diff --git a/src/kosmickrisp/libkk/kk_geometry.cl b/src/kosmickrisp/libkk/kk_geometry.cl
index c54e0059e0e..b240bf2fb5e 100644
--- a/src/kosmickrisp/libkk/kk_geometry.cl
+++ b/src/kosmickrisp/libkk/kk_geometry.cl
@@ -10,21 +10,22 @@
 #include "poly/geometry.h"
 #include "poly/tessellator.h"
 
-KERNEL(1)
+KERNEL(1024)
 libkk_prefix_sum_tess(global struct poly_tess_params *p)
 {
+   local uint scratch[32];
+   poly_prefix_sum(scratch, p->counts, p->nr_patches, 1 /* words */,
+                   0 /* word */, 1024);
+
+   /* After prefix summing, we know the total # of indices, so allocate the
+    * index buffer now. Elect a thread for the allocation.
+    */
+   barrier(CLK_LOCAL_MEM_FENCE);
    if (cl_local_id.x != 0)
       return;
 
    /* The last element of an inclusive prefix sum is the total sum */
-   uint total = 0;
-
-   if (p->nr_patches > 0) {
-      for (uint32_t i = 0u; i < p->nr_patches; ++i) {
-         total += p->counts[i];
-         p->counts[i] = total;
-      }
-   }
+   uint total = p->nr_patches > 0 ? p->counts[p->nr_patches - 1] : 0;
 
    /* Allocate 4-byte indices */
    uint32_t elsize_B = sizeof(uint32_t);
diff --git a/src/kosmickrisp/vulkan/kk_cmd_draw.c b/src/kosmickrisp/vulkan/kk_cmd_draw.c
index b94fb5cbd1d..201a145a7ab 100644
--- a/src/kosmickrisp/vulkan/kk_cmd_draw.c
+++ b/src/kosmickrisp/vulkan/kk_cmd_draw.c
@@ -1768,7 +1768,7 @@ kk_launch_tess(struct kk_cmd_buffer *cmd, struct kk_draw_data draw,
                     state);
    mtl_memory_barrier_with_scope(enc, MTL_BARRIER_SCOPE_BUFFERS);
 
-   libkk_prefix_sum_tess(cmd, kk_grid_1d(1u), true, state);
+   libkk_prefix_sum_tess(cmd, kk_grid_1d(1024u), true, state);
    mtl_memory_barrier_with_scope(enc, MTL_BARRIER_SCOPE_BUFFERS);
 
    libkk_tessellate(cmd, grid_tess, true, info.mode, POLY_TESS_MODE_WITH_COUNTS,