Locality Sensitive Distributed Computing Exercise Set 1

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Locality Sensitive Distributed ComputingExercise
Set 1

• David PelegWeizmann Institute

• Exercises
• Basic complexity considerations
• Global function computation pipeline
• Termination detection for Dijkstras BFS
• Fast DFS
• Tgap in synchronizers a and b
• 3-coloring bounded-degree graphs
• MIS / coloring on unoriented rings

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Basic complexity issues

• Prove or disprove
• In a graph G(V,E),
• if there are at least k edge-disjoint paths
• of length ? d between the nodes v and w,
• then it is possible to send m msgs
• from v to w in time O(dm/k).

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Exercises (cont)

• Prove or disprove
• In a graph G(V,E),
• if dist(v,w)k and there are k2 edge-disjoint
  paths between the nodes v and w,
• then it is possible to send k2 msgs
• from v to w in time O(k).

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Global function computation
Goal Compute global function f(Xv1,...,Xvn)
where each node v holds input Xv Semigroup
function f 1. Well-defined for any input
subset 2. Associative and commutative ? Efficien
tly computable on tree T by convergecast
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Global function computation

• During the process
• The value sent upwards by each v in T
• value of function on inputs of its subtree Tv
• fv f(Yv) where Yv Xw w ? Tv
• Converge(f,X) process
• Leaf v sends Xv to parent
• Intermediate v with k children w1,...,wk
• - receives values fwi f(Ywi) from all children
• applies fv ? (Xv,fw1,...,fwk)

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Example Addition
105
49
50
13
29
13
29
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Example Maximum
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Global function computation

• Claim
• Assume f(Y) is represented in O(p) bits
• for every input set Y
• On tree T
• Message(Converge(f)) ?
• Time(Converge(f)) ?

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Pipelining
Separate broadcast / convergecast operations can
be efficiently pipelined Example Pipelining 3
Converge(max) operations to get Mi max Xi(v)
v leaf for i1,2,3
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Pipelining
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Pipelining
Lemma k global semi-group functions can be
computed on tree in time ?
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Level-synchronized BFS (Dijkstra)
Q. Prove the tightness of the message complexity
analysis of Dijkstra's algorithm, by
establishing the following Lower bound For
integers n and 1 ? D ? n-1, there exists n-node,
D-diameter graph G(V,E) on which the execution
of Dijkstra's algorithm requires ?(nDE)
messages.
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Level-synchronized BFS (Dijkstra)
Termination detection Modify the Distributed
Dijkstra algorithm so that the root can tell when
the process is completed (and the entire graph is
spanned by the constructed BFS tree)
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Distributed Depth-First Search
DFS Search process on G, traversing all
vertices, progressing over edges, with preference
to visiting new vertices
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Distributed Depth-First Search

• DFS algorithm
• Search starts at origin v0
• Whenever search reaches vertex v
• - If v has neighbors not visited so far,
• then visit one of them next.
• - Else return to the vertex
  from which visited first
• - If v v0 then end

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Distributed Depth First Search

• Fact DFS process visits every vertex in G.
• Search defines DFS tree, with v0 as root
• v's parent node from which v was visited first
• Sequential time complexity O(E)

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Direct distributed implementation

• Completely sequential
• one activity locus at any time.
• - Control carried via single message (token)
• traversing G in depth-first fashion

Note For v to know whether neighbor w was
visited or not, it must send message over edge
(v,w)
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Direct distributed implementation
? every edge must be explored
? both time and message complexities Q(E)
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Exercise

• Q.
• Modify the DFS algorithm to allow the traveler to
  complete the tour (visiting all nodes)
• faster than O(E).
• Analyze the time complexity of the modified
  algorithm and prove your bound.

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Synchronizers
Consider a 15-processor asynchronous G(V,E),
V0,,14, constantly running a synchronizer. v,
v' nodes in G. At time t, pulse counter Pv
27. What is the range of possible pulse numbers
at Pv' in following cases
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Synchronizers (cont)

1. G ring (with nodes arranged in order), v11,
   v'2, synchronizer used a.
2. G full balanced binary tree (4 levels), v
   root, v' one of the leaves, synchronizer used
   b.
3. The same as in (2), except both v and v' are
   leaves.

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Synchronizers (cont)

• Synchronizer used g.
• Clusters, spanning tree of each cluster,
  inter-cluster edges and locations of v, v' are as
  follows

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Synchronizer gaps
Gap of synchronizer n Tgap(n) maxv,p
t(v,p1)-t(v,p) (max length of period t(v,p)
that some processor v stays in some pulse p)
Tgap(n)
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Synchronizer gaps (cont)

• How large could Tgap be for
• synchronizer a
• synchronizer b
• synchronizer g
• For synchronizer a, design a scenario realizing
  the worst-case waiting time, Tgap(a)

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3-coloring bounded-degree graphs
Goal Color arbitrary bounded degree G
(D(G)O(1)) with D1 colors in time O(logn)
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Consistent orientation and MIS
Consistent orientation the ring edges are
oriented in a consistent manner (each node
identifies its left and right neighbors, and
each edge e (v,w) is marked as going left by
one of its endpoints and as going right by the
other. We have seen Given an MIS on the ring,
the nodes can be colored with 3 colors in a
single round.
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Consistent orientation and MIS

• Show that algorithm might fail if the ring does
  not enjoy consistent orientation
• Prove that on an anonymous ring (no IDs) without
  a consistent orientation,
• it is impossible to deterministically 3-color
  the nodes, even given an MIS.
• Prove that on a non-anonymous ring
• without a consistent orientation,
• it is still possible to deterministically
  3-color the vertices in a constant number of
  rounds given an MIS. (Try to use the smallest
  number of rounds.)

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Consistent orientation and MIS

1. What lower bound can be proved for the number of
   rounds required for computing MIS on a ring
   without consistent orientation? Specify precise
   constants (not only asymptotic lower bound).
2. What is the smallest n for which the lower bound
   you got in the previous question is greater than
   1? What is the smallest n for which the lower
   bound on 3-coloring is greater than 1?