This Lecture was Given at

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This Lecture was Given at

• Micha Sharirs Geomtery Seminar, Tel-Aviv
  University, March 8, 2006
• Haifa Interdiciplinary Graph Theory Workshop,
  Haifa University, May 29, 2006
• SoCG 06, June 5, 2006

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Lecture Abstract

• Title Colored Range Searching via Matrix
  Multiplication
• Abstract
• In a Colored Range Searching Problem, one is
  given a set of points, each
• point colored in one of c colors, and a set of
  hypercubes, each colored in
• one of the same c colors. The goal is to report
  all pairs of colors
• (c_1,c_2) such that a point of color c_1 is
  contained in a cube of color
• c_2. The same question can be asked for
  halfspaces, triangles, etc.
• In a Sparse Rectangular Matrix Multiplication
  Problem, one is given a
• matrix A with n rows and at most m non-zero
  entries (the number of columns
• is not significant). The goal is to compute
  AAT, i.e. the product of A
• and its transpose.
• It is relatively easy to see that the Colored
  Range Searching Problem is
• harder than the Sparse Rectangular Matrix
  Multiplication Problem, in the
• sense that one can solve the latter using the
  former. In this talk we show
• that this relation goes both ways. In other
  words, we prove that the
• optimal time it takes to solve the former problem
  can be stated in terms

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Slide with more info - History of Colored Problems

• Defined by Gupta, Janardan, Smid in beginning of
  90s.
• Some survey articles by GJS
• Mostly geometrical techniques
• Try to transform to an input where each pair of
  color intersects only once
• Some tricks to allow output-sensitivity.
• A paper on the batched problem by Bozanis,
  Kitsios,Makris and Tsakalidis
• (always O(nxky) where x2y2. Our algorithms
  probably reduce to their when using omega3) is
  this true?

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Important to understand

• The difference between the so-called single-shot
  problem and the other problems is not that one is
  single-shot and the others are not. Indeed, the
  others have single-shot versions too, and the
  single-shot problem can have a DS version. The
  different is that the single-shot problem is
  colors vs. colors (and so the output size os
  O(c2)), and the other problems are colors vs.
  objects (and the output size is e.g. O(n) in the
  counting version)

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Research Subjects

• Output-sensitivity
• What happens if every column has sqrt(n) 1s? Find
  a computational model to get a lower bound.
• VC-dim, finite-representation objects, etc. Put a
  reference to the compression of graphs papers

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I. SMM CI

• Suppose that the hypercubes are degenerate so
  they are also points.
• So report colored points vs. colored points (no
  geometry!)
• Remember m points, n colors

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SMM CI

• This is like computing union of bi-cliques over n
  vertices with total size (number of vertices) m

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SMM CI

• Computing union of bi-cliques SMM

Yellow gives
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SMM CI

• Computing union of bi-cliques SMM

Yellow gives
10
SMM CI

• In general SMM union of bi-cliques over n
  vertices with total size m
• (we will use this again later for the upper
  bound!)

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II. CI SMM

• Method
• Let G(S,T) be the intersection graph
• Suppose we could prove that G(S,T) can always be
  written as the (non-disjoint) union of bi-cliques
  with total size m. ( an efficient algorithm to
  generate this partition). Then we are finished.
• Theorem The intersection graph of m points and m
  hypercubes can be written as the disjoint union
  of bi-cliques with total size mlogdm

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CI SMM

• Method
• Let G(S,T) be the intersection graph
• Compute decomposition of G
• For each bi-clique compress the colors in the two
  sides now the number of vertices is n, and the
  total size is still m.
• Now that we have bi-cliques over the colors
  rather than over the geometrical objects
  compute union of bi-cliques

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CI SMM

• Theorem The intersection graph of m points and m
  hypercubes can be written as the disjoint union
  of bi-cliques with total size mlogdm
• Proof
• Compute range tree over points.
• For each hypercube, find canonical subset
• The bi-cliques are defined by the vertices

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Proof for 1 dimension
p5, p6, p7, p8
p1, p2
p1
p2
p3
p4
p5
p6
p7
p8
1-d range tree
p1
p2
p3
p4
p5
p6
p7
p8
15
p3 I1
p3, p4 I2
p5, p6, p7, p8 I3, I2
p2 I1,I2
p1
p2
p3
p4
p5
p6
p7
p8
1-d range tree
p1
p2
p3
p4
p5
p6
p7
p8
I3
I1
I2
16
p3 I1
p3, p4 I2
p3
I2
p4
p5, p6, p7, p8 I3, I2
p2 I1,I2
p5
p1
p2
p3
p4
p5
p6
p7
p8
I3
p6
p7
I2
These are the bi-cliques. Total number of labels
O(mlogm)! QED.
p8