Communication for a 512×12 factorisation on 16 processors
At its defaults it draws communication for a 512×12 factorisation on 16 processors. Two counts for the same factorisation. Rounds on the critical path: 48 for the column sweep, 4 for the reduction tree, 4 for Cholesky QR — a factor of n between the first and the other two. Words sent: 1170, 1170 and 2160 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
rounds-vs-words is one function in lib/figures/parallel.js —
messages — the count words cannot make, and what the fewest of them costs. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Two counts for the same factorisation. Rounds on the critical path: 48 for the column sweep, 4 for the reduction tree, 4 for Cholesky QR — a factor of n between the first and the other two. Words sent: 1170, 1170 and 2160 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 16
The arguments are the ones Memory bought with messages passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two counts for the same factorisation. Rounds on the critical path: 48 for the column sweep, 4 for the reduction tree, 4 for Cholesky QR — a factor of n between the first and the other two. Words sent: 1170, 1170 and 2160 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 2
The arguments are the ones Memory bought with messages passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two counts for the same factorisation. Rounds on the critical path: 12 for the column sweep, 1 for the reduction tree, 1 for Cholesky QR — a factor of n between the first and the other two. Words sent: 78, 78 and 144 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 32
The arguments are the ones Memory bought with messages passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two counts for the same factorisation. Rounds on the critical path: 60 for the column sweep, 5 for the reduction tree, 5 for Cholesky QR — a factor of n between the first and the other two. Words sent: 2418, 2418 and 4464 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 8
The arguments are the ones Memory bought with messages passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two counts for the same factorisation. Rounds on the critical path: 36 for the column sweep, 3 for the reduction tree, 3 for Cholesky QR — a factor of n between the first and the other two. Words sent: 546, 546 and 1008 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
p: 4
The arguments are the ones Memory bought with messages passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two counts for the same factorisation. Rounds on the critical path: 24 for the column sweep, 2 for the reduction tree, 2 for Cholesky QR — a factor of n between the first and the other two. Words sent: 234, 234 and 432 — the sweep and the tree send the same number, and the method with the fewest rounds sends the most.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
8 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a processor count the reduction tree is defined for
and Cholesky QR takes one reduction
each row block is at least as tall as it is wide
each slice is at least as tall as it is wide
matmul shapes agree
the sweep takes n log₂p rounds
the tree takes log₂p
while the one-reduction method sends more words than the tree
Against the rule
It draws a decomposition and prints its residual. It calls
choleskyQR, tsqrDistributed, householderDistributed,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Memory bought with messages
Holding four copies of the data instead of one is supposed to cut a matrix multiplication's communication by √4. Measured on a machine of 64 processors it costs 14% more traffic; at 576 it saves 44%, which is 72% of what the law promises. The memory is exactly four times, and that part is not asymptotic.
Where the flop count stopped predicting the timeThe message and the word
Three factorisations of one matrix on sixteen processors: 48 communication rounds, 4, and 4. The words sent are 1,170, 1,170 and 2,160 — so the method with the fewest rounds sends the most words, and the count that separates the three is the one no operation count can see.