Approximation Algorithms for Representative Points Problem of Clusters

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Approximation Algorithms for Representative
Points Problem of Clusters

• Sanpawat Kantabutra
• The Theory of Computation Group
• Computer Science Dept.
• Chiang Mai University

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Outline

• Definitions
• Non-Existence of Absolute Performance Guarantee
• MST-Based Approximation Algorithm
• Greedy Approximation Algorithm
• Performance Guarantees
• Open Problems

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Dissimilarity

• Let d(x, y) denote the Euclidean distance of
  points x and y.

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?-Clustering Problem

• Let S be a set of n d-dimensional points and ? a
  real number.
• Want to partition S into l clusters C1, C2,,Cl
  s. t.?x,y?Ci, ? z1,z2,,zm ?Ci, x??y, s. t.
• d(x,z1)lt ?,d(zt,zt1)lt
  ?,d(zm,y)lt ?
• where Ci is a maximal cluster having this
  property, ??Ci S, Ci?? Cj ?? when i ?? j

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Example of ?-Clusters
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?-Clustering Problem of Order K

• Let S be a set of n d-dimensional points and ? a
  real number.
• Want to partition S into l clusters C1, C2,,Cl,
  ? Ci ?? S, s. t. for each Ci ? Gj, ?1 ? j ? m,
  there exists a path of subsets G1,G2,,Gm and 1m
  constitutes all the Gj so that ?x?Gj, ?k-1
  distinct y ?Gj d(x,y)lt?where Gj ?Gj1 ? ? and
  k gt 1, and ?x?Ci, ?y?Cj, i ? j, d(x,y) ? ?.

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Example of ?-Clusters of Order 3
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Representative Points

• Let S ?Ci of d-dimensional points. Given
  disjoint clusters C1,C2,,Cl and sets R1,R2,,Rl,
  Ri ? Ci, we then say that Ri represent Ci iff
• ?x?Ci, ?ri?Ri, ?rj?Rj, i?j, d(ri,x)ltd(rj,x)

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Example of Representative Points
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No Absolute Performance Guarantee

• Theorem 1. If P?NP, no polynomial time
  approximation algorithm A for any instance I can
  solve ?-REP with A(I)-OPT(I) ? r, for any fixed
  r and any optimum solution OPT(I), where OPT(I),
  A(I)?N, and r?N.

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MST-Based Approximation Algorithm (I)

• Modified Depth-First Search(v,R)
• 1. If ((Pred(v) ? R) AND (v is not the root))
• 2. R R ? v
• 3. Mark v as visited
• 4. For all nodes i adjacent to v not visited
• 5. Modified Depth-First Search(i,R)

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MST-Based Approximation Algorithm (II)

• INPUT l ?-clusters (of order k) C1,C2,. . . ,Cl
• OUTPUT A set R Ri of representative points
• MST-Based Approximation Algorithm
• 1. Let Ri ? for all i 1l
• 2. For each Ci
• 3. Compute distances of all pairs of points
  in Ci
• 4. Find a minimum-cost spanning tree MSTi
• from Ci where points and distances
  become
• nodes and edges respectively

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MST-Based Approximation Algorithm (III)

• 5. For each MSTi
• 6. Let v be a leaf node in MSTi and the root
  of DFS
• 7. Modified Depth-First Search(v,Ri)
• 8. Replace every leaf node x ? Ri with
  Pred(x)
• 9. Output R Ri

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Algorithms Correctness

• Proposition 1. The MST-based approximation
• algorithm produces a set R Ri of
  representative points of size in the worst
  case where n is the number of input points.

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Single Cluster Representation Problem

• Given a set S ? Ci of n points in
  d-dimensional space where Ci is a ?-cluster and a
  cluster number h, the single cluster
  representation problem is to find a
  representative set Rh ? Ch such that
• ?y?Ch ?r?Rh ?x?S - Ch d(r,y)ltd(x,y)

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Heuristic Representative Algorithm (I)

• INPUT ?-clusters Ci and h where 1 ? h ? l
• OUTPUT Rep. set Rh
• Rh ?
• While (Ch ? ?)
• Pmax ?
• For all r ? Ch
• P ?

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Heuristic Representative Algorithm (II)

• P ?
• For all y ? Ch
• If (CheckCond(r,y,S-Ch))
• P P ? y
• If (PgtPmax)
• Pmax P
• rmax r
• ChCh-Pmax, SS-Pmax, RhRh ? rmax
• return Rh

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Algorithms Correctness

• Theorem 2. The heuristic representative
  algorithm finds a representative set Rh ? Ch for
  a single cluster representation problem according
  to Definition 5.

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Greedy Approximation Algorithm

• INPUT ?-clusters C1,C2,. . . ,Cl S ?Ci
• OUTPUT A set T of representative sets Ri
• T ?
• For h 1 to l
• Heuristic Representative Algorithm(Ci,h,Rh)
• T T ? Rh
• Return T

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Algorithms Correctness

• Theorem 3. This algorithm finds a set of
  representatives Ri for all ?-clusters Ci, 1 ?
  i ? l, according to the Definition 4.

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Performance Guarantee

• Theorem 4. Let Agreedy denote the greedy
  approximation algorithm and Rgreedy the
  performance ratio. Then, for all input instances
  I,
• Rgreedy(I) ?
  
• where Cmax is the largest ?-cluster and k is
  the order of the ?-clusters.

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Tight Instance
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Open Problems

• MST-Based Approximation is in NC?
• Is the greedy-based approximation scheme the best
  one? What if we change the analysis of the lower
  bound or the lower bound itself?
• Does the constant relative performance guarantee
  exist for this problem?
• Is the greedy approximation problem is
  P-complete?

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Questions and Answers
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