
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Async Bellman-Ford – worst case (cont)
b:0 d:4 f:6
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a:0 c:4 e:6
b:0 d:0 f:2
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c:0 e:2
b:0 d:0 f:2
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c:0 e:2
b:0 d:0 f:0
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c:0 e:0
b:0 d:0 f:0
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c:0 e:0
Shapes of the shortest-so-far cost paths ←→ base 2 num√bers Exponential message complexity ≥ 2k = 2(N−2)/2 = Ω( 2N) Exponential time complexity if FIFO – congestion on black edge
b:0
d:4 f:6
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y:7 z 1112 a:0 y:6 z 1102 a:0 y:5 z 1012 a:0
y:4 z 1002 a:0
y:3 0112 y:2 0102 y:1 0012 y:0 0002
a:0 c:4 e:6
b:0
d:4 f:4
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a:0 c:4 e:4
b:0
d:4 f:4
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a:0 c:4 e:4
Dotted blue arrows : messages still in transit
1:1
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Async Bellman-Ford – worst case
• How to explain the time complexity?
• Time complexity without FIFO pileups = O(|V|)?
• Time complexity with FIFO pileups = exponential?
• If the last edge, before the rightmost node, is slow enough, say t, then all 2k messages can pile-up in this segment
• If we consider congestion, then these piled-up messages can be successively delivered at t intervals
• Total delivery time on the last edge: 2kt = 2O(N) time units, i.e. exponential time complexity [Lynch §15.4]
• This argument fails, if we do NOT consider FIFO congestion, because then even the slowest message will not be affected by the others, and will only take a maximum 1 time unit.
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Echo and Bellman-Ford – complexity highlights
• Sync Echo (aka Sync BFS) : BFS ST, no link changes, fast • Async Echo : arbitrary ST, no link changes, but not so fast • Sync BF : shortest paths ST, many link changes, not so fast • Async BF : shortest paths ST, exponential link changes
• if FIFO: worst case exponential time
• Why the worst case argument does NOT apply to Sync BF?
• emulating slow links by extra edges exponentially increases V !
• the formula is still exponential on k, but not anymore on N = exp(k)!
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Maximal Independent Set
• Independent = no two neighbours
• Maximal = cannot be extended
• Maximal is not necessarily largest
• Here we assume that nodes do not have IDs
• Impossible to solve with conventional means in an absolutely symmetric case!
• Luby’s algorithm can still break the ties with randomization techniques
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Luby’s algorithm
• Sync algorithm, which works in stages, each consisting of the following 3 rounds
1 Each processes chooses a new random value and sends it to its neighbours. Processes that have values greater than all their neighbours become winners
2 Winners notify their neighbours. Processes that receive such messages from their neighbours become losers
3 Losers notify their neighbours. All processes make a conceptual reshaping of the graph. Both winners and losers are conceptually disconnected from further participation. Remaining nodes are still competing and will regenerate new random values in their next stage!
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Luby’s algorithm
• Luby’s algorithm will stop with probability 1, expected O(log n) rounds -– proof quite hard
• The random values are best chosen in the range 1, 2, …N4, will be likely distinct (but this is not necessary)
• Our diagrams will be sketched only, we won’t detail all these rounds individually
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
MIS – Example
• One possible solution, formed by {5, 2}
• Another solution could be {5, 1, 3} (which is larger)
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Luby – Stage #1, Round #1
• green = hopeful
• blue = loser
• red = winner
• we only show most relevant messages
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Luby – Stage #1, Rounds #2
• one winner so far: 5
• losers: 4 and 6
• losers notify their neighbours
• winners and losers are conceptually disconnected
• still competing: 1, 2 and 3
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DFS ClassicalDFS CidonDFS§1 CidonDFS§2 Bellman-Ford MIS
Luby – Stage #2
• a new winner: 2
• new losers, this winners’ neighbours: 1 and 3
• the end
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More about MIS
• Minimum Maximal Independent Set vs. Maximum Maximal Independent Set
• Related readings (NOT required) – S. Butenko, PhD Thesis https://ufdc.ufl.edu/UFE0001011/00001/pdf
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