In my learnings to understand B+trees I now would like to see what it takes to modify this existing B+ index tree (which has the additional constraints of having every array be a power of 2 size in length: 1, 2, 4, 8, 16, or 32), and turn it into a Stack, and turn it into a Queue. Here is the original B+tree code from the wonderful linked answer:
class Node {
constructor(capacity) {
// Mimic fixed-size array (avoid accidentally growing it)
this.children = Object.seal(Array(capacity).fill(null));
this.childCount = 0; // Number of used slots in children array
this.treeSize = 0; // Total number of values in this subtree
// Maintain back-link to parent.
this.parent = null;
// Per level in the tree, maintain a doubly linked list
this.prev = this.next = null;
}
setCapacity(capacity) {
if (capacity < 1) return;
// Here we make a new array, and copy the data into it
let children = Object.seal(Array(capacity).fill(null));
for (let i = 0; i < this.childCount; i++) children[i] = this.children[i];
this.children = children;
}
isLeaf() {
return !(this.children[0] instanceof Node);
}
index() {
return this.parent.children.indexOf(this);
}
updateTreeSize(start, end, sign=1) {
let sum = 0;
if (this.isLeaf()) {
sum = end - start;
} else {
for (let i = start; i < end; i++) sum += this.children[i].treeSize;
}
if (!sum) return;
sum *= sign;
// Apply the sum change to this node and all its ancestors
for (let node = this; node; node = node.parent) {
node.treeSize += sum;
}
}
wipe(start, end) {
this.updateTreeSize(start, end, -1);
this.children.copyWithin(start, end, this.childCount);
for (let i = this.childCount - end + start; i < this.childCount; i++) {
this.children[i] = null;
}
this.childCount -= end - start;
// Reduce allocated size if possible
if (this.childCount * 2 <= this.children.length) this.setCapacity(this.children.length / 2);
}
moveFrom(neighbor, target, start, count=1) {
// Note: `start` can have two meanings:
// if neighbor is null, it is the value/Node to move to the target
// if neighbor is a Node, it is the index from where value(s) have to be moved to the target
// Make room in target node
if (this.childCount + count > this.children.length) this.setCapacity(this.children.length * 2);
this.children.copyWithin(target + count, target, Math.max(target + count, this.childCount));
this.childCount += count;
if (neighbor !== null) {
// Copy the children
for (let i = 0; i < count; i++) {
this.children[target + i] = neighbor.children[start + i];
}
// Remove the original references
neighbor.wipe(start, start + count);
} else {
this.children[target] = start; // start is value to insert
}
this.updateTreeSize(target, target + count, 1);
// Set parent link(s)
if (!this.isLeaf()) {
for (let i = 0; i < count; i++) {
this.children[target + i].parent = this;
}
}
}
moveToNext(count) {
this.next.moveFrom(this, 0, this.childCount - count, count);
}
moveFromNext(count) {
this.moveFrom(this.next, this.childCount, 0, count);
}
basicRemove(index) {
if (!this.isLeaf()) {
// Take node out of the level's linked list
let prev = this.children[index].prev;
let next = this.children[index].next;
if (prev) prev.next = next;
if (next) next.prev = prev;
}
this.wipe(index, index + 1);
}
basicInsert(index, value) {
this.moveFrom(null, index, value);
if (value instanceof Node) {
// Insert node in the level's linked list
if (index > 0) {
value.prev = this.children[index-1];
value.next = value.prev.next;
} else if (this.childCount > 1) {
value.next = this.children[1];
value.prev = value.next.prev;
}
if (value.prev) value.prev.next = value;
if (value.next) value.next.prev = value;
}
}
pairWithSmallest() {
return this.prev && (!this.next || this.next.childCount > this.prev.childCount)
? [this.prev, this] : [this, this.next];
}
toString() {
return "[" + this.children.map(v => v??"-").join() + "]";
}
}
class Tree {
constructor(nodeCapacity=32) {
this.nodeCapacity = nodeCapacity;
this.root = new Node(1);
this.first = this.root; // Head of doubly linked list at bottom level
}
locate(offset) {
let node = this.root;
// Normalise argument
offset = offset < 0 ? Math.max(0, node.treeSize + offset) : Math.min(offset, node.treeSize);
while (!node.isLeaf()) {
let index = 0;
let child = node.children[index];
while (offset > child.treeSize || offset === child.treeSize && child.next) {
offset -= child.treeSize;
child = node.children[++index];
}
node = child;
}
return [node, offset];
}
getItemAt(offset) {
let [node, index] = this.locate(offset);
if (index < node.childCount) return node.children[index];
}
setItemAt(offset, value) {
let [node, index] = this.locate(offset);
if (index < node.childCount) node.children[index] = value;
}
removeItemAt(offset) {
let [node, index] = this.locate(offset);
if (index >= node.childCount) return;
while (true) {
console.assert(node.isLeaf() || node.children[index].treeSize === 0);
node.basicRemove(index);
// Exit when node's fill ratio is fine
if (!node.parent || node.childCount * 2 > this.nodeCapacity) return;
// Node has potentially too few children, we should either merge or redistribute
let [left, right] = node.pairWithSmallest();
if (!left || !right) { // A node with no siblings? Must become the root!
this.root = node;
node.parent = null;
return;
}
let sumCount = left.childCount + right.childCount;
let childCount = sumCount >> 1;
// Check whether to merge or to redistribute
if (sumCount > this.nodeCapacity) { // redistribute
// Move some data from the bigger to the smaller node
let shift = childCount - node.childCount;
if (!shift) { // Boundary case: when a redistribution would bring no improvement
console.assert(node.childCount * 2 === this.nodeCapacity && sumCount === this.nodeCapacity + 1);
return;
}
if (node === left) { // move some children from right to left
left.moveFromNext(shift);
} else { // move some children from left to right
left.moveToNext(shift);
}
return;
}
// Merge:
// Move all data from the right to the left
left.moveFromNext(right.childCount);
// Prepare to delete right node
node = right.parent;
index = right.index();
}
}
insertItemAt(offset, value) {
let [node, index] = this.locate(offset);
while (node.childCount === this.nodeCapacity) { // No room here
if (index === 0 && node.prev && node.prev.childCount < this.nodeCapacity) {
return node.prev.basicInsert(node.prev.childCount, value);
}
// Check whether we can redistribute (to avoid a split)
if (node !== this.root) {
let [left, right] = node.pairWithSmallest();
let joinedIndex = left === node ? index : left.childCount + index;
let sumCount = left.childCount + right.childCount + 1;
if (sumCount <= 2 * this.nodeCapacity) { // redistribute
let childCount = sumCount >> 1;
if (node === right) { // redistribute to the left
let insertInLeft = joinedIndex < childCount;
left.moveFromNext(childCount - left.childCount - +insertInLeft);
} else { // redistribute to the right
let insertInRight = index >= sumCount - childCount;
left.moveToNext(childCount - right.childCount - +insertInRight);
}
if (joinedIndex > left.childCount ||
joinedIndex === left.childCount && left.childCount > right.childCount) {
right.basicInsert(joinedIndex - left.childCount, value);
} else {
left.basicInsert(joinedIndex, value);
}
return;
}
}
// Cannot redistribute: split node
let childCount = node.childCount >> 1;
// Create a new node that will later become the right sibling of this node
let sibling = new Node(childCount);
// Move half of node node's data to it
sibling.moveFrom(node, 0, childCount, childCount);
// Insert the value in either the current node or the new one
if (index > node.childCount) {
sibling.basicInsert(index - node.childCount, value);
} else {
node.basicInsert(index, value);
}
// Is this the root?
if (!node.parent) {
// ...then first create a parent, which is the new root
this.root = new Node(2);
this.root.basicInsert(0, node);
}
// Prepare for inserting the sibling node into the tree
index = node.index() + 1;
node = node.parent;
value = sibling;
}
node.basicInsert(index, value);
}
/* Below this point: these methods are optional */
* [Symbol.iterator]() { // Make tree iterable
let i = 0;
for (let node = this.first; node; node = node.next) {
for (let i = 0; i < node.childCount; i++) yield node.children[i];
}
}
print() {
console.log(this.root && this.root.toString());
}
verify() {
// Raise an error when the tree violates one of the required properties
if (!this.root) return; // An empty tree is fine.
if (this.root.parent) throw "root should not have a parent";
// Perform a breadth first traversal
let q = [this.root];
while (q.length) {
if (q[0].isLeaf() && this.first !== q[0]) throw "this.first is not pointing to first leaf";
let level = [];
let last = null;
for (let parent of q) {
if (!(parent instanceof Node)) throw "parent is not instance of Node";
if (parent.children.length > this.nodeCapacity) throw "node's children array is too large";
if (parent.childCount > 0 && parent.childCount * 2 <= parent.children.length) throw "node's fill ratio is too low";
for (let i = parent.childCount; i < parent.children.length; i++) {
if (parent.children[i] !== null) throw "child beyond childCount should be null but is not";
}
let treeSize = parent.treeSize;
if (parent.isLeaf()) {
for (let value of parent.children.slice(0, parent.childCount)) {
if (value === null) throw "leaf has a null as value";
if (value instanceof Node) throw "leaf has a Node as value";
}
if (parent.treeSize !== parent.childCount) throw "leaf has mismatch in treeSize and childCount";
} else {
for (let node of parent.children.slice(0, parent.childCount)) {
if (node === null) throw "internal node has a null as value";
if (!(node instanceof Node)) throw "internal node has a non-Node as value";
if (node.parent !== parent) throw "wrong parent";
if (node.prev !== last) throw "prev link incorrect";
if (last && last.next !== node) throw "next link incorrect";
if (last && last.children.length + node.children.length <= this.nodeCapacity) {
throw "two consecutive siblings have a total number of children that is too small";
}
if (node.childCount * 2 < this.nodeCapacity) {
throw "internal node is too small: " + node;
}
level.push(node);
last = node;
treeSize -= node.treeSize;
}
if (treeSize) throw "internal node treeSize sum mismatches";
}
}
if (last && last.next) throw "last node in level has a next reference";
q = level;
}
}
test(count=100, option=3) {
// option:
// 0 = always insert & delete at left side (offset 0)
// 1 = always insert & delete at right side
// 2 = always insert & delete at middle
// 3 = insert & delete at random offsets
// Create array to perform the same operations on it as on the tree
let arr = [];
// Perform a series of insertions
for (let i = 0; i < count; i++) {
// Choose random insertion index
let index = Array.isArray(option) ? option[i] : [0, i, i >> 1, Math.floor(Math.random() * (i+1))][option];
// Perform same insertion in array and tree
arr.splice(index, 0, i);
this.insertItemAt(index, i);
// Verify tree consistency and properties
this.verify();
// Verify the order of values in the array is the same as in the tree
if (arr+"" !== [...this]+"") throw i + ": tree not same as array";
}
// Perform a series of updates
for (let i = 0; i < count; i++) {
// Choose random update index
let index = Math.floor(Math.random() * count);
// Perform same insertion in array and tree
arr[index] += count;
this.setItemAt(index, this.getItemAt(index) + count);
// Verify tree consistency and properties
this.verify();
// Verify the order of values in the array is the same as in the tree
if (arr+"" !== [...this]+"") throw "tree not same as array";
}
// Perform a series of deletions
for (let i = arr.length - 1; i >= 0; i--) {
// Choose random deletion index
let index = [0, i, i >> 1, Math.floor(Math.random() * (i+1))][option];
// Perform same deletion in array and tree
arr.splice(index, 1);
this.removeItemAt(index);
// Verify tree consistency and properties
this.verify();
// Verify the order of values in the array is the same as in the tree
if (arr+"" !== [...this]+"") throw "tree not same as array";
}
}
}
// Perform 1000 insertions, 1000 updates, and 1000 deletions on a tree with node capacity of 8
new Tree(8).test(1000);
console.log("all tests completed");A stack is a last-in-first-out LIFO data structure, while a queue is a first-in-first-out FIFO one. A stack has a push and pop method (in addition to getItemAt(index) and other basic array methods, which this B+tree already implements). A queue has a push and shift method, where shift removes "from the front" of the array. So already they are similar, just add items to the end of the array, or remove from the start or end of the array.
The way I would do the push operation with the existing B+tree is to simply keep track of the length of the array (which you can do with tree.root.treeSize), and insertItemAt(tree.root.treeSize, val) it at that position. But maybe there is a more optimized way of doing this?
Tree.prototype.push = function(val) { this.insertItemAt(this.root.treeSize, val) }
For the pop, I would simply do:
Tree.prototype.pop = function() {
let val = this.getItemAt(this.root.treeSize - 1)
this.removeItemAt(this.root.treeSize - 1)
return val
}
Finally, for the shift, I would do:
Tree.prototype.shift = function() {
let val = this.getItemAt(0)
this.removeItemAt(0)
return val
}
But my question is, is there a better more optimized way to do this? Rather than traversing the whole tree to find the first or last item, maybe we could cache them? Not sure exactly the best approach here. What is the way to make this most optimal given the structure of the B+tree (as having these power of two constraints, that's pretty much it)? For example, on the "pluck" operations (pop and shift), there are two traversals, maybe this could be one (or none even)? How could this B+tree be modified to make these operations optimal?
