diff --git a/Documentation/technical/commit-graph.txt b/Documentation/technical/commit-graph.txt
index 0550c6d0dc..e1a883eb46 100644
--- a/Documentation/technical/commit-graph.txt
+++ b/Documentation/technical/commit-graph.txt
@@ -77,6 +77,29 @@ in the commit graph. We can treat these commits as having "infinite"
 generation number and walk until reaching commits with known generation
 number.
 
+We use the macro GENERATION_NUMBER_INFINITY = 0xFFFFFFFF to mark commits not
+in the commit-graph file. If a commit-graph file was written by a version
+of Git that did not compute generation numbers, then those commits will
+have generation number represented by the macro GENERATION_NUMBER_ZERO = 0.
+
+Since the commit-graph file is closed under reachability, we can guarantee
+the following weaker condition on all commits:
+
+    If A and B are commits with generation numbers N amd M, respectively,
+    and N < M, then A cannot reach B.
+
+Note how the strict inequality differs from the inequality when we have
+fully-computed generation numbers. Using strict inequality may result in
+walking a few extra commits, but the simplicity in dealing with commits
+with generation number *_INFINITY or *_ZERO is valuable.
+
+We use the macro GENERATION_NUMBER_MAX = 0x3FFFFFFF to for commits whose
+generation numbers are computed to be at least this value. We limit at
+this value since it is the largest value that can be stored in the
+commit-graph file using the 30 bits available to generation numbers. This
+presents another case where a commit can have generation number equal to
+that of a parent.
+
 Design Details
 --------------
 
@@ -98,18 +121,14 @@ Future Work
 - The 'commit-graph' subcommand does not have a "verify" mode that is
   necessary for integration with fsck.
 
-- The file format includes room for precomputed generation numbers. These
-  are not currently computed, so all generation numbers will be marked as
-  0 (or "uncomputed"). A later patch will include this calculation.
-
 - After computing and storing generation numbers, we must make graph
   walks aware of generation numbers to gain the performance benefits they
   enable. This will mostly be accomplished by swapping a commit-date-ordered
   priority queue with one ordered by generation number. The following
   operations are important candidates:
 
-    - paint_down_to_common()
     - 'log --topo-order'
+    - 'tag --merged'
 
 - Currently, parse_commit_gently() requires filling in the root tree
   object for a commit. This passes through lookup_tree() and consequently
diff --git a/alloc.c b/alloc.c
index cf4f8b61e1..e8ab14f4a1 100644
--- a/alloc.c
+++ b/alloc.c
@@ -94,6 +94,7 @@ void *alloc_commit_node(void)
 	c->object.type = OBJ_COMMIT;
 	c->index = alloc_commit_index();
 	c->graph_pos = COMMIT_NOT_FROM_GRAPH;
+	c->generation = GENERATION_NUMBER_INFINITY;
 	return c;
 }
 
diff --git a/builtin/merge.c b/builtin/merge.c
index fb4faca250..4a4c09496c 100644
--- a/builtin/merge.c
+++ b/builtin/merge.c
@@ -1186,14 +1186,15 @@ int cmd_merge(int argc, const char **argv, const char *prefix)
 	branch = branch_to_free = resolve_refdup("HEAD", 0, &head_oid, NULL);
 	if (branch)
 		skip_prefix(branch, "refs/heads/", &branch);
+
+	init_diff_ui_defaults();
+	git_config(git_merge_config, NULL);
+
 	if (!branch || is_null_oid(&head_oid))
 		head_commit = NULL;
 	else
 		head_commit = lookup_commit_or_die(&head_oid, "HEAD");
 
-	init_diff_ui_defaults();
-	git_config(git_merge_config, NULL);
-
 	if (branch_mergeoptions)
 		parse_branch_merge_options(branch_mergeoptions);
 	argc = parse_options(argc, argv, prefix, builtin_merge_options,
diff --git a/commit-graph.c b/commit-graph.c
index 4c6127088f..b63a1fc85e 100644
--- a/commit-graph.c
+++ b/commit-graph.c
@@ -1,5 +1,6 @@
 #include "cache.h"
 #include "config.h"
+#include "dir.h"
 #include "git-compat-util.h"
 #include "lockfile.h"
 #include "pack.h"
@@ -248,6 +249,13 @@ static struct commit_list **insert_parent_or_die(struct commit_graph *g,
 	return &commit_list_insert(c, pptr)->next;
 }
 
+static void fill_commit_graph_info(struct commit *item, struct commit_graph *g, uint32_t pos)
+{
+	const unsigned char *commit_data = g->chunk_commit_data + GRAPH_DATA_WIDTH * pos;
+	item->graph_pos = pos;
+	item->generation = get_be32(commit_data + g->hash_len + 8) >> 2;
+}
+
 static int fill_commit_in_graph(struct commit *item, struct commit_graph *g, uint32_t pos)
 {
 	uint32_t edge_value;
@@ -265,6 +273,8 @@ static int fill_commit_in_graph(struct commit *item, struct commit_graph *g, uin
 	date_low = get_be32(commit_data + g->hash_len + 12);
 	item->date = (timestamp_t)((date_high << 32) | date_low);
 
+	item->generation = get_be32(commit_data + g->hash_len + 8) >> 2;
+
 	pptr = &item->parents;
 
 	edge_value = get_be32(commit_data + g->hash_len);
@@ -293,31 +303,40 @@ static int fill_commit_in_graph(struct commit *item, struct commit_graph *g, uin
 	return 1;
 }
 
+static int find_commit_in_graph(struct commit *item, struct commit_graph *g, uint32_t *pos)
+{
+	if (item->graph_pos != COMMIT_NOT_FROM_GRAPH) {
+		*pos = item->graph_pos;
+		return 1;
+	} else {
+		return bsearch_graph(g, &(item->object.oid), pos);
+	}
+}
+
 int parse_commit_in_graph(struct commit *item)
 {
+	uint32_t pos;
+
 	if (!core_commit_graph)
 		return 0;
 	if (item->object.parsed)
 		return 1;
-
 	prepare_commit_graph();
-	if (commit_graph) {
-		uint32_t pos;
-		int found;
-		if (item->graph_pos != COMMIT_NOT_FROM_GRAPH) {
-			pos = item->graph_pos;
-			found = 1;
-		} else {
-			found = bsearch_graph(commit_graph, &(item->object.oid), &pos);
-		}
-
-		if (found)
-			return fill_commit_in_graph(item, commit_graph, pos);
-	}
-
+	if (commit_graph && find_commit_in_graph(item, commit_graph, &pos))
+		return fill_commit_in_graph(item, commit_graph, pos);
 	return 0;
 }
 
+void load_commit_graph_info(struct commit *item)
+{
+	uint32_t pos;
+	if (!core_commit_graph)
+		return;
+	prepare_commit_graph();
+	if (commit_graph && find_commit_in_graph(item, commit_graph, &pos))
+		fill_commit_graph_info(item, commit_graph, pos);
+}
+
 static struct tree *load_tree_for_commit(struct commit_graph *g, struct commit *c)
 {
 	struct object_id oid;
@@ -440,6 +459,8 @@ static void write_graph_chunk_data(struct hashfile *f, int hash_len,
 		else
 			packedDate[0] = 0;
 
+		packedDate[0] |= htonl((*list)->generation << 2);
+
 		packedDate[1] = htonl((*list)->date);
 		hashwrite(f, packedDate, 8);
 
@@ -572,6 +593,45 @@ static void close_reachable(struct packed_oid_list *oids)
 	}
 }
 
+static void compute_generation_numbers(struct packed_commit_list* commits)
+{
+	int i;
+	struct commit_list *list = NULL;
+
+	for (i = 0; i < commits->nr; i++) {
+		if (commits->list[i]->generation != GENERATION_NUMBER_INFINITY &&
+		    commits->list[i]->generation != GENERATION_NUMBER_ZERO)
+			continue;
+
+		commit_list_insert(commits->list[i], &list);
+		while (list) {
+			struct commit *current = list->item;
+			struct commit_list *parent;
+			int all_parents_computed = 1;
+			uint32_t max_generation = 0;
+
+			for (parent = current->parents; parent; parent = parent->next) {
+				if (parent->item->generation == GENERATION_NUMBER_INFINITY ||
+				    parent->item->generation == GENERATION_NUMBER_ZERO) {
+					all_parents_computed = 0;
+					commit_list_insert(parent->item, &list);
+					break;
+				} else if (parent->item->generation > max_generation) {
+					max_generation = parent->item->generation;
+				}
+			}
+
+			if (all_parents_computed) {
+				current->generation = max_generation + 1;
+				pop_commit(&list);
+
+				if (current->generation > GENERATION_NUMBER_MAX)
+					current->generation = GENERATION_NUMBER_MAX;
+			}
+		}
+	}
+}
+
 void write_commit_graph(const char *obj_dir,
 			const char **pack_indexes,
 			int nr_packs,
@@ -584,7 +644,6 @@ void write_commit_graph(const char *obj_dir,
 	struct hashfile *f;
 	uint32_t i, count_distinct = 0;
 	char *graph_name;
-	int fd;
 	struct lock_file lk = LOCK_INIT;
 	uint32_t chunk_ids[5];
 	uint64_t chunk_offsets[5];
@@ -695,24 +754,14 @@ void write_commit_graph(const char *obj_dir,
 	if (commits.nr >= GRAPH_PARENT_MISSING)
 		die(_("too many commits to write graph"));
 
+	compute_generation_numbers(&commits);
+
 	graph_name = get_commit_graph_filename(obj_dir);
-	fd = hold_lock_file_for_update(&lk, graph_name, 0);
-
-	if (fd < 0) {
-		struct strbuf folder = STRBUF_INIT;
-		strbuf_addstr(&folder, graph_name);
-		strbuf_setlen(&folder, strrchr(folder.buf, '/') - folder.buf);
-
-		if (mkdir(folder.buf, 0777) < 0)
-			die_errno(_("cannot mkdir %s"), folder.buf);
-		strbuf_release(&folder);
-
-		fd = hold_lock_file_for_update(&lk, graph_name, LOCK_DIE_ON_ERROR);
-
-		if (fd < 0)
-			die_errno("unable to create '%s'", graph_name);
-	}
+	if (safe_create_leading_directories(graph_name))
+		die_errno(_("unable to create leading directories of %s"),
+			  graph_name);
 
+	hold_lock_file_for_update(&lk, graph_name, LOCK_DIE_ON_ERROR);
 	f = hashfd(lk.tempfile->fd, lk.tempfile->filename.buf);
 
 	hashwrite_be32(f, GRAPH_SIGNATURE);
diff --git a/commit-graph.h b/commit-graph.h
index 260a468e73..96cccb10f3 100644
--- a/commit-graph.h
+++ b/commit-graph.h
@@ -17,6 +17,14 @@ char *get_commit_graph_filename(const char *obj_dir);
  */
 int parse_commit_in_graph(struct commit *item);
 
+/*
+ * It is possible that we loaded commit contents from the commit buffer,
+ * but we also want to ensure the commit-graph content is correctly
+ * checked and filled. Fill the graph_pos and generation members of
+ * the given commit.
+ */
+void load_commit_graph_info(struct commit *item);
+
 struct tree *get_commit_tree_in_graph(const struct commit *c);
 
 struct commit_graph {
diff --git a/commit.c b/commit.c
index 2d4431a21f..298ad747c6 100644
--- a/commit.c
+++ b/commit.c
@@ -344,7 +344,7 @@ const void *detach_commit_buffer(struct commit *commit, unsigned long *sizep)
 	return ret;
 }
 
-int parse_commit_buffer(struct commit *item, const void *buffer, unsigned long size)
+int parse_commit_buffer(struct commit *item, const void *buffer, unsigned long size, int check_graph)
 {
 	const char *tail = buffer;
 	const char *bufptr = buffer;
@@ -399,6 +399,9 @@ int parse_commit_buffer(struct commit *item, const void *buffer, unsigned long s
 	}
 	item->date = parse_commit_date(bufptr, tail);
 
+	if (check_graph)
+		load_commit_graph_info(item);
+
 	return 0;
 }
 
@@ -425,7 +428,7 @@ int parse_commit_gently(struct commit *item, int quiet_on_missing)
 		return error("Object %s not a commit",
 			     oid_to_hex(&item->object.oid));
 	}
-	ret = parse_commit_buffer(item, buffer, size);
+	ret = parse_commit_buffer(item, buffer, size, 0);
 	if (save_commit_buffer && !ret) {
 		set_commit_buffer(item, buffer, size);
 		return 0;
@@ -653,6 +656,24 @@ static int compare_commits_by_author_date(const void *a_, const void *b_,
 	return 0;
 }
 
+int compare_commits_by_gen_then_commit_date(const void *a_, const void *b_, void *unused)
+{
+	const struct commit *a = a_, *b = b_;
+
+	/* newer commits first */
+	if (a->generation < b->generation)
+		return 1;
+	else if (a->generation > b->generation)
+		return -1;
+
+	/* use date as a heuristic when generations are equal */
+	if (a->date < b->date)
+		return 1;
+	else if (a->date > b->date)
+		return -1;
+	return 0;
+}
+
 int compare_commits_by_commit_date(const void *a_, const void *b_, void *unused)
 {
 	const struct commit *a = a_, *b = b_;
@@ -800,11 +821,14 @@ static int queue_has_nonstale(struct prio_queue *queue)
 }
 
 /* all input commits in one and twos[] must have been parsed! */
-static struct commit_list *paint_down_to_common(struct commit *one, int n, struct commit **twos)
+static struct commit_list *paint_down_to_common(struct commit *one, int n,
+						struct commit **twos,
+						int min_generation)
 {
-	struct prio_queue queue = { compare_commits_by_commit_date };
+	struct prio_queue queue = { compare_commits_by_gen_then_commit_date };
 	struct commit_list *result = NULL;
 	int i;
+	uint32_t last_gen = GENERATION_NUMBER_INFINITY;
 
 	one->object.flags |= PARENT1;
 	if (!n) {
@@ -823,6 +847,15 @@ static struct commit_list *paint_down_to_common(struct commit *one, int n, struc
 		struct commit_list *parents;
 		int flags;
 
+		if (commit->generation > last_gen)
+			BUG("bad generation skip %8x > %8x at %s",
+			    commit->generation, last_gen,
+			    oid_to_hex(&commit->object.oid));
+		last_gen = commit->generation;
+
+		if (commit->generation < min_generation)
+			break;
+
 		flags = commit->object.flags & (PARENT1 | PARENT2 | STALE);
 		if (flags == (PARENT1 | PARENT2)) {
 			if (!(commit->object.flags & RESULT)) {
@@ -871,7 +904,7 @@ static struct commit_list *merge_bases_many(struct commit *one, int n, struct co
 			return NULL;
 	}
 
-	list = paint_down_to_common(one, n, twos);
+	list = paint_down_to_common(one, n, twos, 0);
 
 	while (list) {
 		struct commit *commit = pop_commit(&list);
@@ -929,6 +962,7 @@ static int remove_redundant(struct commit **array, int cnt)
 		parse_commit(array[i]);
 	for (i = 0; i < cnt; i++) {
 		struct commit_list *common;
+		uint32_t min_generation = array[i]->generation;
 
 		if (redundant[i])
 			continue;
@@ -937,8 +971,12 @@ static int remove_redundant(struct commit **array, int cnt)
 				continue;
 			filled_index[filled] = j;
 			work[filled++] = array[j];
+
+			if (array[j]->generation < min_generation)
+				min_generation = array[j]->generation;
 		}
-		common = paint_down_to_common(array[i], filled, work);
+		common = paint_down_to_common(array[i], filled, work,
+					      min_generation);
 		if (array[i]->object.flags & PARENT2)
 			redundant[i] = 1;
 		for (j = 0; j < filled; j++)
@@ -1048,14 +1086,21 @@ int in_merge_bases_many(struct commit *commit, int nr_reference, struct commit *
 {
 	struct commit_list *bases;
 	int ret = 0, i;
+	uint32_t min_generation = GENERATION_NUMBER_INFINITY;
 
 	if (parse_commit(commit))
 		return ret;
-	for (i = 0; i < nr_reference; i++)
+	for (i = 0; i < nr_reference; i++) {
 		if (parse_commit(reference[i]))
 			return ret;
+		if (reference[i]->generation < min_generation)
+			min_generation = reference[i]->generation;
+	}
 
-	bases = paint_down_to_common(commit, nr_reference, reference);
+	if (commit->generation > min_generation)
+		return ret;
+
+	bases = paint_down_to_common(commit, nr_reference, reference, commit->generation);
 	if (commit->object.flags & PARENT2)
 		ret = 1;
 	clear_commit_marks(commit, all_flags);
diff --git a/commit.h b/commit.h
index e48f05dffc..cb943013d0 100644
--- a/commit.h
+++ b/commit.h
@@ -10,6 +10,9 @@
 #include "pretty.h"
 
 #define COMMIT_NOT_FROM_GRAPH 0xFFFFFFFF
+#define GENERATION_NUMBER_INFINITY 0xFFFFFFFF
+#define GENERATION_NUMBER_MAX 0x3FFFFFFF
+#define GENERATION_NUMBER_ZERO 0
 
 struct commit_list {
 	struct commit *item;
@@ -33,6 +36,7 @@ struct commit {
 	 */
 	struct tree *maybe_tree;
 	uint32_t graph_pos;
+	uint32_t generation;
 	unsigned int index;
 };
 
@@ -72,7 +76,7 @@ struct commit *lookup_commit_reference_by_name(const char *name);
  */
 struct commit *lookup_commit_or_die(const struct object_id *oid, const char *ref_name);
 
-int parse_commit_buffer(struct commit *item, const void *buffer, unsigned long size);
+int parse_commit_buffer(struct commit *item, const void *buffer, unsigned long size, int check_graph);
 int parse_commit_gently(struct commit *item, int quiet_on_missing);
 static inline int parse_commit(struct commit *item)
 {
@@ -341,6 +345,7 @@ extern int remove_signature(struct strbuf *buf);
 extern int check_commit_signature(const struct commit *commit, struct signature_check *sigc);
 
 int compare_commits_by_commit_date(const void *a_, const void *b_, void *unused);
+int compare_commits_by_gen_then_commit_date(const void *a_, const void *b_, void *unused);
 
 LAST_ARG_MUST_BE_NULL
 extern int run_commit_hook(int editor_is_used, const char *index_file, const char *name, ...);
diff --git a/object.c b/object.c
index f7f4de3aaf..d33e2a5cc8 100644
--- a/object.c
+++ b/object.c
@@ -210,7 +210,7 @@ struct object *parse_object_buffer(const struct object_id *oid, enum object_type
 	} else if (type == OBJ_COMMIT) {
 		struct commit *commit = lookup_commit(oid);
 		if (commit) {
-			if (parse_commit_buffer(commit, buffer, size))
+			if (parse_commit_buffer(commit, buffer, size, 1))
 				return NULL;
 			if (!get_cached_commit_buffer(commit, NULL)) {
 				set_commit_buffer(commit, buffer, size);
diff --git a/ref-filter.c b/ref-filter.c
index 01c1a82075..fa3685d91f 100644
--- a/ref-filter.c
+++ b/ref-filter.c
@@ -16,6 +16,7 @@
 #include "trailer.h"
 #include "wt-status.h"
 #include "commit-slab.h"
+#include "commit-graph.h"
 
 static struct ref_msg {
 	const char *gone;
@@ -1662,12 +1663,13 @@ static int in_commit_list(const struct commit_list *want, struct commit *c)
 }
 
 /*
- * Test whether the candidate or one of its parents is contained in the list.
+ * Test whether the candidate is contained in the list.
  * Do not recurse to find out, though, but return -1 if inconclusive.
  */
 static enum contains_result contains_test(struct commit *candidate,
 					  const struct commit_list *want,
-					  struct contains_cache *cache)
+					  struct contains_cache *cache,
+					  uint32_t cutoff)
 {
 	enum contains_result *cached = contains_cache_at(cache, candidate);
 
@@ -1683,6 +1685,10 @@ static enum contains_result contains_test(struct commit *candidate,
 
 	/* Otherwise, we don't know; prepare to recurse */
 	parse_commit_or_die(candidate);
+
+	if (candidate->generation < cutoff)
+		return CONTAINS_NO;
+
 	return CONTAINS_UNKNOWN;
 }
 
@@ -1698,8 +1704,18 @@ static enum contains_result contains_tag_algo(struct commit *candidate,
 					      struct contains_cache *cache)
 {
 	struct contains_stack contains_stack = { 0, 0, NULL };
-	enum contains_result result = contains_test(candidate, want, cache);
+	enum contains_result result;
+	uint32_t cutoff = GENERATION_NUMBER_INFINITY;
+	const struct commit_list *p;
 
+	for (p = want; p; p = p->next) {
+		struct commit *c = p->item;
+		load_commit_graph_info(c);
+		if (c->generation < cutoff)
+			cutoff = c->generation;
+	}
+
+	result = contains_test(candidate, want, cache, cutoff);
 	if (result != CONTAINS_UNKNOWN)
 		return result;
 
@@ -1717,7 +1733,7 @@ static enum contains_result contains_tag_algo(struct commit *candidate,
 		 * If we just popped the stack, parents->item has been marked,
 		 * therefore contains_test will return a meaningful yes/no.
 		 */
-		else switch (contains_test(parents->item, want, cache)) {
+		else switch (contains_test(parents->item, want, cache, cutoff)) {
 		case CONTAINS_YES:
 			*contains_cache_at(cache, commit) = CONTAINS_YES;
 			contains_stack.nr--;
@@ -1731,7 +1747,7 @@ static enum contains_result contains_tag_algo(struct commit *candidate,
 		}
 	}
 	free(contains_stack.contains_stack);
-	return contains_test(candidate, want, cache);
+	return contains_test(candidate, want, cache, cutoff);
 }
 
 static int commit_contains(struct ref_filter *filter, struct commit *commit,
diff --git a/sha1-file.c b/sha1-file.c
index 695e5c6276..de4839e634 100644
--- a/sha1-file.c
+++ b/sha1-file.c
@@ -1801,7 +1801,7 @@ static void check_commit(const void *buf, size_t size)
 {
 	struct commit c;
 	memset(&c, 0, sizeof(c));
-	if (parse_commit_buffer(&c, buf, size))
+	if (parse_commit_buffer(&c, buf, size, 0))
 		die("corrupt commit");
 }
 
diff --git a/t/t5318-commit-graph.sh b/t/t5318-commit-graph.sh
index a380419b65..77d85aefe7 100755
--- a/t/t5318-commit-graph.sh
+++ b/t/t5318-commit-graph.sh
@@ -221,4 +221,13 @@ test_expect_success 'write graph in bare repo' '
 graph_git_behavior 'bare repo with graph, commit 8 vs merge 1' bare commits/8 merge/1
 graph_git_behavior 'bare repo with graph, commit 8 vs merge 2' bare commits/8 merge/2
 
+test_expect_success 'perform fast-forward merge in full repo' '
+	cd "$TRASH_DIRECTORY/full" &&
+	git checkout -b merge-5-to-8 commits/5 &&
+	git merge commits/8 &&
+	git show-ref -s merge-5-to-8 >output &&
+	git show-ref -s commits/8 >expect &&
+	test_cmp expect output
+'
+
 test_done