What cannot be computed locally

1
What cannot be computed locally!

• by Fabian Kuhn, Thomas Moscibroda, and Roger
  Wattenhofer
• (PODC 2004)
• Presenter Yongwook Choi

2
Basic Problems

• Minimum vertex cover (MVC)
• A vertex cover for a graph G(V,E) is a subset of
  nodes V such that, for each edge, at least one
  of the end-points belongs to V.
• Minimum dominating set (MDS)
• A dominating set S is a subset of the nodes of a
  graph G such that all nodes of G are either in S
  or they have a neighbor in S.
• Maximal matching (MM)
• A maximal matching of a graph G is a maximal set
  of edges which do not share common end-points.
• Maximal independent set (MIS)
• A maximal independent set of a graph G is a
  maximal set of non-adjacent nodes.

3
Main Results

• Lower bounds for the distributed approximation of
  MVC and MDS.
• In k communication rounds, MVC and MDS can only
  be approximated by factors
• The number of rounds required in order to achieve
  a constant or even only polylogarithmic
  approximation ratio is at least
• The same time lower bounds for MM and MIS can be
  shown by a simple reduction.
• Previously the best known lower bound was O(logn)

4
Main Results

• Related work
• Kuhn and Wattenhofer, 2003
• In k rounds, MVC can be approximated by a factor
  O(?1/k).
• For a polylogarithmic approximation ratio,
• kO(log?/loglog?)
• The lower bound is tight.
• For a constant approximation ratio,
• kO(log?)
• The lower bound is tight up to a factor
  O(loglog?).

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Model

• Classic message passing model
• Undirected graph G(V,E)
• V processors
• E communication channels
• Each node can send an arbitrarily long message to
  each of its neighbors
• Local computations are for free
• Initially, nodes have no knowledge about the
  network graph
• Each nodes only knows its own unique identifier

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Model

• In k rounds, a node v may collect the IDs and
  interconnections of all nodes up to distance k
  from v
• Tv,k Topology seen by v after k rounds
• L(Tv,k) Labelling (or assignment of IDs) of Tv,k

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Model

• The best algorithm
• Deterministic algorithms
• Collect its k-neighborhood
• Make a decision based on (Tv,k,L(Tv,k))
• Randomized algorithms
• Also depend on the randomness

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Model

• This model is the strongest possible model when
  proving lower bounds for local computations
• Focuses entirely on the locality
• Abstracts away other issues
• Need for small messages
• Fast local computations
• The lower bounds are true consequence of locality

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Outline of the Proof

• Construct a graph Gk(V,E) for each positive
  integer k.
• Gk contains a bipartite subgraph S.
• The goal is to construct Gk in such a way that
  all nodes in S see the same topology within
  distance k.
• In a globally optimal solution
• All edges of S may be covered by nodes in C1.
• In a local algorithm
• The decision of whether a node joins the vertex
  cover depends only on its local view.
• Because adjacent nodes in S see the same
  topology, every algorithm adds a large portion of
  nodes in C0 to its VC.

Gk
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The Cluster Tree

• The nodes of graph Gk can be grouped into
  disjoint sets (clusters) which are linked to each
  other as bipartite graphs.
• Cluster tree CTk (C,A)
• C set of clusters
• A doubly labelled arcs
• e.g. L(C0,C1) (d0,d1)

C0 d0 C1 d1
CT1
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The Cluster Tree

• CTk is recursively defined
• Given CTk-1, we obtain CTk in two steps.
• For each inner-cluster Ci, add a new leaf-cluster
  Ci with L(Ci,Ci)(dk,dk1)
• For each leaf-cluster Ci with L(Ci,Ci)(dp,dp1),
  add k new leaf-clusters Cj with
  L(Ci,Cj)(dj,dj1) for j0k, j?p1

• Level of a cluster
• Distance to C0
• CTk defines just the number of neighbors for each
  node in each cluster.
• CTk does not specify the actual adjacencies.

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The Lower Bound Graph

• Later, we will prove that the topologies seen by
  nodes in C0 and C1 are identical.
• The proof is greatly simplified if each nodes
  topology is a tree (rather than a general graph)
  because we do not need to worry about cycles.
• g(G) the girth of a graph G
• Length of the shortest cycle in G
• We want to construct Gk with girth at least 2k1.
• All nodes see a tree in k communication rounds.

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The Graph Family D(r,q)

• Proposed in Explicit construction of graphs with
  an arbitrary large girth and of large size
• by Lazebnik and Ustimenko
• For given an integer r1 and a prime power q,
• D(r,q) defines a bipartite graph with 2qr nodes
  and girth g(D(r,q))r5.
• The nodes in P and L are labelled by the
  r-vectors over the finite field Fq

D(r,q)


P


L
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The Graph Family D(r,q)

• Lemma 3.2
• For all (p) and l1, there is exactly one l such
  that l1 is the first coordinate of l and that
  (p) and l are connected by an edge in D(r,q).
• Proof
• r-1 equations define a linear system for the r-1
  unknown coordinates of l.
• The matrix corresponding to this system is a
  lower triangular matrix with non-zero elements in
  the diagonal.
• The solution of this matrix is unique.

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The Construction of Gk

• Construct an instance Gk of the cluster tree
• Gk may have the minimum possible girth 4.
• Gk is a bipartite graph with odd-level clusters
  in one set(V1) and even-level clusters in the
  other set(V2).
• m the number of nodes in the larger of the two
  partitions of Gk.
• Choose the smallest prime power qm.
• In both partitions of Gk, uniquely label all
  nodes over Fq.

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The Construction of Gk

• Set r2k-4.
• Construct D(r,q)
• D(r,q) is a bipartite graph with 2qr nodes and
  girth at least 2k1.
• Gk is constructed as a subgraph of D(r,q).
• For each p1 (in V1) in Gk, put qr-1 nodes
  (p)(p1,) in Gk.
• Put an edge between (p)(p1,) and ll1, if
  both of the followings hold.
• (p) and l are connected in D(r,q).
• p1 in V1 and l1 in V2 are connected in Gk.
• Remove all the nodes without incident edges.

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The Properties of Gk

• Lemma 3.3 first part
• Gk has at most 2mq2k-5 nodes and girth at least
  2k1.
• Proof
• Each node in Gk, qr-1 nodes are created.
• The number of nodes in Gk is at most 2m.
• Gk is a subgraph of D(r,q).
• g(D(r,q))2k1.

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The Properties of Gk

• Lemma 3.3 second part
• Gk is a cluster tree with the same degrees di as
  in Gk.
• Proof
• In Gk, consider two neighboring clusters C1 in
  V1 and C2 in V2.
• The clusters C1 and C2 consist of all nodes (p)
  and l which have their first coordinates equal
  to the labels of the nodes in C1 and C2,
  respectively.
• If each node in C1 have d neighbors in C2, then
  nodes in C1 have d neighbors in C2 by Lemma 3.2.

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Equality of Views

• In Gk, two adjacent nodes in clusters C0 and C1
  see exactly the same topology (same tree) within
  distance k.

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Equality of Views
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G3
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G3
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Analysis

• We can derive the lower bounds on the
  approximation ratio of k-local MVC algorithms.
• OPT optimal solution for MVC
• ALG solution computed by any algorithm
• Main observation
• Adjacent nodes in C0 and C1 have the same view.
• Every algorithm treats the nodes the same way.
• ALG contains a significant portion of the nodes
  in C0.
• OPT may not have any node in C0.

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Analysis

• Lemma 3.13
• Let ALG be the solution of any k-local MVC
  algorithm.
• When applied to Gk, ALG contains at least half of
  the nodes of C0 in the worst case (in expectation
  for randomized algorithms).
• Proof for deterministic algorithms
• The decision whether a given node enters the
  vertex cover depends solely on its topology and
  the labelling.
• Assume that the labelling is chosen uniformly at
  random.
• Let v0 and v1 be two adjacent nodes from C0 and
  C1, respectively.
• Let pi denote the probability that vi ends up in
  VC when algorithm A operates on the randomly
  chosen labelling.
• p0p1 because they see the same topology and the
  same distribution on the labelling.

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Analysis

• Proof for deterministic algorithms (cont.)
• For a feasible solution, at least one of the two
  nodes must be in VC.
• This implies p0p11 and p01/2.
• At least half of the nodes in C0 are chosen by
  algorithm A in average by the linearity of
  expectation.
• Therefore, for every deterministic algorithm,
  there is at least one labelling for which at
  least half of the nodes of C0 are in VC.
• Proof for randomized algorithms
• Use Yaos minimax principle
• The expected number of nodes chosen by a
  randomized algorithm cannot be smaller than the
  expected number of nodes chosen by an optimal
  deterministic algorithm for an arbitrarily chosen
  distribution on the labels.

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Analysis

• ALG n0/2
• OPT n-n0
• This is because the nodes of cluster C0 are not
  necessary to obtain a feasible vertex cover.
• We define didi (i0,,k1) for some value d.
• Lemma 3.14

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Analysis

• Proof
• The number of nodes per cluster decreases for
  each additional level by a factor d.
• A cluster on level l contains n0/dl nodes.
• Each cluster has at most k1 neighboring clusters
  on a higher level.
• nl number of nodes on level l

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Analysis

• Relationship between d and n0
• n0 size of C0 in Gk
• Set n0 d2k1 so that there is no conflict
  during the construction of Gk.
• If we assume that k1d/2, we have n2n0 by
  Lemma 3.14.
• Applying the construction of Gk, we get n0
  n0q2k-5, where q is the smallest prime power
  such that qn.
• q lt 2n 4n0
• Finally, we get

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Analysis

• Theorem 3.15
• There are graphs G, such that in k communication
  rounds, every distributed algorithm for the MVC
  problem on G has approximation ratios at least
• for some constant c1/4, where n denotes the
  number of nodes and ? denotes the highest degree
  in G.
• Proof

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Analysis

• Theorem 3.16
• In order to obtain a polylogarithmic or even
  constant approximation ratio, every distributed
  algorithm for the MVC problem requires at least
• communication rounds.

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Analysis

• Proof

by theorem 3.15

• For every polylogarithmic term a(n), there is a
  constant ß such that approximation ratio is at
  least a(n).
• We can show the second lower bound similarly.

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Lower Bounds by Reductions

• Using the lower bound for MVC, we can obtain
  lower bounds for several other graph problems.
• Time lower bounds
• For the construction of maximal matchings (MM)
  and maximal independent sets (MIS)
• For the approximation of minimum dominating set
  (MDS)
• Use reductions

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Lower Bounds for MM and MIS

• Theorem 4.1
• There are graphs G on which every distributed,
  possibly randomized algorithm requires time
• to compute a maximal matching.
• The same lower bounds hold for the construction
  of maximal independent sets.
• Proof for MM
• The set of all end-points of the edges of a MM
  form a 2-approximation for MVC.
• This means that computing a MM achieves a
  constant approximation ratio for MVC.
• Thus, the lower bound directly follows from
  theorem 3.16.

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Lower Bounds for MM and MIS

• Proof for MIS
• Consider the line graph L(Gk) of Gk.
• L(G) line graph of G
• Nodes of L(G) are the edges of G.
• Two nodes in L(G) are connected if two
  corresponding edges in G are incident.
• The MM problem on G is equivalent to the MIS
  problem on L(G).
• k rounds on L(G) can be simulated in kO(1)
  rounds on G.
• The number of nodes in L(Gk) is less than n2/2.
• The maximum degree of L(Gk) is less than 2?.
• Therefore, the lower bounds have the same order.

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Lower Bounds for MDS

• Theorem 4.2
• There are graphs G, such that in k communication
  rounds, every distributed algorithm for the
  minimum dominating set problem on G has
  approximation ratios at least
• for some constant c, where n and ? denote the
  number of nodes and the highest degree in G,
  respectively.

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Lower Bounds for MDS

• Proof
• We show that every MVC instance can be seen as a
  MDS instance with the same locality.
• G(V,E) graph for MVC problem
• Construct G(V,E) for MDS problem
• k communication rounds on one of the two graphs
  can be simulated by kO(1) rounds on the other
  graph.
• MDS on G is exactly as hard as MVC on G.

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Lower Bounds for MDS

• Corollary 4.3
• To obtain a polylogarithmic or constant
  approximation ratio for minimum dominating set,
  there are graphs on which every distributed
  algorithm needs time
• Proof
• The same as the proof for theorem 3.16

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Questions or Comments?
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References

• Fabian Kuhn, Thomas Moscibroda, and Roger
  Wattenhofer. What cannot be computed locally!,
  In Proc. of the 23rd ACM Symposium on the
  Principles of Distributed Computing (PODC), 2004.
• F. Lazebnik and V. A. Ustimenko. Explicit
  Construction of Graphs with an Arbitrary Large
  Girth and of Large Size, Discrete Applied
  Mathematics, 1995.
• N. Linial. Locality in Distributed Graph
  Algorithms, SIAM Journal on Computing, 1992.
• M. Naor and L. Stockmeyer. What Can Be Computed
  Locally?, In Proc. of the 25th Annual ACM Symp.
  on Theory of Computing (STOC), 1993.