CS 290H Lecture 8 Directed graphs and triangular solves

1
CS 290H Lecture 8Directed graphs and triangular
solves

• Pick up course reader in CS department office
  Friday
• Reading assignment
• Sparse matrices in Matlab (course reader 1)
• Sparse partial pivoting in time proportional to
  arithmetic ops (course reader 3).
• Final project
• Either an algorithms experiment, or an
  application experiment, or a survey paper
  and talk
• Experiments can be solo or group surveys are
  solo
• Suggested term projects on course web site
• Choose a project by next week email me or come
  talk about it

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Symmetric positive definite systems ALLT

• Preorder
• Independent of numerics
• Symbolic Factorization
• Elimination tree
• Nonzero counts
• Supernodes
• Nonzero structure of R
• Numeric Factorization
• Static data structure
• Supernodes use BLAS3 to reduce memory traffic
• Triangular Solves

O(nonzeros in L)
O(flops)

• Result
• Modular gt Flexible
• Sparse Dense in terms of time/flop

3
Triangular solve x L \ b

• Column oriented
• x(1n) b(1n)for j 1 n x(j)
  x(j) / L(j, j)
• x(j1n) x(j1n)
  L(j1n, j) x(j) end

• Row oriented
• for i 1 n x(i) b(i)
• for j 1 (i-1)
• x(i) x(i) L(i, j) x(j) end
  x(i) x(i) / L(i, i)end

• Either way works in O(nnz(L)) time
• If b and x are dense, flops nnz(L) so no
  problem
• If b and x are sparse, how do it in O(flops) time?

4
Directed Graph
A
G(A)

• A is square, unsymmetric, nonzero diagonal
• Edges from rows to columns
• Symmetric permutations PAPT

5
Directed Acyclic Graph
A
G(A)

• If A is triangular, G(A) has no cycles
• Lower triangular gt edges from higher to lower s
• Upper triangular gt edges from lower to higher s

6
Directed Acyclic Graph
A
G(A)

• If A is triangular, G(A) has no cycles
• Lower triangular gt edges from higher to lower s
• Upper triangular gt edges from lower to higher s

7
Depth-first search and postorder

• dfs (starting vertices)
• marked(1 n) false
• p 1
• for each starting vertex v do visit(v)
• visit (v)
• if marked(v) then return marked(v)
  true
• for each edge (v, w) do
  visit(w)
• postorder(v) p p p 1
• When G is acyclic, postorder(v) gt postorder(w)
  for every edge (v, w)

8
Sparse Triangular Solve
G(LT)
L
x
b

• Symbolic
• Predict structure of x by depth-first search from
  nonzeros of b
• Numeric
• Compute values of x in topological order

Time O(flops)
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Sparse-sparse triangular solve x L \ b

• Column oriented
• dfs in G(LT) to predict nonzeros of xx(1n)
  b(1n)for j nonzero indices of x in
  topological order x(j) x(j) / L(j, j)
• x(j1n) x(j1n) L(j1n, j) x(j)
  end

• Depth-first search calls visit once per flop
• Runs in O(flops) time even if its less than
  nnz(L) or n
• Except for one-time O(n) SPA setup

10
Structure Prediction for Sparse Solve

G(A)
A
x
b

• Given the nonzero structure of b, what is the
  structure of x?

• Vertices of G(A) from which there is a path to a
  vertex of b.