diff --git a/Misc/NEWS.d/next/Core and Builtins/2019-05-28-17-02-46.bpo-37029.MxpgfJ.rst b/Misc/NEWS.d/next/Core and Builtins/2019-05-28-17-02-46.bpo-37029.MxpgfJ.rst
new file mode 100644
index 0000000..c18f5d2
--- /dev/null
+++ b/Misc/NEWS.d/next/Core and Builtins/2019-05-28-17-02-46.bpo-37029.MxpgfJ.rst
@@ -0,0 +1 @@
+Freeing a great many small objects could take time quadratic in the number of arenas, due to using linear search to keep ``obmalloc.c``'s list of usable arenas sorted by order of number of free memory pools.  This is accomplished without search now, leaving the worst-case time linear in the number of arenas.  For programs where this quite visibly matters (typically with more than 100 thousand small objects alive simultaneously), this can greatly reduce the time needed to release their memory.
\ No newline at end of file
diff --git a/Objects/obmalloc.c b/Objects/obmalloc.c
index f54856d..fc7bef6 100644
--- a/Objects/obmalloc.c
+++ b/Objects/obmalloc.c
@@ -918,6 +918,11 @@
 #define POOL_SIZE               SYSTEM_PAGE_SIZE        /* must be 2^N */
 #define POOL_SIZE_MASK          SYSTEM_PAGE_SIZE_MASK
 
+#define MAX_POOLS_IN_ARENA  (ARENA_SIZE / POOL_SIZE)
+#if MAX_POOLS_IN_ARENA * POOL_SIZE != ARENA_SIZE
+#   error "arena size not an exact multiple of pool size"
+#endif
+
 /*
  * -- End of tunable settings section --
  */
@@ -1155,6 +1160,18 @@
 
 Note that an arena_object associated with an arena all of whose pools are
 currently in use isn't on either list.
+
+Changed in Python 3.8:  keeping usable_arenas sorted by number of free pools
+used to be done by one-at-a-time linear search when an arena's number of
+free pools changed.  That could, overall, consume time quadratic in the
+number of arenas.  That didn't really matter when there were only a few
+hundred arenas (typical!), but could be a timing disaster when there were
+hundreds of thousands.  See bpo-37029.
+
+Now we have a vector of "search fingers" to eliminate the need to search:
+nfp2lasta[nfp] returns the last ("rightmost") arena in usable_arenas
+with nfp free pools.  This is NULL if and only if there is no arena with
+nfp free pools in usable_arenas.
 */
 
 /* Array of objects used to track chunks of memory (arenas). */
@@ -1172,6 +1189,9 @@
  */
 static struct arena_object* usable_arenas = NULL;
 
+/* nfp2lasta[nfp] is the last arena in usable_arenas with nfp free pools */
+static struct arena_object* nfp2lasta[MAX_POOLS_IN_ARENA + 1] = { NULL };
+
 /* How many arena_objects do we initially allocate?
  * 16 = can allocate 16 arenas = 16 * ARENA_SIZE = 4MB before growing the
  * `arenas` vector.
@@ -1281,8 +1301,7 @@
     /* pool_address <- first pool-aligned address in the arena
        nfreepools <- number of whole pools that fit after alignment */
     arenaobj->pool_address = (block*)arenaobj->address;
-    arenaobj->nfreepools = ARENA_SIZE / POOL_SIZE;
-    assert(POOL_SIZE * arenaobj->nfreepools == ARENA_SIZE);
+    arenaobj->nfreepools = MAX_POOLS_IN_ARENA;
     excess = (uint)(arenaobj->address & POOL_SIZE_MASK);
     if (excess != 0) {
         --arenaobj->nfreepools;
@@ -1478,22 +1497,32 @@
         }
         usable_arenas->nextarena =
             usable_arenas->prevarena = NULL;
+        assert(nfp2lasta[usable_arenas->nfreepools] == NULL);
+        nfp2lasta[usable_arenas->nfreepools] = usable_arenas;
     }
     assert(usable_arenas->address != 0);
 
+    /* This arena already had the smallest nfreepools value, so decreasing
+     * nfreepools doesn't change that, and we don't need to rearrange the
+     * usable_arenas list.  However, if the arena becomes wholly allocated,
+     * we need to remove its arena_object from usable_arenas.
+     */
+    assert(usable_arenas->nfreepools > 0);
+    if (nfp2lasta[usable_arenas->nfreepools] == usable_arenas) {
+        /* It's the last of this size, so there won't be any. */
+        nfp2lasta[usable_arenas->nfreepools] = NULL;
+    }
+    /* If any free pools will remain, it will be the new smallest. */
+    if (usable_arenas->nfreepools > 1) {
+        assert(nfp2lasta[usable_arenas->nfreepools - 1] == NULL);
+        nfp2lasta[usable_arenas->nfreepools - 1] = usable_arenas;
+    }
+
     /* Try to get a cached free pool. */
     pool = usable_arenas->freepools;
     if (pool != NULL) {
         /* Unlink from cached pools. */
         usable_arenas->freepools = pool->nextpool;
-
-        /* This arena already had the smallest nfreepools
-         * value, so decreasing nfreepools doesn't change
-         * that, and we don't need to rearrange the
-         * usable_arenas list.  However, if the arena has
-         * become wholly allocated, we need to remove its
-         * arena_object from usable_arenas.
-         */
         --usable_arenas->nfreepools;
         if (usable_arenas->nfreepools == 0) {
             /* Wholly allocated:  remove. */
@@ -1501,7 +1530,6 @@
             assert(usable_arenas->nextarena == NULL ||
                    usable_arenas->nextarena->prevarena ==
                    usable_arenas);
-
             usable_arenas = usable_arenas->nextarena;
             if (usable_arenas != NULL) {
                 usable_arenas->prevarena = NULL;
@@ -1709,7 +1737,23 @@
     ao = &arenas[pool->arenaindex];
     pool->nextpool = ao->freepools;
     ao->freepools = pool;
-    nf = ++ao->nfreepools;
+    nf = ao->nfreepools;
+    /* If this is the rightmost arena with this number of free pools,
+     * nfp2lasta[nf] needs to change.  Caution:  if nf is 0, there
+     * are no arenas in usable_arenas with that value.
+     */
+    struct arena_object* lastnf = nfp2lasta[nf];
+    assert((nf == 0 && lastnf == NULL) || 
+           (nf > 0 && 
+            lastnf != NULL &&
+            lastnf->nfreepools == nf &&
+            (lastnf->nextarena == NULL ||
+             nf < lastnf->nextarena->nfreepools)));
+    if (lastnf == ao) {  /* it is the rightmost */
+        struct arena_object* p = ao->prevarena;
+        nfp2lasta[nf] = (p != NULL && p->nfreepools == nf) ? p : NULL;
+    }
+    ao->nfreepools = ++nf;
 
     /* All the rest is arena management.  We just freed
      * a pool, and there are 4 cases for arena mgmt:
@@ -1777,6 +1821,9 @@
             usable_arenas->prevarena = ao;
         usable_arenas = ao;
         assert(usable_arenas->address != 0);
+        if (nfp2lasta[1] == NULL) {
+            nfp2lasta[1] = ao;
+        }
 
         goto success;
     }
@@ -1788,14 +1835,23 @@
      * a few un-scientific tests, it seems like this
      * approach allowed a lot more memory to be freed.
      */
-    if (ao->nextarena == NULL ||
-                 nf <= ao->nextarena->nfreepools) {
+    /* If this is the only arena with nf, record that. */
+    if (nfp2lasta[nf] == NULL) {
+        nfp2lasta[nf] = ao;
+    } /* else the rightmost with nf doesn't change */
+    /* If this was the rightmost of the old size, it remains in place. */
+    if (ao == lastnf) {
         /* Case 4.  Nothing to do. */
         goto success;
     }
-    /* Case 3:  We have to move the arena towards the end
-     * of the list, because it has more free pools than
-     * the arena to its right.
+    /* If ao were the only arena in the list, the last block would have
+     * gotten us out.
+     */
+    assert(ao->nextarena != NULL);
+
+    /* Case 3:  We have to move the arena towards the end of the list,
+     * because it has more free pools than the arena to its right.  It needs
+     * to move to follow lastnf.
      * First unlink ao from usable_arenas.
      */
     if (ao->prevarena != NULL) {
@@ -1809,24 +1865,13 @@
         usable_arenas = ao->nextarena;
     }
     ao->nextarena->prevarena = ao->prevarena;
-
-    /* Locate the new insertion point by iterating over
-     * the list, using our nextarena pointer.
-     */
-    while (ao->nextarena != NULL && nf > ao->nextarena->nfreepools) {
-        ao->prevarena = ao->nextarena;
-        ao->nextarena = ao->nextarena->nextarena;
-    }
-
-    /* Insert ao at this point. */
-    assert(ao->nextarena == NULL || ao->prevarena == ao->nextarena->prevarena);
-    assert(ao->prevarena->nextarena == ao->nextarena);
-
-    ao->prevarena->nextarena = ao;
+    /* And insert after lastnf. */
+    ao->prevarena = lastnf;
+    ao->nextarena = lastnf->nextarena;
     if (ao->nextarena != NULL) {
         ao->nextarena->prevarena = ao;
     }
-
+    lastnf->nextarena = ao;
     /* Verify that the swaps worked. */
     assert(ao->nextarena == NULL || nf <= ao->nextarena->nfreepools);
     assert(ao->prevarena == NULL || nf > ao->prevarena->nfreepools);