plotter/docs/data-and-performance.md

Data and performance

The sample stores

All series derive from plot::data::Series (include/plot/data/series.hpp), which describes the sample-indexing contract; a uniform-grid series additionally has sample i at x = x0() + i * dx() (ArraySeries, StreamSeries), while XYSeries carries its own sorted, ascending x array for data that is not on a uniform grid.

class Series {
 public:
  virtual u64 size() const;
  virtual f64 x0() const;                 // x of sample 0 (the oldest retained sample)
  virtual f64 dx() const;                 // 0 for XYSeries, which owns its x array
  virtual Range x_bounds() const;
  virtual Range y_bounds() const;
  virtual const MinMaxPyramid& pyramid() const;
  virtual bool streaming() const noexcept;                 // default false
  virtual f64 x_at(u64 i) const;                           // x0() + i*dx(), or x()[i]
  virtual bool neighbours(f64 x, u64& i0, u64& i1) const;  // bracketing samples
  virtual u64 snap_index(f64 x) const;                     // nearest sample
  virtual void gather(u32 level, u64 begin, u64 end, std::vector<f32>& out_xy,
                      GatherMode mode = GatherMode::Envelope) const;

  Range reduce(u64 begin, u64 end) const;                  // exact min/max over [begin, end)
  f64   sum(u64 begin, u64 end) const;                     // exact sum over [begin, end)
  f64   sumsq(u64 begin, u64 end) const;                   // exact sum of squares
};

Indices are the series' own window: index 0 is the oldest sample it currently holds, so size(), x0(), reduce(), gather() and pyramid() are all in the same coordinates.

ArraySeries (array_series.hpp) owns a std::vector<f32> of y samples plus an x0/dx grid. Construct with ArraySeries(std::vector<f32> y, f64 x0, f64 dx). The vector is taken by value and owned, so the samples cannot change behind the series' back; the store changes only through reset(y, x0, dx) (a whole new vector, then a rebuild), commit(), which rebuilds the pyramid and cached bounds from the current y, or push(y), which appends one sample incrementally — the pyramid folds it in amortised O(1) and the bounds extend in place, NaN-aware, instead of rebuilding.

StreamSeries (stream_series.hpp) is the single-producer / single-consumer streaming store: StreamSeries(u64 capacity, f64 x0, f64 dx). capacity is rounded up to a power of two by the ring, size() saturates at that rounded capacity() (a StreamSeries(100) retains up to 128 samples), and the pyramid is reserved for the rounded value up front. freeze() copies the window published by the last acquire() into a fresh ArraySeries (consumer thread only): a stable snapshot that no later production can move.

Non-finite samples

The library-wide NaN policy: a non-finite sample is excluded from every aggregate — min, max, sum, sum of squares and the pyramid's per-bucket finite count — and from the cached bounds. A window with no finite samples at all aggregates to a zero-span Range / zero sums, never to NaN. The policy is applied everywhere an aggregate is built: MinMaxPyramid::append (a NaN occupies its absolute position, so bucket boundaries stay positional, but folds nothing), the edge folds in Series::reduce, sum and sumsq, ArraySeries::commit and push bounds, and StreamSeries::acquire bounds. A bucket whose samples were all non-finite is stored as an empty aggregate (count 0, zero sums) that every reader skips — it is absent, not zero-valued. gather() follows suit: Envelope and Mean emit nothing for such a bucket, and Nearest draws its centre sample raw. Hosts that want gaps in the data should draw them with NaN samples and will get exactly that: excluded from statistics, present in raw geometry.

Gather x is relative, and what each mode emits

Series::gather emits x relative to x0() — exactly i * dx for a uniform series, x_at(i) - x0() for XYSeries — never the absolute coordinate. y is unchanged. The renderer reconstructs absolute data x in f64 (x0() + emitted) before transforming, so no precision is lost at the f32 narrowing: the staged value is O(visible pixels), not O(epoch). (Staging an absolute epoch x as f32 would quantise it to a grid coarser than the pixel — f32 carries 24 bits, about 1 s at 1.7e9 — while the relative value is small and exact.) The emitted geometry always spans [begin, end) exactly and never reaches outside it: at level ≥ 1 the bucket edges are selected ceil/floor (first = first bucket starting at or after begin, last = one past the last bucket ending at or before end) and the two partial edge spans are drawn from raw samples, which is what keeps an unaligned window envelope-exact (previously the buckets straddling the range leaked samples from outside it into the geometry). level is clamped to kMaxPyramidLevel (57 — the deepest level whose bucket span fits a u64), so any u32 level is safe.

The GatherMode decides what each bucket contributes, at the same level the renderer already picked:

• Envelope (the default): one (min, max) pair per bucket, min drawn at the bucket's first sample and max at its last. The dense, extreme-reading geometry a time-series trace wants.
• Nearest: one raw sample per bucket — the centre sample, the pick pyramid().snap() would make — drawn at its own x. Thinner geometry for traces whose shape matters more than their extremes.
• Mean: one point per bucket, y = the bucket's finite samples' mean (the exact bucket_sum() the pyramid stores divided by its finite count), x at the bucket centre. A smoothed trace for noisy data.

Level 0 is the raw samples for every mode.

The SPSC contract

One producer thread, one consumer thread, and the division is strict (stream_series.hpp):

• The producer may only call push(f32 y), push(const f32* ys, u64 n) and produced(). Those touch the ring buffer and nothing else: no lock, no pyramid, no GL. The acquisition thread can therefore never stall the renderer.
• The consumer calls acquire(), which snapshots the produced count and brings the pyramid and bounds up to it, and then queries the published window. size(), reduce(), sum(), sumsq(), gather() and pyramid() all use the window published by the last acquire().

acquire() costs O(samples produced since the last acquire), not O(capacity): the pyramid is indexed in absolute sample space, so a sliding window only drops the leading buckets the ring overwrote and appends what is new; aggregates still in the window are never recomputed. The one exception is a producer that laps the consumer (more than capacity() samples between two acquires), where the missing samples cannot be represented and the pyramid restarts at the new base, re-reading the whole window.

The ring overwrites the oldest samples — that is what keeps a live stream showing the newest data — so it is not a frozen window. Those slots are relaxed atomics, so this is not a data race and no value tears; but if the producer pushes k samples while a read is in flight, the first k positions of that read can hold one of those k newer samples rather than the one the snapshot named. Ingest with no reuse underneath a read at all needs the producer to stop before unread data (drop-on-full) or a claim protocol — a different contract from this one.

streaming() returns true for StreamSeries and false for everything else. Only a streaming series has a live edge (the newest x it holds, which moves on its own), and it is that property XFollow::Live follows (see interaction.md).

The min/max LOD pyramid

data::MinMaxPyramid (minmax_pyramid.hpp) is the index that makes per-frame cost depend on the visible pixels rather than on the sample count.

• Samples are grouped into buckets of kBucket = 64 (kBucket is a fixed constant, so all series share one LOD geometry). Level l bucket k covers the absolute samples [k * (kBucket << l), (k + 1) * (kBucket << l)), and bucket boundaries are fixed for the pyramid's life — appending never re-buckets existing aggregates, and a window can slide over the same buckets.
• Each level holds its complete buckets in a std::deque (O(1) push_back and pop_front) plus a base. append() folds a sample into the in-progress level-0 bucket and, when it completes, pushes it and walks up, stopping at the first level that gains no entry: an append costs amortised O(1). drop_before(u) slides the window's leading edge, bounding the memory by the retained window rather than by everything ever appended.
• An entry is 32 bytes (MinMax plus the exact sum/sumsq the exact range queries fold over and the finite-sample count the mean folds by), so level 0 costs 0.5n bytes and the geometric tail brings the whole pyramid to about 1n bytes — roughly 25% of the 4n bytes of raw samples (f32). A min/max-only index would be 0.25n (~6%); the extra words are the price of answering mean/RMS/integral from whole buckets instead of raw samples.

Queries on the pyramid are bucket-grained and named *_buckets (reduce_buckets, sum_buckets, sumsq_buckets): a pyramid stores aggregates, so it can only answer exactly for whole buckets, and partial edge buckets are excluded rather than approximated. Exact range queries live on Series, which owns the raw samples.

level_for(u64 n, f64 samples_per_pixel) returns the pyramid level whose buckets hold about samples_per_pixel samples for a window of n samples (clamped to levels that exist, to levels with more than one bucket, and to levels that give the window at least two buckets).

The renderer uses this: a trace's x range is mapped onto sample indices, and Series::gather is asked for that window at pyramid().level_for(n, samples_per_pixel) whenever a pixel covers Style::dense_threshold_spp samples or more (default 1.5). A 100-million-sample series therefore stages O(visible pixels) vertices, not O(samples), and FrameStats records which path each trace took. Below the threshold the raw samples are drawn instead; both sides go through the same instanced anti-aliased quad pipeline.

Dense traces are reduced onto the pixel grid

A gathered window denser than the axis is not staged as a polyline. For each integer column of layout.axes, Renderer::draw_traces emits one exactly vertical segment from the lowest to the highest screen y any gathered point put in that column, flushed in framebuffer pixels with u_snap on. FrameStats::column_traces counts the traces that took this path.

It exists because a polyline over such a window moirés, and three things compound to make it. MinMaxPyramid::level_for() picks the level whose buckets hold about samples_per_pixel samples, so a bucket spans one to two pixel columns; GatherMode::Envelope emits two points per bucket, so each bucket contributes a near-vertical min→max hairline and a near-vertical connector to the next — about 3.6 near-vertical hairlines per column; and each of those lands at a fractional x, so with ordinary src-alpha compositing a column's brightness is a non-linear function of the sub-pixel phases overlapping it. The phase pattern repeats at the beat between the bucket pitch and the pixel pitch, which is a visible banding, and panning sweeps the beat so it crawls. Reducing to one segment per column leaves the trace one sub-pixel phase, so there is nothing left to beat, and the snap then puts that one phase on the grid.

Four properties of the reduction are load-bearing:

• It is gated on the gathered density, not on the level. It runs when the window is GatherMode::Envelope and carries at least two gathered points per column of the axes — the raw path at 1.5–64 samples per pixel has the same beat and the same cure. The density term is also what keeps it away from a sparse window, whose trace is drawn by the two segments that enter and leave the frame (a_sparse_window_between_samples_still_inks_both_margins); Nearest and Mean are excluded because they are one-point-per-bucket modes a band defeats.
• A hole stays a hole. A column no gathered point landed in emits nothing. Bridging it would draw a vertical line over x the series says nothing about.
• Disjoint columns are joined. Two neighbouring columns' extremes need not overlap — a fast edge rises through a whole column — and two disjoint vertical segments read as a broken trace, which is what the polyline's connector prevented. The later column is extended to meet the earlier one's nearest bound, so the join is made inside the column that moved.
• A span shorter than the stroke is drawn as the stroke. The line fragment shader ramps on the distance to the segment's axis and does not cap its ends, so a vertical segment shorter than its own width rasterises a sliver of itself, and a flat trace's column, where the two bounds are equal, nothing at all.

This is the one carve-out in u_snap's rule that chrome may be snapped and data may not (see LineProgram::set_snap()): a column is a pixel column and its x is already a screen coordinate, so there is no data position to move it off, and the y bounds the segment reports are untouched by the snap.

a_dense_trace_is_drawn_as_whole_pixel_columns is the guard, and what it measures is the number of distinct coverage levels in the ink: 4 with the reduction against 126 with the polyline, over one sine period across 60 000 samples. A count of partially covered pixels is not the reading and was tried first — a 1.5 px stroke with a 1 px anti-aliasing fringe always spills into its neighbour column, so the two renderings measure 577 and 652, which separates nothing. The frame's whole geometry collapses with it: 627 segments against 60 054.

A reduced column can be graded by dwell

Style::intensity_grading draws each of those columns graded rather than as one flat band: bright where the signal sits, faint where it only passes through, the way a digital-phosphor scope reads. It is off by default — a change of look, not a correction of one — and FrameStats::graded_traces counts the traces that took it, a subset of column_traces, since grading is a way of drawing the reduction and not a third path.

What it puts back is the information the reduction throws away: a column's min and max are the same whether its samples fill the span evenly or pile at its two ends, so a ramp and a pulse are one band. The distribution cannot be recovered from the band, which is why the gather is asked at a level finer than the pixel grid (intensity_sub_columns sub-buckets per column) and the dwell read from those sub-buckets.

Five things carry it, and each was arrived at by being wrong first:

• The measure is dwell per row, each segment depositing one unit spread over the rows it spans (1 / height). Counting +1 per row instead grades a sine's fast middle as brightly as its rails, which is the pattern inverted — the middle is where the signal spends the least time.
• The accumulator is one column wide. Gathered x is monotonic, so points arrive column by column: accumulate, read, clear. A full axes.w × axes.h density image is 33 MB at 4K; one column of axes.h floats is L1-resident. A span is written as a +1/height delta at its first row and the negation one past its last, so it costs two writes whatever its height, and the running sum over the column is the per-row dwell.
• Normalisation is per column, against the column's own peak. A frame-wide scale needs the whole image to exist before anything can be drawn, and reads worse: a column the signal flew through uniformly comes out faint at every row, so the trace fades out exactly where it moves fastest. The cost of the local scale is one extra non-clearing walk of the accumulator to find the peak.
• Alpha is quantised into intensity_steps tiers, one LineBatch and one draw call each. The line pipeline's colour is a uniform, not per-instance (LineInstance is four floats of geometry; only TextInstance carries a_colour), so a per-segment alpha is inexpressible and the tiers are what make the gradient drawable at all. intensity_floor keeps the faintest tier visible — a fast riser is one sub-bucket deep everywhere and must not be dimmed out of existence — and intensity_gamma lifts the faint end. The floor only holds if the riser's rows reach a tier: the "lit at all" cutoff that discards a column's empty rows is min(0.01 × peak, 0.5 × w_min) over the smallest deposit any of that column's own segments made, and not the first term alone. A column holding a flat level and a riser has its peak set by the level — one unit in one row — while the riser spreads a unit over 300, which is three orders of magnitude under 0.01 × peak, so the riser was discarded before the floor could apply to it. Measured on a 2098-sample square wave at 105 samples a column: 2 916 of the ungraded path's 35 884 riser rows, 106 of 116 riser columns gutted entirely, with a 311-row dark run in a 312-row axis; the ten survivors were the phase that put a riser in a column of its own, which is why it read as a riser flickering under a pan rather than as a missing edge. With the second term it is 35 572 rows and 115 of 116 columns, for 48 extra segments (a_graded_column_keeps_a_riser_it_shares_with_a_flat_level). A connector between two disjoint columns is emitted at the faintest tier: it is inferred geometry, not a place the signal was measured to dwell.
• Grading may not undo the reduction. level_for() answers 0 — raw samples — as soon as the asked-for fraction of a column falls under one bucket (kBucket is 64), so the finer ask is floored at level 1 whenever the flat path would have used the pyramid. Without the floor every moderately dense frame became an O(visible) gather: measured in udpscope at 1.3 MS/s over a 2 s window, 172 samples a pixel, the flat path sat on the pyramid and grading dropped the whole figure to raw (raw=1154 lod=0 against raw=710 lod=453) and cost 2.3 → 6.4 ms of draw mean. The floor degrades the look instead — fewer sub-columns than asked for, down to what the coarsest buckets give, which at 64 samples a bucket is still two or three per column. That is also why raising intensity_sub_columns past about 8 buys nothing: the bucket size is the limit, not the number.

a_dense_trace_can_be_graded_by_dwell is the guard, and it draws the same frame both ways. It profiles the mean peak coverage over the band's two outer fifths against its middle fifth: the flat band is flat (191 and 191, one ink) and the graded one is 191 against 131, a middle at 0.69 of the edge. That ratio is a tier ratio and not the dwell ratio — the default gamma 0.45 and floor 0.3 compress it on purpose. The geometry stays bounded by the axis and not by its area: 8303 segments over a 572 px axis, 14.5 runs a column, against the flat path's 627 at one end and 572 × 156 rows at the other. A fourth frame over a window wide enough that the flat path is on the pyramid is what exercises the floor (raw=0 lod=1 … seg=634 visible=200001); the test's first window sits at 105 samples a pixel, where both paths are raw and the floor is never reached.

A sparse polyline is graded by slope

The column reduction stops at dense_threshold_spp (1.5 samples a device pixel), below which the trace is a polyline. Grading used to stop there with it, because it was staged inside the reduction — and that coupling was a defect, not a design. The reduction's gate is a density, and its only purpose is removing the moiré beat between the bucket pitch and the pixel pitch. Dwell has nothing to do with that beat, so zooming past the threshold turned the phosphor model off in one step: the same square-wave edge measured 71 of 191 at 3.4 samples a pixel and 191 of 191 at 1.0, a 2.7× jump across one pixel of zoom, which reads as a discontinuity lighting up the moment you look at it closely. The thing a reader most needs told apart from a real signal excursion is a transition, and it was being drawn at full ink exactly when it filled the screen.

Renderer::stage_trace_slopes() grades the polyline on the same principle, differing only where it must:

• The measure is the segment's own slope, 1 / (1 + |Δy_screen|), which is dwell per row for a single segment. The 1 + is not a guard against a division: a flat segment — span 0, the commonest case in a sparse window — must be the brightest tier, and this makes the measure dimensionless and absolute. There is no per-column peak to normalise against down here, and that is an improvement: a sparse frame's tiers mean the same thing from frame to frame, so panning does not reshade the trace.
• No gather mode is excluded. The reduction is Envelope-only, because turning a bucket's min/max into a band answers a different question from the one a Nearest or Mean host asked. A segment's slope is a property of the polyline itself, so grade is not reused as the gate — only intensity_grading and intensity_steps > 1 are.
• The geometry is bit-for-bit the ungraded polyline's, staged exactly as stage_xy() stages it, with the same snap = false; only the batch a segment lands in differs. A segment with one non-finite end is dropped rather than clamped, which is the conclusion add_polyline_xy() reaches too.
• FrameStats::slope_traces counts it, and it is disjoint from column_traces. A window is either dense enough to reduce onto the pixel grid or it is drawn as a polyline, never both — so a host asserting graded_traces == column_traces keeps that reading unchanged.

a_sparse_trace_is_graded_by_slope_like_a_dense_one is the guard: one window either side of the threshold over the same transition at the same place. The level reads 191 either way (a flat segment dwells in its row, so it is the top tier and full ink) and the riser reads 71 either way, against the 191 the sparse frame gave before. Across the whole sweep from 101 down to 0.1 samples a pixel the riser holds 71 and the level holds 191, so there is no step at the threshold left to see.

A phosphor density accumulates into a screen-sized grid

Plot::set_density() gives a plot a phosphor density: an image of how often its traces have passed, drawn behind them, fading by a time constant or by an acquisition count. It is the library's because the library is the side that knows a trace's device geometry; a host that built one itself would be re-rasterising traces the renderer has already staged.

The split from the grading above is what it costs: grading measures one frame's dwell and throws it away, while the density keeps what it measured — that is the whole point of it, and also why it cannot be a per-frame accumulator.

• The grid is the screen's, not the data's. One f32 cell per device pixel of the axes rect (DensityGrid, plot/plot/density.hpp): the axes of a 640x360 viewport is a 573x313 grid of 179 349 cells and 717 396 bytes, and a thousand acquisitions through it cost the same. A cell's value is a number of sweeps: a trace covering it during one Plot::accumulate_density() leaves 1.0, once however many of its own samples crossed it. Counting the hits instead puts a single dense sweep at the ceiling of the brightness scale and hides the sweeps behind it, and a cell would then mean "how much the beam moved here" rather than "how many acquisitions were here". The dedupe is a stamp per grid row (one u64, a few kB), keyed by (trace, column), so two sweeps agreeing on a cell still count twice and the two segments that share a column boundary do not.
• The deposit is one pass per cell, not one per sample. A trace's visible window is mapped to sample indices and gathered exactly as Renderer::draw_traces() maps it (neighbours(), the same 64-spp gate for the LOD level), then walked as a polyline: for each grid column the segment covers, the rows it spans in that column get +1 each. A window of millions of samples over hundreds of columns therefore costs columns, plus one row of ink per column the trace crossed — the deposit is bounded by the screen, not by the data, which is what makes an accumulation affordable at the trigger's own rate.
• The scale is applied once, on the CPU. DensityConfig::full_scale and gamma map a cell's count to a luminance byte when the grid changed, and the shaders (kDensityVS/kDensityFS) only tint that byte. The one derivation is deliberate: a value a host reads back through DensityGrid::ink()/peak() and the picture it sees come from the same numbers.
• Per frame, it is one texture upload and one draw call -- or one draw call. The upload is one byte per cell (179 kB for the axes above); a frame whose grid did not move skips the conversion and the upload entirely and pays the quad alone (FrameStats::density_upload_bytes reads 0 for it). The decay is what makes the grid move every frame under TimeConstant, and what makes it stand still between acquisitions under PerAcquisition.

a_density_grid_is_the_axes_in_device_pixels reads the resolution and the memory back (584x458 = 267 472 cells, 1 069 888 bytes, ink=584 peak=1 for one flat sweep — the ink of one sweep is the columns it covers), a_density_halves_per_acquisition_and_keeps_the_newest_brightest reads the count schedule (584 → 876 → 1022 on a 584-column grid, i.e. 1 + 1/2 + 1/4 of the width, peak_steady=2.000), a_density_fades_by_its_time_constant reads exp(-dt/tau) off the ink, and a_phosphor_density_is_drawn_under_the_traces reads the frame: cells=179349 uploaded=179349 on the frame the grid moved and uploaded=0 on the identical one after it, with the trace's stroke keeping its own colour over the ink (min_red < 40, where a density drawn on top would put the red at ~190).

One sample or a statistic of many

Every number the library reports is one the samples actually hold. What differs between these two is how many samples answer the question:

• Series::reduce / sum / sumsq are exact over a sample range. They answer "what are the samples' extremes / sum / sum of squares over [begin, end)". Each is implemented once over copy_raw(): the bucket-aligned interior comes from the pyramid (O(log n) bucket aggregates) and at most two partial edge buckets are read raw and scanned, so the cost is O(kBucket) worst case and independent of the series length, and no query allocates. Measure::evaluate and Plot::region_stats are built on these, never on the *_buckets forms, which would silently report wrong extremes for an unaligned interval.
• The crosshair readout is a single sample — the nearest one — by default. crosshair_span(series, x) returns the two samples x lies between (begin, end, t), and CrosshairValue::NearestSample takes whichever of the two is nearer: t <= 0.5 the earlier, otherwise the later. It answers "which datum is under the pointer", not "what is the signal at this instant" — a linearly interpolated y is a number the series does not contain, and on a source whose samples are already averaged or decimated at ingest it is a claim about a signal nobody measured. That is why it is the default; a host whose x is a quantity it is willing to interpolate states CrosshairValue::Interpolated and gets crosshair_interpolate(y_begin, y_end, t) instead, at the crosshair's own x. The lookup is O(1) and allocation-free either way, and it reads level 0, so the value is a stored sample (or a blend of two) and never an LOD bucket's aggregate. Plot::crosshair_point(trace, x, out_x, out_y) exposes both coordinates to a host and crosshair_value() the y alone; CrosshairRow::x/y and its marker are placed from that same pair, so in NearestSample the mark snaps onto the quoted sample and the dot and the number cannot disagree. A row whose trace has no sample at the crosshair's x shows kCrosshairNoValue (an em dash, "—"), and an x outside the series window yields no value rather than clamping to the first or last sample.

Measurement (from Plot::measurement()) documents integral as a left-Riemann sum, sum(begin, end) * dx — the samples taken as the value at the left edge of each step. That is part of the result's contract, not an implementation detail.

Measured frame times

These are the demos' own frame summaries, taken from a linux-release build, all of them llvmpipe (Mesa software rasteriser) under Xvfb — no run touched a GPU. Treat them as upper bounds and as a record of the software path, not as hardware numbers.

demo	window	mean ms/frame	work
GLFW (plot_glfw_demo)	1280×720	0.42	one 1 MS/s StreamSeries
ImGui (plot_imgui_demo)	1280×720	0.37	the same stream, embedded
multiplot (plot_multiplot_demo)	1280×720	0.66	3 rows, 6 traces, 500 kS/s each
Qt6 (plot_qt_demo)	820×588 at exit	3.23	one stream, QOpenGLWidget FBO
GTK4 (plot_gtk_demo)	810×564 at exit	2.85	one stream, GtkGLArea FBO

The multiplot run stages 18 651 segments for ~4.5 million visible samples, which is the pyramid doing its job: the frame is bounded by the pixels, not the samples. The Qt and GTK figures are dominated by their toolkit's texture-backed FBO under llvmpipe, not by the library.