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Merge pull request #637 from dotCooCoo/fix/linesweeper-sweep-doc-left-scan

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linesweeper: the left intersection scan in docs/sweep.typ compares with alpha^(i+1)
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Dylan M. Taylor authored and GitHub committed 2026-11-14 11:19:24 -05:00
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NOTICE
+4 -3
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@@ -254,9 +254,10 @@ applications retain their own document-font selection and licensing policy.
linesweeper
linesweeper 0.5.0 (https://crates.io/crates/linesweeper/0.5.0), MIT OR Apache-2.0, is stored in
vendor/linesweeper with its upstream license files and one local patch. Its manifest names Joe Neeman
as author, and its LICENSE-MIT reads "Copyright (c) 2018 Raph Levien". See
vendor/linesweeper/README.craft.md and vendor/linesweeper/SOURCE.json for provenance, the local patch
vendor/linesweeper with its upstream license files and local patches, to src/sweep/sweep_line.rs and
docs/sweep.typ. Its manifest names Joe Neeman as author, and its LICENSE-MIT reads "Copyright (c) 2018
Raph Levien". See vendor/linesweeper/README.craft.md and vendor/linesweeper/SOURCE.json for
provenance, the local patches
and the files left out.
Platform lifecycle source provenance
vendor/linesweeper/README.craft.md
Vendored
+10 -6
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@@ -1,21 +1,25 @@
# Local linesweeper 0.5.0 copy
This is the crates.io linesweeper 0.5.0 package (MIT OR Apache-2.0) with some files left out and the
local patch listed below. `.cargo_vcs_info.json` names the commit it was packaged from,
`824b0791a750f7b27b8edcbb6c6244d0e629875e`. That commit is not in the GitHub repository the manifest
names, which is archived; its README says development moved off GitHub.
local patches listed below, to `src/sweep/sweep_line.rs` and `docs/sweep.typ`. `.cargo_vcs_info.json`
names the commit it was packaged from, `824b0791a750f7b27b8edcbb6c6244d0e629875e`. That commit is not
in the GitHub repository the manifest names, which is archived; its README says development moved off
GitHub.
Every file here except this one, `SOURCE.json` and the patched file below is byte for byte the file
in the `.crate` archive. `SOURCE.json` records the hash of the `.crate` download
Every file here except this one, `SOURCE.json` and the two patched files below is byte for byte the
file in the `.crate` archive. `SOURCE.json` records the hash of the `.crate` download
(`source_archive_sha256`) and, for every file kept, the hash in the package (`source_sha256`) and the
hash here (`sha256`). `upstream_repository` is null because no public repository this copy can name
holds `upstream_revision`. `local_patches` describes the local patch.
holds `upstream_revision`. `local_patches` describes each local patch.
## Local patches
| File | Change |
|---|---|
| `src/sweep/sweep_line.rs` | The doc comment of `SweepLine::line_segment` says it can return a segment that has exited, which `segment_range` leaves out. Code comments label the change `Craft patch (exited slots)`. |
| `docs/sweep.typ` | The intersection scan to the left compares each segment with `alpha^(i+1)` instead of `alpha^i`. With `alpha^i`, its first pass compared `alpha^i` with itself, and the lemma after it is stated for `alpha^(i+1)`. |
| `docs/sweep.typ` | Two typos are corrected. "condi0.2tions" reads "conditions", and `q(i+i)` reads `q(i+1)`, the first index for which `q` is false. |
| `docs/sweep.typ` | The lemma for the intersection scan to the left asks for the ordering invariant on `(alpha^1, ..., alpha^(i+1))` and the crossing invariant on `(alpha^1, ..., alpha^i)`, which is how the bidirectional lemma uses it. The argument for the order invariant cites the crossing invariant instead of a "third invariant"; the document defines two. |
`omitted_files` lists the 43 files of the package left out here, with their hashes:
vendor/linesweeper/SOURCE.json
Vendored
+5 -2
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@@ -8,7 +8,10 @@
"upstream_revision": "824b0791a750f7b27b8edcbb6c6244d0e629875e",
"upstream_directory": "",
"local_patches": [
"In `src/sweep/sweep_line.rs`, the doc comment of `SweepLine::line_segment` says it can return a segment that has exited, which `segment_range` leaves out."
"In `src/sweep/sweep_line.rs`, the doc comment of `SweepLine::line_segment` says it can return a segment that has exited, which `segment_range` leaves out.",
"In `docs/sweep.typ`, the intersection scan to the left compares each segment with `alpha^(i+1)`, the segment the lemma after it is stated for, instead of `alpha^i`.",
"In `docs/sweep.typ`, the typo \"condi0.2tions\" reads \"conditions\", and `q(i+i)` reads `q(i+1)`, the first index for which `q` is false.",
"In `docs/sweep.typ`, the lemma for the intersection scan to the left asks for the ordering invariant on `(alpha^1, ..., alpha^(i+1))` and the crossing invariant on `(alpha^1, ..., alpha^i)`, as the bidirectional lemma uses it, and the argument for the order invariant cites the crossing invariant instead of a \"third invariant\"."
],
"files": [
{ "path": ".cargo_vcs_info.json", "source_sha256": "47fba6a9dbdf445a5c97f783c9be62093035f9d5e5cdc2cf3216ad8259bcd8fc", "source_git_blob": null, "sha256": "47fba6a9dbdf445a5c97f783c9be62093035f9d5e5cdc2cf3216ad8259bcd8fc" },
@@ -23,7 +26,7 @@
{ "path": "benches/linebender_logo.rs", "source_sha256": "d10867f43439d37f0e1da5d30eaed9ab3bfbe655ed47212f65dc20ab4db65cb3", "source_git_blob": null, "sha256": "d10867f43439d37f0e1da5d30eaed9ab3bfbe655ed47212f65dc20ab4db65cb3" },
{ "path": "benches/painted_dreams.rs", "source_sha256": "073008db93840d380710e13ed3ae38e08cb823f10c12e5f838dac1ccf37eaecf", "source_git_blob": null, "sha256": "073008db93840d380710e13ed3ae38e08cb823f10c12e5f838dac1ccf37eaecf" },
{ "path": "benches/synthetic.rs", "source_sha256": "7ce59bef2e1c4f28b00cb89feb15bc71ca4b098a8d27e7b02ea699aca9bcc810", "source_git_blob": null, "sha256": "7ce59bef2e1c4f28b00cb89feb15bc71ca4b098a8d27e7b02ea699aca9bcc810" },
{ "path": "docs/sweep.typ", "source_sha256": "4766082c8672a7bb4356b5581043489c13b4e3fb8931ba3a5f0a473fab0172f3", "source_git_blob": null, "sha256": "4766082c8672a7bb4356b5581043489c13b4e3fb8931ba3a5f0a473fab0172f3" },
{ "path": "docs/sweep.typ", "source_sha256": "4766082c8672a7bb4356b5581043489c13b4e3fb8931ba3a5f0a473fab0172f3", "source_git_blob": null, "sha256": "c151c2c47032eca41914139b105177d10996b6ece9c93622aaeb31a371cfe17b" },
{ "path": "examples/boolean_op.rs", "source_sha256": "3b38a1ccda7e736cce4b191a3a45ab4824c4c365190e3eb637b2fb39b5996c9f", "source_git_blob": null, "sha256": "3b38a1ccda7e736cce4b191a3a45ab4824c4c365190e3eb637b2fb39b5996c9f" },
{ "path": "examples/bus.rs", "source_sha256": "e206a08be73899f564188052c8ecc15526b54606e52ee2935a64ce6b5723da33", "source_git_blob": null, "sha256": "e206a08be73899f564188052c8ecc15526b54606e52ee2935a64ce6b5723da33" },
{ "path": "examples/curve_metrics.rs", "source_sha256": "0db6a5327fd2119a03a9a66624fea3e04b6bbb519655667ad30487aee6c787d2", "source_git_blob": null, "sha256": "0db6a5327fd2119a03a9a66624fea3e04b6bbb519655667ad30487aee6c787d2" },
vendor/linesweeper/docs/sweep.typ
Vendored
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@@ -221,7 +221,7 @@ The first observation is that the sweep line invariants are maintained as we inc
to the next event:
+ For the order invariant, first note that there is an event whenever $y$ leaves the domain of a segment, so $y$ remains in all domains until the next event.
Moreover, if at any $y' > y$ the ordering breaks, two line segments must have $epsilon$-crossed one another by $y'$.
The third invariant guarantees that there's an event before this happens, so by the contra-positive until an event happens the ordering
The crossing invariant guarantees that there's an event before this happens, so by the contra-positive until an event happens the ordering
constraint is maintained.
+ The crossing invariant is maintained because the set of things to check (i.e. the set of line segments that $epsilon$-cross
one another after $y$) only shrinks as $y$ increases.
@@ -368,13 +368,13 @@ of the other one.
#figure(
pseudocode-list([
+ *for* $j = i$ down to $1$
+ let $w^j$ be the smallest height of any event between $j$ and $i$
+ let $w^j$ be the smallest height of any event between $j$ and $i+1$
+ *if* $(alpha^j_(-,epsilon), alpha^i)$ cross by $w^j$
+ choose $z$ before the crossing, such that $alpha^j approx_(z,epsilon) alpha^i$
+ insert an intersection event for $(alpha^j, alpha^i)$ at $z$
+ *if* $(alpha^j_(-,epsilon), alpha^(i+1))$ cross by $w^j$
+ choose $z$ before the crossing, such that $alpha^j approx_(z,epsilon) alpha^(i+1)$
+ insert an intersection event for $(alpha^j, alpha^(i+1))$ at $z$
+ *if* $alpha^j_+ (z) <= alpha^i (z)$ for all $z in [y, w^j]$
+ *if* $alpha^j_+ (z) <= alpha^(i+1) (z)$ for all $z in [y, w^j]$
+ *break*
]),
caption: "Intersection scan to the left"
@@ -383,8 +383,8 @@ of the other one.
We'll skip the proof of the following lemma, because it's basically the same as the last one.
#lemma[
Suppose that $(alpha^1, ..., alpha^i)$ satisfy the ordering invariant
at $y$, and suppose also that $(alpha^1, ..., alpha^(i+1))$
Suppose that $(alpha^1, ..., alpha^(i+1))$ satisfy the ordering invariant
at $y$, and suppose also that $(alpha^1, ..., alpha^i)$
satisfy the crossing invariant at $y$.
After running an intersection scan to the left,
$(alpha^1, ..., alpha^(i+1))$ satisfy a slightly stronger crossing invariant at $y$: for every $j <= i$,
@@ -555,7 +555,7 @@ height. Note that because the sweep line changes at a discrete set of heights,
at every important height $y$ we actually have two weakly-ordered sweep line:
the "old" one from infinitesmally before $y$ height and the "new" one from $y$
onwards. Because everything is continuous and the weak ordering conditions are
closed, both the old and new sweep lines satisfy the weak ordering condi0.2tions
closed, both the old and new sweep lines satisfy the weak ordering conditions
at $y$.
The algorithm goes like this: at every important height we use
@@ -623,7 +623,7 @@ Let's check that this insertion position is valid: for every $j <= i$ we have
either $j <= i_-$ or $i_- < j <= i$. In the first case, we already observed
that $alpha^j_-(y) <= beta(y)$; in the second case, $q(j)$ returned `true` and
so $alpha^j_-(y) <= beta^j^+ (y)$. In either case, $alpha^j prec.tilde beta$.
On the other hand, $q(i+i)$ being `false` implies that $alpha^(i+1)_- (y) >=
On the other hand, $q(i+1)$ being `false` implies that $alpha^(i+1)_- (y) >=
beta(y)$. This implies that for all $j >= i+1$, $alpha^j_+ (y) >= beta(y)$ and
so $alpha^j succ.tilde_y beta$.