CPSC 689: Discrete Algorithms
for Mobile and Wireless Systems
Spring 2009
Prof. Jennifer Welch
Lecture 23
 Topic:
 Lower Bounds for Dominating Sets and Related
Problems
 Sources:
 Kuhn, Moscibroda and Wattenhofer, "What Cannot
Be Computed Locally!"
 MIT 6.885 Fall 2008 slides
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Locality
 Locality means that nodes only have to
communicate (even indirectly) with nodes
that are close by
 Desirable property of a distributed
algorithm:
 local algorithms have (the possibility of) low
time complexity
 why bother far away nodes?
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Locality
 k communication
rounds means
being restricted to
a locality radius k.
1 rounds
2
3
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Locality
 Can we find local algorithms for various
distributed problems?
 means time complexity (number of rounds) is
independent of network size
 A few positive results, e.g.:
 Naor & Stockmeyer: studied a class of problems called
locally checkable labelings and showed there are nontrivial LCL problems that have local algorithms,
including a variant of dining philosophers
 What about negative results (lower bounds)?
 Linial: coloring on a ring takes (log*n) rounds
 What about for dominating set and related problems?
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Minimum Vertex Cover
 Minimum Vertex Cover problem: Given a graph,
find smallest subset S of vertices (nodes) such
that every edge is "covered" by a node in S (at
least one endpoint is in S)
 NP-complete
 consider polynomial time approximation algorithms
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Overview of [KMW] Results
 Any k-round MVC algorithm has an approximation
ratio that is (nc/k*k/k), where n is number of nodes
and c is a constant > 1/4
 To ensure that the approximation ratio is no more
than poly-log, k has to be at least
((log n / log log n)), which is not local
 Any k-round MVC algorithm has an approximation
ratio that is (1/k /k), where  is the maximum degree
 To ensure that the approximation ratio is no more
than poly-log, k has to be at least
(log  / log log ), which is not local
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Some Special Case Graphs
 Consider a ring:




minimum VC consists of every other node
constant-time approx algorithm is to include every node
approx ratio w.r.t. n is 2
Generalize to a d-regular graph
 Consider a tree:
 minimum VC consists of every other node down each
branch
 constant-time approx algorithm is to include every nonleaf node
 approx ratio w.r.t. n is 2
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Some More Special Case Graphs
 Consider graphs with constant max degree :
 constant time approx alg is to include every node
 approx ratio w.r.t.  is constant
 Consider graphs that contain nodes with high
degree (say, (n)):
 then diameter is small (say, O(1)), so in constant time,
an alg can learn the entire graph and choose exactly
which nodes to include
 approx ratio is 1
 To show non-locality property, need to consider
more complicated graphs…
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Intuition for Locality-Based Lower
Bounds
 In k rounds of communication (time k), every node
can collect information about its k-neighborhood
 Hence, the solution of a node v in a distributed kround computation can only depend on the k-hop
neighborhood of v
 If two nodes u and v have the same k-hop
neighborhoods, they will make the same decision:
the execution of a k-round algorithm looks the
same to both nodes
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Example for Locality-Based Lower
Bound
 How to prove such a lower bound?
 Let’s look at case k=2 to get the basic
intuition
 After 1 round, nodes know their neighbors
 After 2 rounds, nodes know the neighbors
of their neighbors
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Two-Round Lower Bound
m3
…
…
m
…
…
m
…
m2
complete
m nodes
…
m2
m2-1
m nodes
…
same view
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Hint of Proof
 Construct graph Gk for each k > 0 containing a
bipartite subgraph S with node set C0 U C1
 C0 has n0 nodes, each with d0 neighbors in C1
 C1 has n1 nodes, each with d1 neighbors in C0
 n1 = n0*d0/d1
C1, n1 = 8, d1 = 2
C0, n0 = 4, d0 = 4
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Hint of Proof
 In a globally optimal solution, all edges of S (the
bipartite graph) can be covered by choosing all
nodes of C1, and none of C0, to be in the VC
 But in a local algorithm, decision can only be
made based on k-neighborhood
 Construct Gk so that two adjacent nodes (one in
C0 and one in C1) have the same k-neighborhood
and thus do the same thing (both join the VC)
 Since symmetry cannot be broken in only k
rounds, suboptimal local decisions are made and
a suboptimal approximation ratio achieved
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Constructing Gk
 The heart of the paper is recursive
construction of Gk with high degree of
symmetry
 See Appendix for G3
 What do we do with Gk? Have to consider
what happens with node IDs
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Handling Node IDs
 Assume random node ID assignment with IDs
from {1,…,N}
 If nodes u and v see same topology up to
distance k:
 Every possible ID assignment is equally probable
 Probability to see a particular ID assignment equal for u
and v
 u and v make the same decision with the same
probability
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Handling Node IDs
 Deterministic algorithms: there exists a
node assignment for which solution is at
least as bad as expected value with
random IDs
 Randomized algorithms: Same bound
using Yao’s principle
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Hints on Rest of Proof
 Lemma: Any (randomized or deterministic) kround distributed algorithm, when run on Gk, puts
at least half the nodes of C0 into the VC.
 Proof is based on constructed properties of Gk
and previous discussion about handling IDs.
 So approx ratio is at least (n0/2) / (n – n0), since
optimal solution does not need any node in C0
 Do some math to show that the construction of Gk
can be tweaked to ensure that n0 is sufficiently
large relative to n to show the claimed lower
bounds w.r.t. n and .
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Relationship to Dominating Sets
 Theorem: Every (randomized or deterministic) kround distributed algorithm for MDS has same
asymptotic lower bounds on approx ratio as for
MVC: (nc/k*k/k) and (1/k/k).
 Proof: By reduction.
 Let A be a k-round alg for MDS with approx ratio R.
 Show how to use ADS as a subroutine in algorithm AVC
to approximate MVC with only a constant number of
extra rounds
 Analyze the resulting approx ratio for the MVC problem
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Reduction
Here is algorithm AVC:
1. Suppose input graph for MVC is G'
2. Simulate another graph G
 see next slide
3. Call MDS approx alg ADS on G
4. ADS returns some set of nodes S
5. Return S as an approx MVC for G'
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Reduction
 Transform G' into G:
a
b
d
c
a
ab
b
ad
bd
bc
d
cd
c
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VC to DS
 Any VC of G' is a DS of G:
a
b
d
c
red nodes
cover all edges
a
ab
b
ad
bd
bc
d
cd
c
red nodes
cover all nodes
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DS to VC
 Take any DS of G, replace any green node with a nongreen neighbor; result is a VC of G'
a
b
d
c
red nodes
cover all edges
a
ab
b
ad
bd
bc
d
cd
c
red nodes
cover all nodes
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Relating Quality of Approximations
 Algorithm ADS returns S, a DS of G that is
at most R times as large as an optimal DS
of G
 Size of optimal DS of G is ≤ size of optimal
VC on G'
 since every VC of G' forms a DS of G
 Thus S is a VC of G' that is at most R times
as large as an optimal VC of G'
 By MVC lower bound R must be at least …
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Summary: Lower Bound
 Lower bound shows that the time-approximation
trade-off of the existing algorithm is not too far off the
optimum (there still is a significant gap…)
 By a reduction, the time lower bound for polylog
approximations also holds for the apparently unrelated
problem of computing a maximal independent set
 Remark: The lower bound is obtained by using very
special graphs. This is definitely not how wireless
network graphs look! In fact, for special graph classes,
we can do better.
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