diff --git a/lib/nghttp3_conn.c b/lib/nghttp3_conn.c
index 49273ab..dcb2ba0 100644
--- a/lib/nghttp3_conn.c
+++ b/lib/nghttp3_conn.c
@@ -273,7 +273,7 @@ static int conn_new(nghttp3_conn **pconn, int server, int callbacks_version,
     nghttp3_pq_init(&conn->sched[i].spq, cycle_less, mem);
   }
 
-  nghttp3_idtr_init(&conn->remote.bidi.idtr, server, mem);
+  nghttp3_idtr_init(&conn->remote.bidi.idtr, !server, mem);
 
   conn->callbacks = *callbacks;
   conn->local.settings = *settings;
diff --git a/lib/nghttp3_gaptr.c b/lib/nghttp3_gaptr.c
index 88cb49a..20eed5f 100644
--- a/lib/nghttp3_gaptr.c
+++ b/lib/nghttp3_gaptr.c
@@ -37,14 +37,8 @@ void nghttp3_gaptr_init(nghttp3_gaptr *gaptr, const nghttp3_mem *mem) {
 
 static int gaptr_gap_init(nghttp3_gaptr *gaptr) {
   nghttp3_range range = {0, UINT64_MAX};
-  int rv;
-
-  rv = nghttp3_ksl_insert(&gaptr->gap, NULL, &range, NULL);
-  if (rv != 0) {
-    return rv;
-  }
 
-  return 0;
+  return nghttp3_ksl_insert(&gaptr->gap, NULL, &range, NULL);
 }
 
 void nghttp3_gaptr_free(nghttp3_gaptr *gaptr) {
@@ -82,7 +76,9 @@ int nghttp3_gaptr_push(nghttp3_gaptr *gaptr, uint64_t offset,
       nghttp3_ksl_remove_hint(&gaptr->gap, &it, &it, &k);
       continue;
     }
+
     nghttp3_range_cut(&l, &r, &k, &m);
+
     if (nghttp3_range_len(&l)) {
       nghttp3_ksl_update_key(&gaptr->gap, &k, &l);
 
@@ -95,23 +91,23 @@ int nghttp3_gaptr_push(nghttp3_gaptr *gaptr, uint64_t offset,
     } else if (nghttp3_range_len(&r)) {
       nghttp3_ksl_update_key(&gaptr->gap, &k, &r);
     }
+
     nghttp3_ksl_it_next(&it);
   }
+
   return 0;
 }
 
 uint64_t nghttp3_gaptr_first_gap_offset(nghttp3_gaptr *gaptr) {
   nghttp3_ksl_it it;
-  nghttp3_range r;
 
   if (nghttp3_ksl_len(&gaptr->gap) == 0) {
     return 0;
   }
 
   it = nghttp3_ksl_begin(&gaptr->gap);
-  r = *(nghttp3_range *)nghttp3_ksl_it_key(&it);
 
-  return r.begin;
+  return ((nghttp3_range *)nghttp3_ksl_it_key(&it))->begin;
 }
 
 nghttp3_range nghttp3_gaptr_get_first_gap_after(nghttp3_gaptr *gaptr,
@@ -136,7 +132,6 @@ int nghttp3_gaptr_is_pushed(nghttp3_gaptr *gaptr, uint64_t offset,
                             uint64_t datalen) {
   nghttp3_range q = {offset, offset + datalen};
   nghttp3_ksl_it it;
-  nghttp3_range k;
   nghttp3_range m;
 
   if (nghttp3_ksl_len(&gaptr->gap) == 0) {
@@ -145,8 +140,7 @@ int nghttp3_gaptr_is_pushed(nghttp3_gaptr *gaptr, uint64_t offset,
 
   it = nghttp3_ksl_lower_bound_compar(&gaptr->gap, &q,
                                       nghttp3_ksl_range_exclusive_compar);
-  k = *(nghttp3_range *)nghttp3_ksl_it_key(&it);
-  m = nghttp3_range_intersect(&q, &k);
+  m = nghttp3_range_intersect(&q, (nghttp3_range *)nghttp3_ksl_it_key(&it));
 
   return nghttp3_range_len(&m) == 0;
 }
diff --git a/lib/nghttp3_gaptr.h b/lib/nghttp3_gaptr.h
index 7c83c84..867368b 100644
--- a/lib/nghttp3_gaptr.h
+++ b/lib/nghttp3_gaptr.h
@@ -40,8 +40,9 @@
  * nghttp3_gaptr maintains the gap in the range [0, UINT64_MAX).
  */
 typedef struct nghttp3_gaptr {
-  /* gap maintains the range of offset which is not received
-     yet. Initially, its range is [0, UINT64_MAX). */
+  /* gap maintains the range of offset which is not pushed
+     yet. Initially, its range is [0, UINT64_MAX).  "gap" is the range
+     that is not pushed yet. */
   nghttp3_ksl gap;
   /* mem is custom memory allocator */
   const nghttp3_mem *mem;
@@ -58,8 +59,7 @@ void nghttp3_gaptr_init(nghttp3_gaptr *gaptr, const nghttp3_mem *mem);
 void nghttp3_gaptr_free(nghttp3_gaptr *gaptr);
 
 /*
- * nghttp3_gaptr_push adds new data of length |datalen| at the stream
- * offset |offset|.
+ * nghttp3_gaptr_push pushes the range [offset, offset + datalen).
  *
  * This function returns 0 if it succeeds, or one of the following
  * negative error codes:
@@ -77,7 +77,7 @@ uint64_t nghttp3_gaptr_first_gap_offset(nghttp3_gaptr *gaptr);
 
 /*
  * nghttp3_gaptr_get_first_gap_after returns the first gap which
- * overlaps or comes after |offset|.
+ * includes or comes after |offset|.
  */
 nghttp3_range nghttp3_gaptr_get_first_gap_after(nghttp3_gaptr *gaptr,
                                                 uint64_t offset);
diff --git a/lib/nghttp3_idtr.c b/lib/nghttp3_idtr.c
index dc34841..0e95989 100644
--- a/lib/nghttp3_idtr.c
+++ b/lib/nghttp3_idtr.c
@@ -42,8 +42,7 @@ void nghttp3_idtr_free(nghttp3_idtr *idtr) {
 }
 
 /*
- * id_from_stream_id translates |stream_id| to id space used by
- * nghttp3_idtr.
+ * id_from_stream_id translates |stream_id| to an internal ID.
  */
 static uint64_t id_from_stream_id(int64_t stream_id) {
   return (uint64_t)(stream_id >> 2);
@@ -52,8 +51,8 @@ static uint64_t id_from_stream_id(int64_t stream_id) {
 int nghttp3_idtr_open(nghttp3_idtr *idtr, int64_t stream_id) {
   uint64_t q;
 
-  assert((idtr->server && (stream_id % 2)) ||
-         (!idtr->server && (stream_id % 2)) == 0);
+  assert((idtr->server && (stream_id & 1)) ||
+         (!idtr->server && !(stream_id & 1)));
 
   q = id_from_stream_id(stream_id);
 
@@ -67,14 +66,10 @@ int nghttp3_idtr_open(nghttp3_idtr *idtr, int64_t stream_id) {
 int nghttp3_idtr_is_open(nghttp3_idtr *idtr, int64_t stream_id) {
   uint64_t q;
 
-  assert((idtr->server && (stream_id % 2)) ||
-         (!idtr->server && (stream_id % 2)) == 0);
+  assert((idtr->server && (stream_id & 1)) ||
+         (!idtr->server && !(stream_id & 1)));
 
   q = id_from_stream_id(stream_id);
 
   return nghttp3_gaptr_is_pushed(&idtr->gap, q, 1);
 }
-
-uint64_t nghttp3_idtr_first_gap(nghttp3_idtr *idtr) {
-  return nghttp3_gaptr_first_gap_offset(&idtr->gap);
-}
diff --git a/lib/nghttp3_idtr.h b/lib/nghttp3_idtr.h
index ea3346c..772303d 100644
--- a/lib/nghttp3_idtr.h
+++ b/lib/nghttp3_idtr.h
@@ -39,8 +39,10 @@
  * nghttp3_idtr tracks the usage of stream ID.
  */
 typedef struct nghttp3_idtr {
-  /* gap maintains the range of ID which is not used yet. Initially,
-     its range is [0, UINT64_MAX). */
+  /* gap maintains the range of an internal ID which is not used yet.
+     Initially, its range is [0, UINT64_MAX).  The internal ID and
+     stream ID are in the different number spaces.  See
+     id_from_stream_id to convert a stream ID to an internal ID. */
   nghttp3_gaptr gap;
   /* server is nonzero if this object records server initiated stream
      ID. */
@@ -50,7 +52,7 @@ typedef struct nghttp3_idtr {
 /*
  * nghttp3_idtr_init initializes |idtr|.
  *
- * If this object records server initiated ID (even number), set
+ * If this object records server initiated stream ID (odd number), set
  * |server| to nonzero.
  */
 void nghttp3_idtr_init(nghttp3_idtr *idtr, int server, const nghttp3_mem *mem);
@@ -61,30 +63,21 @@ void nghttp3_idtr_init(nghttp3_idtr *idtr, int server, const nghttp3_mem *mem);
 void nghttp3_idtr_free(nghttp3_idtr *idtr);
 
 /*
- * nghttp3_idtr_open claims that |stream_id| is in used.
+ * nghttp3_idtr_open claims that |stream_id| is in use.
  *
  * It returns 0 if it succeeds, or one of the following negative error
  * codes:
  *
  * NGHTTP3_ERR_STREAM_IN_USE
- *     ID has already been used.
+ *     |stream_id| has already been used.
  * NGHTTP3_ERR_NOMEM
  *     Out of memory.
  */
 int nghttp3_idtr_open(nghttp3_idtr *idtr, int64_t stream_id);
 
 /*
- * nghttp3_idtr_open tells whether ID |stream_id| is in used or not.
- *
- * It returns nonzero if |stream_id| is used.
+ * nghttp3_idtr_open returns nonzero if |stream_id| is in use.
  */
 int nghttp3_idtr_is_open(nghttp3_idtr *idtr, int64_t stream_id);
 
-/*
- * nghttp3_idtr_first_gap returns the first id of first gap.  If there
- * is no gap, it returns UINT64_MAX.  The returned id is an id space
- * used in this object internally, and not stream ID.
- */
-uint64_t nghttp3_idtr_first_gap(nghttp3_idtr *idtr);
-
 #endif /* NGHTTP3_IDTR_H */
diff --git a/lib/nghttp3_ksl.c b/lib/nghttp3_ksl.c
index fce7952..5c41564 100644
--- a/lib/nghttp3_ksl.c
+++ b/lib/nghttp3_ksl.c
@@ -268,9 +268,9 @@ static void ksl_insert_node(nghttp3_ksl *ksl, nghttp3_ksl_blk *blk, size_t i,
   ++blk->n;
 }
 
-static size_t ksl_bsearch(nghttp3_ksl *ksl, nghttp3_ksl_blk *blk,
-                          const nghttp3_ksl_key *key,
-                          nghttp3_ksl_compar compar) {
+static size_t ksl_search(nghttp3_ksl *ksl, nghttp3_ksl_blk *blk,
+                         const nghttp3_ksl_key *key,
+                         nghttp3_ksl_compar compar) {
   size_t i;
   nghttp3_ksl_node *node;
 
@@ -307,7 +307,7 @@ int nghttp3_ksl_insert(nghttp3_ksl *ksl, nghttp3_ksl_it *it,
   }
 
   for (;;) {
-    i = ksl_bsearch(ksl, blk, key, ksl->compar);
+    i = ksl_search(ksl, blk, key, ksl->compar);
 
     if (blk->leaf) {
       if (i < blk->n &&
@@ -549,7 +549,7 @@ int nghttp3_ksl_remove(nghttp3_ksl *ksl, nghttp3_ksl_it *it,
   }
 
   for (;;) {
-    i = ksl_bsearch(ksl, blk, key, ksl->compar);
+    i = ksl_search(ksl, blk, key, ksl->compar);
 
     if (i == blk->n) {
       if (it) {
@@ -613,43 +613,7 @@ int nghttp3_ksl_remove(nghttp3_ksl *ksl, nghttp3_ksl_it *it,
 
 nghttp3_ksl_it nghttp3_ksl_lower_bound(nghttp3_ksl *ksl,
                                        const nghttp3_ksl_key *key) {
-  nghttp3_ksl_blk *blk = ksl->head;
-  nghttp3_ksl_it it;
-  size_t i;
-
-  if (!blk) {
-    nghttp3_ksl_it_init(&it, ksl, &null_blk, 0);
-    return it;
-  }
-
-  for (;;) {
-    i = ksl_bsearch(ksl, blk, key, ksl->compar);
-
-    if (blk->leaf) {
-      if (i == blk->n && blk->next) {
-        blk = blk->next;
-        i = 0;
-      }
-      nghttp3_ksl_it_init(&it, ksl, blk, i);
-      return it;
-    }
-
-    if (i == blk->n) {
-      /* This happens if descendant has smaller key.  Fast forward to
-         find last node in this subtree. */
-      for (; !blk->leaf; blk = nghttp3_ksl_nth_node(ksl, blk, blk->n - 1)->blk)
-        ;
-      if (blk->next) {
-        blk = blk->next;
-        i = 0;
-      } else {
-        i = blk->n;
-      }
-      nghttp3_ksl_it_init(&it, ksl, blk, i);
-      return it;
-    }
-    blk = nghttp3_ksl_nth_node(ksl, blk, i)->blk;
-  }
+  return nghttp3_ksl_lower_bound_compar(ksl, key, ksl->compar);
 }
 
 nghttp3_ksl_it nghttp3_ksl_lower_bound_compar(nghttp3_ksl *ksl,
@@ -665,7 +629,7 @@ nghttp3_ksl_it nghttp3_ksl_lower_bound_compar(nghttp3_ksl *ksl,
   }
 
   for (;;) {
-    i = ksl_bsearch(ksl, blk, key, compar);
+    i = ksl_search(ksl, blk, key, compar);
 
     if (blk->leaf) {
       if (i == blk->n && blk->next) {
@@ -703,7 +667,7 @@ void nghttp3_ksl_update_key(nghttp3_ksl *ksl, const nghttp3_ksl_key *old_key,
   assert(ksl->head);
 
   for (;;) {
-    i = ksl_bsearch(ksl, blk, old_key, ksl->compar);
+    i = ksl_search(ksl, blk, old_key, ksl->compar);
 
     assert(i < blk->n);
     node = nghttp3_ksl_nth_node(ksl, blk, i);
diff --git a/lib/nghttp3_range.c b/lib/nghttp3_range.c
index e9c5cee..af810a2 100644
--- a/lib/nghttp3_range.c
+++ b/lib/nghttp3_range.c
@@ -36,9 +36,11 @@ nghttp3_range nghttp3_range_intersect(const nghttp3_range *a,
   nghttp3_range r = {0, 0};
   uint64_t begin = nghttp3_max_uint64(a->begin, b->begin);
   uint64_t end = nghttp3_min_uint64(a->end, b->end);
+
   if (begin < end) {
     nghttp3_range_init(&r, begin, end);
   }
+
   return r;
 }
 
diff --git a/lib/nghttp3_range.h b/lib/nghttp3_range.h
index 20dab69..f14d132 100644
--- a/lib/nghttp3_range.h
+++ b/lib/nghttp3_range.h
@@ -59,7 +59,7 @@ uint64_t nghttp3_range_len(const nghttp3_range *r);
 
 /*
  * nghttp3_range_eq returns nonzero if |a| equals |b|, such that
- * a->begin == b->begin, and a->end == b->end hold.
+ * a->begin == b->begin and a->end == b->end hold.
  */
 int nghttp3_range_eq(const nghttp3_range *a, const nghttp3_range *b);