Final Project

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Final Project

• On algorithms for permuting large entries to the
  diagonal of a sparse matrix
• Iain S. Duff and Jacko Koster

Arturo Gonzalez-Gutierrez
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Agenda

• Introduction
• Matching
• Weighted Matching
• Boottleneck Matching
• Scaling
• Experimental Results

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Introduction
Bipartite Matching Algorithms ( 5 3 1 2 4)
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Objective

• Compute a matching that corresponds to
• a permutation of a sparse matrix such that the
  product (or sum) of the diagonal entries is
  maximized.
• a permutation of rows and columns such that the
  smallest ratio between the absolute value of a
  diagonal entry and the maximum absolute value in
  its column is maximized (a permutation of the
  matrix such that the smallest diagonal entry is
  maximized)

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Matrix/Maximum Matching
M7lt9
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M-Alternating path/M-Augmenting path
M7
M8lt9
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Non-singular Matrix/Matching
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Non-singular Matrix/Perfect Matching
M9
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If Anxn is non-singular then there exists a
perfect matching M for GA and a permutation
matrix P.
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AP, PA

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If M is a matching and P is an M-augmenting path
then M?P(M\P)?(P\M) is a matching.
Proof
M7
M8
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• If P is an M-augmenting path then M?PM1
• If P is an M-alternating cycle path then
  M?PM

Proof
(a)
(b)
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Algorithm
j0 unmatched
j0 unmatched
i0 unmatched
i0
j1
M-augmenting path
i1
j2
ik-2
Jk-1
ik-1
Jk
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Algorithm

• for j0?Vc do
• jj0 pjnull iapnull
• B?
• repeat
• if there exists i?COL(j) and i is unmatched
  then
• iapi
• else
• if there exists i?COL(j)\B then
• BBi
• kmi pkj jk
• else
• jpj
• end if
• end if
• until iap?null or jnull
• if iap?null then augmented along path from node
  iap to node j0
• end for

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Weighted Matching
MAXIMIZATION MULTIPLICATIVE PROBLEM Find M that
corresponds to a permutation s that maximizes
MINIMIZATION ADDITIVE PROBLEM
min
BIPARTITE WEIGHTED MATCHING PROBLEM Find a
minimum weight perfect matching in an edge
weighted bipartite graph
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Bipartite Weighted Matching Problem

• C(cij) ? ?nxn ? GC(Vr,Vc,E) bipartite graph
  where each edge (i,j) ? E has weight cij
• If M is a minimum weight perfect matching then
  c(M)?c(M) for all possible maximum matching M.
• An M-augmenting path P starting at an unmatched
  column node j is called shortest if
  c(M?P)?c(M?P) for all other possible
  M-augmenting paths P starting at node j.

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Length of alternating path Pl(P)
c(M?P)-c(M)c(P\M)-c(M?P)
1
1
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3
3
l(P)c9,7-c5,7c5,6-c7,6c7,9
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9
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Bipartite Weighted Matching Problem

• M is extreme matching ? M doesnt allow any
  alternative cycle path with negative length
• If M is extreme then M is a perfect matching with
  minimum weight.
• If M is extreme and P is a shortest M-augmenting
  path then M?P is extreme also.

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Bipartite Weighted Matching Problem (LP)

• Find matrix

Minimize
Subject to
If there is a solution to LP then there is one
for which xij?0,1 and there exists a
permutation matrix X such that M(i,j)xij1 is
a minimum weight perfect matching.
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• M has minimum weight ?? dual variables ui and vj
  with
• uivi?cij for (i,j)?E
• uivicij for (i,j)?M
• Reduced weight matrix
• Reduced weight of matching M
• Reduced length of any M-alternating path P
• If (M?P) is a matching then its reduced weight

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Weighted Matching
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• Since for every edge (i,j)?M and
  contains no alternating paths P with negative
  length then for every
  principal leading subpath P of P.
• Shortest Augmenting Paths in a weighted bipartite
  graph G(Vr,Vc,E) can be obtained by a shortest
  alternating path tree.
• A shortest alternating path tree T is an
  alternating tree where each of whose paths is a
  short path in G.

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• For any node i?Vr?Vc, di is the length of the
  shortest path in T from node i to root.
• T is a shortest alternating path tree ?

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Dijkstra Algorithm (sparse variant)
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Complexity of the Algorithm

• Dijkstras Algorithm
• for dense graphs
• O(n2)
• for sparse problems
• O(n(n logk n t logk n n k logk n))
• O(n(tnk) logk n)

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Bottleneck matchingPermute rows and columns of
a sparse matrix A

• MAXIMIZATION PROBLEM
• Compute a permutation s that maximizes

MINIMIZATION PROBLEM
min
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• Given a matching M in the bipartite graph
  GC(Vr,Vc,E), the bottleneck of M is defined as
  c(M)max(i,j)?M cij
• Find Perfect (or maximum) bottleneck matching M
  for which c(M) is minimal c(M)?c(M) for all
  possible maximum matchings M.
• M is extreme matching ?M does not allow any
  alternating cycle path P for which c(M?P)ltc(M)

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A large initial extreme matching
j unmatched
i matched
j1 matched
i1 unmatched
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Dijkstra Algorithm (sparse variant)
j0 unmatched
j0 unmatched
j0 unmatched
j
j
j with lsp
i matched max(dj , )ltdi
i max(dj , )gtlsap
i unmatched max(dj , )ltlsap
new shorter alternating path
new shorter augmenting path
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Scaling

• By using the dual solution produced by the
  weighted matching algorithm
• uivi?cij for (i,j)?E
• uivicij for (i,j)?M

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• Reduce the amount of pivoting in the context of
  dense Gaussian elimination the more dominant the
  diagonal, the higher the chance that diagonal
  entries are stable enough to serve as pivots for
  elimination
• For iterative methods (Gershgorins theorem)
• This guarantees that iterative methods like
  Jacobi and Gauss Seidel will converge

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Time (secs) for matching algorithms
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IMPCOL E Chemical Engineering Plant Models,
Hydrocarbon separation problem. 225x225, 1308
entries.
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MAHINDASAustralian Economic Models. 1258x1258,
7682 entries.
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WEST0497 Chemical engineering plant
modelsRigorous flash unit with recycling.
497x497, 1727 entries.
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Number of iterations required by preconditioned
iterative methods after permutation
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Number of iterations required by some
ILUT-preconditioned iterative methods after the
matching reordering with and without reverse
Cuthill-McKee
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Reverse Cuthill-McKee ordering