CS 290N/219: Sparse matrix algorithms: Homework 3
Assigned October 19, 2009
Due by class Wednesday, October 28
1. [20 points]
Let
G
be the graph of the
n
vertex model problem, that is, a
k
by
k
grid graph
with
n
=
k
2
vertices.
Prove that there is some constant
c >
0 such that for
every
elimination
ordering on
G
, the filled graph
G
+
contains a complete subgraph with at least
c
√
n
vertices. (A
complete subgraph
is a set of vertices such that every pair is joined by an edge.)
Hint: Suppose you’re given an ordering for the vertices of
G
. Think of playing the graph game
in the given order, and consider the first time that you’ve either marked all the vertices in any
single row of the entire grid or else marked all the vertices in any single column.
2. [40 points]
(see Davis problem 6.13). An
incomplete
LU
factorization
is an approximate
factorization
A
≈
LU
, in which
L
and
U
are lower and upper triangular matrices whose product
is “approximately”
A
in some sense, but
L
and
U
have fewer nonzeros than the actual
LU
factors
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 Fall '09
 Chong
 Ilu, diagonal element ujj, socalled ILU factorization

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