More parallel backdrop propagation

This is a nice improvement but still not great on tiger.

piet-gpu/shader/backdrop.comp

@@ -5,7 +5,8 @@
 
 #include "setup.h"
 
-#define BACKDROP_WG 256
+#define LG_BACKDROP_WG 8
+#define BACKDROP_WG (1 << LG_BACKDROP_WG)
 
 layout(local_size_x = BACKDROP_WG, local_size_y = 1) in;
 
@@ -29,28 +30,62 @@ layout(set = 0, binding = 2) buffer TileBuf {
 #include "annotated.h"
 #include "tile.h"
 
+shared uint sh_row_count[BACKDROP_WG];
+shared uint sh_row_base[BACKDROP_WG];
+shared uint sh_row_width[BACKDROP_WG];
+
 void main() {
+    uint th_ix = gl_LocalInvocationID.x;
     uint element_ix = gl_GlobalInvocationID.x;
     AnnotatedRef ref = AnnotatedRef(element_ix * Annotated_size);
 
-    uint tag = Annotated_Nop;
+    uint row_count = 0;
     if (element_ix < n_elements) {
-        tag = Annotated_tag(ref);
-    }
-    if (tag == Annotated_Fill) {
-        PathRef path_ref = PathRef(element_ix * Path_size);
-        Path path = Path_read(path_ref);
-        uint width = path.bbox.z - path.bbox.x;
-        uint height = path.bbox.w - path.bbox.y;
-        // slightly handrolling the tile structure here...
-        uint tile_el_ix = (path.tiles.offset >> 2) + 1;
-        for (uint y = 0; y < height; y++) {
-            uint sum = 0;
-            for (uint x = 0; x < width; x++) {
-                sum += tile[tile_el_ix];
-                tile[tile_el_ix] = sum;
-                tile_el_ix += 2;
+        uint tag = Annotated_tag(ref);
+        if (tag == Annotated_Fill) {
+            PathRef path_ref = PathRef(element_ix * Path_size);
+            Path path = Path_read(path_ref);
+            sh_row_width[th_ix] = path.bbox.z - path.bbox.x;
+            row_count = path.bbox.w - path.bbox.y;
+            if (row_count == 1) {
+                // Note: this can probably be expanded to width = 2 as
+                // long as it doesn't cross the left edge.
+                row_count = 0;
             }
+            sh_row_base[th_ix] = (path.tiles.offset >> 2) + 1;
+        }
+    }
+
+    sh_row_count[th_ix] = row_count;
+    // Prefix sum of sh_row_count
+    for (uint i = 0; i < LG_BACKDROP_WG; i++) {
+        barrier();
+        if (th_ix >= (1 << i)) {
+            row_count += sh_row_count[th_ix - (1 << i)];
+        }
+        barrier();
+        sh_row_count[th_ix] = row_count;
+    }
+    barrier();
+    uint total_rows = sh_row_count[BACKDROP_WG - 1];
+    for (uint row = th_ix; row < total_rows; row += BACKDROP_WG) {
+        // Binary search to find element
+        uint el_ix = 0;
+        for (uint i = 0; i < LG_BACKDROP_WG; i++) {
+            uint probe = el_ix + ((BACKDROP_WG / 2) >> i);
+            if (row >= sh_row_count[probe - 1]) {
+                el_ix = probe;
+            }
+        }
+        uint seq_ix = row - (el_ix > 0 ? sh_row_count[el_ix - 1] : 0);
+        uint width = sh_row_width[el_ix];
+        // Process one row sequentially
+        uint tile_el_ix = sh_row_base[el_ix] + seq_ix * 2 * width;
+        uint sum = tile[tile_el_ix];
+        for (uint x = 1; x < width; x++) {
+            tile_el_ix += 2;
+            sum += tile[tile_el_ix];
+            tile[tile_el_ix] = sum;
         }
     }
 }