+ assert(W.LastCluster - W.FirstCluster + 1 >= 2 && "Too small to split!");
+
+ // Balance the tree based on branch weights to create a near-optimal (in terms
+ // of search time given key frequency) binary search tree. See e.g. Kurt
+ // Mehlhorn "Nearly Optimal Binary Search Trees" (1975).
+ CaseClusterIt LastLeft = W.FirstCluster;
+ CaseClusterIt FirstRight = W.LastCluster;
+ uint32_t LeftWeight = LastLeft->Weight;
+ uint32_t RightWeight = FirstRight->Weight;
+
+ // Move LastLeft and FirstRight towards each other from opposite directions to
+ // find a partitioning of the clusters which balances the weight on both
+ // sides. If LeftWeight and RightWeight are equal, alternate which side is
+ // taken to ensure 0-weight nodes are distributed evenly.
+ unsigned I = 0;
+ while (LastLeft + 1 < FirstRight) {
+ if (LeftWeight < RightWeight || (LeftWeight == RightWeight && (I & 1)))
+ LeftWeight += (++LastLeft)->Weight;
+ else
+ RightWeight += (--FirstRight)->Weight;
+ I++;
+ }
+
+ for (;;) {
+ // Our binary search tree differs from a typical BST in that ours can have up
+ // to three values in each leaf. The pivot selection above doesn't take that
+ // into account, which means the tree might require more nodes and be less
+ // efficient. We compensate for this here.
+
+ unsigned NumLeft = LastLeft - W.FirstCluster + 1;
+ unsigned NumRight = W.LastCluster - FirstRight + 1;
+
+ if (std::min(NumLeft, NumRight) < 3 && std::max(NumLeft, NumRight) > 3) {
+ // If one side has less than 3 clusters, and the other has more than 3,
+ // consider taking a cluster from the other side.
+
+ if (NumLeft < NumRight) {
+ // Consider moving the first cluster on the right to the left side.
+ CaseCluster &CC = *FirstRight;
+ unsigned RightSideRank = caseClusterRank(CC, FirstRight, W.LastCluster);
+ unsigned LeftSideRank = caseClusterRank(CC, W.FirstCluster, LastLeft);
+ if (LeftSideRank <= RightSideRank) {
+ // Moving the cluster to the left does not demote it.
+ ++LastLeft;
+ ++FirstRight;
+ continue;
+ }
+ } else {
+ assert(NumRight < NumLeft);
+ // Consider moving the last element on the left to the right side.
+ CaseCluster &CC = *LastLeft;
+ unsigned LeftSideRank = caseClusterRank(CC, W.FirstCluster, LastLeft);
+ unsigned RightSideRank = caseClusterRank(CC, FirstRight, W.LastCluster);
+ if (RightSideRank <= LeftSideRank) {
+ // Moving the cluster to the right does not demot it.
+ --LastLeft;
+ --FirstRight;
+ continue;
+ }
+ }
+ }
+ break;
+ }
+
+ assert(LastLeft + 1 == FirstRight);
+ assert(LastLeft >= W.FirstCluster);
+ assert(FirstRight <= W.LastCluster);