Slides: MST.

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Minimum Spanning Trees
Gallagher-Humblet-Spira (GHS)
Algorithm
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Weighted Graph G=(V,E), |V|=n, |E|=m
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w (e )  6
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Spanning tree
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Any tree T=(V,E’) (connected acyclic graph)
spanning all the nodes of G
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(MST)
Minimum-weight spanning tree
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A spanning tree s.t. the sum of its weights is
minimized:
For MST T : w (T )  w (e ) is minimized
eT
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Spanning tree fragment:
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Any (connected) sub-tree of a MST
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Minimum weight outgoing edge (MWOE)
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An edge adjacent to the fragment with
smallest weight and that does not create a
cycle
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Two important properties for building MST
Property 1: The union of a fragment and any
of its MWOE is a fragment of
some MST (so called blue rule).
Property 2: If the weights are distinct
then the MST is unique
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Property 1: The union of a fragment FT and
any of its MWOE is a fragment
of some MST.
Proof: Distinguish two cases:
1. the MWOE belongs to T
2. the MWOE does not belong to T
In both cases, we can prove the
claim.
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Case 1: e T
Fragment
F
MST T
e
MWOE
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Trivially, if e T then F  {e } is a fragment
Fragment
F   F  {e }
MST T
e
MWOE
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Case 2: e T
Fragment
F
MST T
MWOE
x
e
w (e )  w (x )
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If e T then add
e
to T and remove x
Fragment
F
MST T
x
e
w (e )  w (x )
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Fragment
F
Obtain T’
and since
w (e )  w (x )
x
e
 w(T' )  w(T)
But w(T’)  w(T),
since T is an MST
 w(T’)=w(T), i.e., T’ is an MST
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Fragment
F   F  {e }
e
MST T’
thus F  {e } is a fragment of T’
END OF PROOF
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Property 2: If the weights are distinct
then the MST is unique
Proof:
Basic Idea:
Suppose there are two MSTs
Then there is another MST
of smaller weight
 contradiction!
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Suppose there are two MSTs
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Take the smallest weight edge
not in intersection
e
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Cycle in RED MST
e
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Cycle in RED MST
e
e
e’: any red edge not in BLUE MST
( since blue tree is acyclic)
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Cycle in RED MST
e
e
Since e  is not in the intersection,w (e )  w (e )
(the weight of e is the smallest)
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w (e )  w (e )
Cycle in RED MST
e
e
Delete e  and add e in RED MST
 we obtain a new tree with smaller weight
 contradiction!
END OF PROOF 21
Prim’s Algorithm (sequential version)
Start with a node as an initial fragment F
Repeat
Augment fragment F with a MWOE
Until no other edge can be added to F
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Fragment F
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Fragment F
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MWOE
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Fragment F
MWOE
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Fragment F
MWOE
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Fragment F
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Theorem:
Proof:
Prim’s algorithm gives an MST
Use Property 1 repeatedly
END OF PROOF
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Prim’s algorithm (distributed version)
•
Works by repeatedly applying the blue rule to a single
fragment, to yield the MST for G
• Works with both asynchronous and synchronous nonanonymous, uniform models (and also with non-distinct
weights)
Algorithm (asynchronous high-level version):
Let vertex r be the root as well as the first fragment
REPEAT
•
•
•
•
r broadcasts a message on the current fragment to search for the
MWOE of the fragment (each vertex in the fragment searches
for its local (i.e., adjacent) MWOE)
convergecast (i.e., reverse broadcast towards r) the MWOE of the
appended subfragment (i.e., the minimum of the local MWOEs of
itself and its descendents)
the MWOE of the fragment is then selected by r and added to the
fragment, by sending an add-edge message on the appropriate path
then, the root and nodes adjacent to that just entered in the
fragment are notified the edge has been added
UNTIL there is only one fragment left
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Local description of asynchronous Prim
Each processor stores:
1. The state of any of its incident edges,
which can be either of {basic, branch,
reject}
2. Its own state, which can be either {in, out}
3. Local MWOE
4. MWOE for each branch-down edge
5. Parent channel (route towards the root)
6. MWOE channel (route towards the MWOE
of its appended subfragment)
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Type of messages in asynchronous Prim
1. Search MWOE: coordination message initiated by the
root
2. Test: check the status of a basic edge
3. Reject, Accept: response to Test
4. Report(weight): report to the parent node the MWOE
of the appended subfragment
5. Add edge: say to the fragment node adjacent to the
fragment’s MWOE to add it
6. Connect: perform the union of the found MWOE to the
fragment (this changes the status of the corresponding
end-node from out to in)
7. Connected: notify the root and adjacent nodes that
connection has taken place
Message complexity = O(n2)
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Synchronous Prim
It will work in O(n2) rounds…think to it by
yourself…
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Kruskal’s Algorithm (sequential version)
Initially, each node is a fragment
Repeat
• Find the smallest MWOE e of all fragments
• Merge the two fragments adjacent to e
Until there is only one fragment left
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Initially, every node is a fragment
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Find the smallest MWOE
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Merge the two fragments
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Find the smallest MWOE
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Merge the two fragments
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Merge the two fragments
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Resulting MST
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Theorem:
Kruskal’s algorithm gives an MST
Proof: Use Property 1, and observe that no
cycle is created.
END OF PROOF
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Synchronous GHS Algorithm
Distributed version of Kruskal’s Algorithm
• Works by repeatedly applying the blue rule to
multiple fragments
• Works with non-uniform models, distinct weights,
synchronous start
Initially, each node is a fragment
Repeat in parallel: (Synchronous Phase)
• Each fragment – coordinated by a fragment
root node - finds its MWOE
• Merge fragments adjacent to MWOE’s
Until there is only one fragment left
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Local description of syncr. GHS
Each processor stores:
1. The state of any of its incident edges, which
can be either of {basic, branch, reject}
2. Identity of its fragment (the weigth of a core
edge – for single-node fragments, the proc. id )
3. Local MWOE
4. MWOE for each branching-out edge
5. Parent channel (route towards the root)
6. MWOE channel (route towards the MWOE of
its appended subfragment)
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Type of messages
1. New fragment(identity): coordination message
sent by the root at the end of a phase
2. Test(identity): for checking the status of a
basic edge
3. Reject, Accept: response to Test
4. Report(weight): for reporting to the parent node the
MWOE of the appended subfragment
5. Merge: sent by the root to the node incident to
the MWOE to activate union of fragments
6. Connect(My Id): sent by the node incident to
the MWOE to perform the union
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Phase 0: Initially, every node is a fragment…
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… and every node is a root of a fragment
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Phase 1: Find the MWOE for each node
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Phase 1: Merge the nodes and select a new root
Root
Root
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Root
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Root
Symmetric MWOE
Asymmetric MWOE
The new root is adjacent to a symmetric MWOE
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Rule for selecting a new root in a fragment
Fragment 2
Fragment 1
root
MWOE
root
Merging 2 fragments
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Rule for selecting a new root in a fragment
Merged Fragment
root
Higher ID Node
on MWOE
(non-anonymity)
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Rule for selecting a new root in a fragment
F2
root
root
F3
F1
F7
F4
root
F5
root
F6
root
root
root
Merging more than 2 fragments
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Rule for selecting a new root in a fragment
Merged Fragment
Root
Higher ID Node
on symmetric MWOE
asymmetric
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Remark: In merged fragments there is
exactly one symmetric MWOE (n-1 edges
vs n arrows)
 two
zero
F1
F2
F3
F4
F1
F2
F3
F4
F7
F5
F5
F6
Impossible
Creates a fragment
with no MWOE
F6
F8
Impossible
Creates a fragment
with two MWOEs
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After merging has taken place, the new root
broadcasts New fragment(w(e)) to the new
fragment, and afterwards a new phase starts
w (e )
e is the
symmetric
MWOE of
the
merged
fragments
w (e )
e
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
w (e )
is the identity of the new fragment
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In our example, at the end of phase 1 each
fragment has its new identity.
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Root7
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Root 14
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Root
End of phase 1
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Root
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At the beginning of each new phase each
node in fragment finds its MWOE

MWOE
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


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
MWOE
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

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

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

12
4


MWOE


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
19
MWOE

22

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To discover its own MWOE, each node sends a
Test message containing its identity over its basic
edge of min weight, until it receives an Accept
test()

3

10
accept

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
test()

6
reject





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Then it knows its local MWOE

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MWOE
3









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Then each node sends a Report with the MWOE of the
appended subfragment to the root with convergecast (the
global minimum survives in propagation)

MWOE
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10
3



3
15

3
3
4


MWOE
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
19

3
MWOE
7


25
22


12


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
19
MWOE

22

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The root selects the minimum MWOE and sends along
the appropriate path a Merge message, which will
become a Connect message at the proper node
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

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
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


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

3

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

12
4


MWOE


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
19
22


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Phase 2 of our example: After receiving the
new identity, find again the MWOE for each
fragment
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Phase 2: Merge the fragments
Root7
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Root
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At the end of phase 2 each fragment
has its own unique identity.
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Root7
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15
13
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5
12
9
End of phase 2
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18
Root
8
3
2
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9
9
9
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Phase 3: Find the MWOE for each fragment
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Phase 3: Merge the fragments
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Root
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Phase 3: New fragment
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FINAL MST
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Correctness
• To guarantee correctness, phases must be syncronized
• But at the beginning of a phase, each fragment can have a
different number of nodes, and thus the MWOE selection
is potentially asynchronous…
• But each fragment can have at most n nodes, has height at
most n-1, and each node has at most n-1 incident edges…
• So, the MWOE selection requires at most 3n rounds, and
the Merge message requires at most n rounds.
• Then, the Connect message must be sent at round 4n+1 of
a phase, and so at round 4n+2 a node knows whether it is a
new root
• Finally, the New fragment message will require at most n
rounds.
 A fixed number of 5n+2 total rounds can be used to
complete each phase (in some rounds nodes do nothing…)!
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Complexity
Phase
Smallest Fragment size
(#nodes)
0
1
1
2
2
4


i
i
2
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Algorithm Time Complexity
Maximum possible fragment size 2i  n
Number of nodes
Maximum # phases:
i  log2 n

Total time = Phase time • #phases = O (n logn )
O (n )
logn
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Algorithm Message Complexity
Thr: Synchronous GHS requires O(m+n logn) msgs.
Proof: We have the following messages:
1. Test-Reject msgs: at most 2 for each edge;
2. Each node sends/receives at most a single:
New Fragment, Test-Accept, Report,
Merge, Connect message for each phase.
Since from previous lemma we have at most log n
phases, the claim follows.
END OF PROOF
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Asynchronous Version of GHS Algorithm
•Simulates the synchronous version
•Works with uniform models, distinct
weights, asynchronous start
•Every fragment F has a level L(F)≥0: at the
beginning, each node is a fragment of level 0
•Two types of merges: absorption and join
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Local description of asyncr. GHS
Like the synchronous, but:
1. Identity of a fragment is now given by an edge
weight plus the level of the fragment;
2. A node has its own status, which can be either
of {sleeping, finding, found}
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Type of messages
Like the synchronous, but now:
1. New fragment(weight,level,status):
coordination message sent just after a
merge
2. Test(weight,level): to test an edge
3. Connect(weight,level): to perform the
union
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Absorption
Fragment
Fragment
F1
MWOE
F2
If L(F1 )  L(F2 ) then F1 is absorbed by F2
(cost of merging proportional to F1 )
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The combined level is L(F )  L(F2 )
New fragment
F1
MWOE
F2
and a New fragment(weight,level,status)
message is broadcasted to nodes of F1 by the
node of F2 on which the merge took place
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Join
Fragment
Fragment
F1
MWOE
F2
If L(F1 )  L(F2 ) and F1 and F2 have a
symmetric MWOE, then F1 joins with F2
(cost of merging proportional to F1  F2 )
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The combined level is L( F )  L( F1,2 )  1
New fragment
F1
MWOE
F2
and a New fragment(weight,L(F2)+1,finding)
message is broadcasted to all nodes of F1 and
F2, where weigth is that of the edge on which
the merge took place
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Remark: a joining requires that fragment
levels are equal…how to control this?
 If L(F1)>L(F2), then F1 waits until L(F1)=
L(F2) (this is obtained by letting F2 not
replying to Test messages from F1 )
Fragment
Fragment
F1
Test
F2
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Algorithm Message Complexity
Lemma: A fragment of level L contains at least 2L
nodes.
Proof: By induction. For L=0 is trivial. Assume it is
true up to L=k-1, and let F be of level k. But
then, either:
1. F was obtained by joining two fragments of level
k-1, each containing at least 2k-1 nodes by
inductive hypothesis  F contains at least 2k-1 +
2k-1 = 2k nodes;
2. F was obtained after absorbing another
fragment F’ of level<k  apply recursively to
F\F’, until case (1) applies.
END OF PROOF
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Algorithm Message Complexity (2)
Thr: Asynchronous GHS requires O(m+n logn) msgs.
Proof: We have the following messages:
1. Connect msgs: at most 2 for each edge;
2. Test-Reject msgs: at most 2 for each edge;
3. Each node sends/receives at most a single:
New Fragment, Test-Accept, Merge,
Report message each time the level of its
fragment increases;
and since from previous lemma each node can
change at most log n levels, the claim follows.
END OF PROOF
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Homework
Execute asynchronous GHS on the following graph:
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2
3
4
2
4
1
17
11
9
7
8
8
24
14
5
10
6
6
1
9
7
12
assuming that system is pseudosynchronous: Start from 1
and 5, and messages sent from odd (resp., even) nodes are
received after 1 (resp., 2) round(s)
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