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// see: https://en.wikipedia.org/wiki/Order_statistic_tree // based on: https://algs4.cs.princeton.edu/33balanced/RedBlackBST.java.html sclass LogNArray extends RandomAccessAbstractList { // A symbol table implemented using a left-leaning red-black BST. // This is the 2-3 version. private static final boolean RED = true; private static final boolean BLACK = false; private Node root; // root of the BST // BST helper node data type private class Node { private Value val; // associated data private Node left, right; // links to left and right subtrees private boolean color; // color of parent link private int size; // subtree count public Node(Value val, boolean color, int size) { this.val = val; this.color = color; this.size = size; } } // is node x red; false if x is null ? private boolean isRed(Node x) { if (x == null) return false; return x.color == RED; } // number of node in subtree rooted at x; 0 if x is null private int size(Node x) { if (x == null) return 0; return x.size; } public int size() { return size(root); } public boolean isEmpty() { return root == null; } public void add(int idx, Value val) { if (idx < 0 || idx > size()) throw new IndexOutOfBoundsException(idx + " / " + size()); root = add(root, idx, val); root.color = BLACK; ifdef LogNArray_debug assertTrue(check()); endifdef } // insert the value pair in index k of subtree rooted at h private Node add(Node h, int k, Value val) { if (h == null) ret new Node(val, RED, 1); int t = size(h.left); if (k < t) // replace / fit in left child h.left = add(h.left, k, val); else if (k == t) { // move value to right child, replace value h.right = add(h.right, 0, h.val); h.val = val; } else // replace / fit in right child h.right = add(h.right, k-t-1, val); // fix-up any right-leaning links if (isRed(h.right) && !isRed(h.left)) h = rotateLeft(h); if (isRed(h.left) && isRed(h.left.left)) h = rotateRight(h); if (isRed(h.left) && isRed(h.right)) flipColors(h); h.size = size(h.left) + size(h.right) + 1; ifdef LogNArray_debug print("LogNArray.add: Node " + h.val + " new size: " + h.size); endifdef ret h; } public Value remove(int idx) { if (idx < 0 || idx >= size()) throw new IndexOutOfBoundsException(idx + " / " + size()); Value oldValue = nodeAtIndex(idx); // if both children of root are black, set root to red if (!isRed(root.left) && !isRed(root.right)) root.color = RED; root = remove(root, idx); if (root != null) root.color = BLACK; ifdef LogNArray_debug assertTrue(check()); endifdef ret oldValue; } private Node remove(Node h, int k) { int t = size(h.left); if (k < t) { // remove from left child if (!isRed(h.left) && !isRed(h.left.left)) h = moveRedLeft(h); h.left = remove(h.left, k); } else { // remove from center or right child if (isRed(h.left)) { h = rotateRight(h); t = size(h.left); } if (h.right == null) null; // drop whole node if (!isRed(h.right) && !isRed(h.right.left)) { h = moveRedRight(h); t = size(h.left); } if (k == t) { // replace center value with min of right child h.val = nodeAtIndex(h.right, 0).val; h.right = remove(h.right, 0); } else h.right = remove(h.right, k-t-1); } ret balance(h); } // make a left-leaning link lean to the right private Node rotateRight(Node h) { // assert (h != null) && isRed(h.left); Node x = h.left; h.left = x.right; x.right = h; x.color = x.right.color; x.right.color = RED; x.size = h.size; h.size = size(h.left) + size(h.right) + 1; return x; } // make a right-leaning link lean to the left private Node rotateLeft(Node h) { // assert (h != null) && isRed(h.right); Node x = h.right; h.right = x.left; x.left = h; x.color = x.left.color; x.left.color = RED; x.size = h.size; h.size = size(h.left) + size(h.right) + 1; return x; } // flip the colors of a node and its two children private void flipColors(Node h) { // h must have opposite color of its two children // assert (h != null) && (h.left != null) && (h.right != null); // assert (!isRed(h) && isRed(h.left) && isRed(h.right)) // || (isRed(h) && !isRed(h.left) && !isRed(h.right)); h.color = !h.color; h.left.color = !h.left.color; h.right.color = !h.right.color; } // Assuming that h is red and both h.left and h.left.left // are black, make h.left or one of its children red. private Node moveRedLeft(Node h) { // assert (h != null); // assert isRed(h) && !isRed(h.left) && !isRed(h.left.left); flipColors(h); if (isRed(h.right.left)) { h.right = rotateRight(h.right); h = rotateLeft(h); flipColors(h); } return h; } // Assuming that h is red and both h.right and h.right.left // are black, make h.right or one of its children red. private Node moveRedRight(Node h) { // assert (h != null); // assert isRed(h) && !isRed(h.right) && !isRed(h.right.left); flipColors(h); if (isRed(h.left.left)) { h = rotateRight(h); flipColors(h); } return h; } // restore red-black tree invariant private Node balance(Node h) { // assert (h != null); if (isRed(h.right)) h = rotateLeft(h); if (isRed(h.left) && isRed(h.left.left)) h = rotateRight(h); if (isRed(h.left) && isRed(h.right)) flipColors(h); h.size = size(h.left) + size(h.right) + 1; return h; } /** * Returns the height of the BST (for debugging). * @return the height of the BST (a 1-node tree has height 0) */ public int height() { return height(root); } private int height(Node x) { if (x == null) return -1; return 1 + Math.max(height(x.left), height(x.right)); } public Value get(int k) { ret nodeAtIndex(k).val; } public Node nodeAtIndex(int k) { if (k < 0 || k >= size()) { throw new IndexOutOfBoundsException(k + " / " + size()); } ret nodeAtIndex(root, k); } // the key of rank k in the subtree rooted at x private Node nodeAtIndex(Node x, int k) { // assert x != null; // assert k >= 0 && k < size(x); int t = size(x.left); if (t > k) return nodeAtIndex(x.left, k); else if (t < k) return nodeAtIndex(x.right, k-t-1); else return x; } private boolean check() { if (!isSizeConsistent()) println("Subtree counts not consistent"); if (!is23()) println("Not a 2-3 tree"); if (!isBalanced()) println("Not balanced"); return isSizeConsistent() && is23() && isBalanced(); } // are the size fields correct? private boolean isSizeConsistent() { return isSizeConsistent(root); } private boolean isSizeConsistent(Node x) { if (x == null) return true; if (x.size != size(x.left) + size(x.right) + 1) return false; return isSizeConsistent(x.left) && isSizeConsistent(x.right); } // Does the tree have no red right links, and at most one (left) // red links in a row on any path? private boolean is23() { return is23(root); } private boolean is23(Node x) { if (x == null) return true; if (isRed(x.right)) return false; if (x != root && isRed(x) && isRed(x.left)) return false; return is23(x.left) && is23(x.right); } // do all paths from root to leaf have same number of black edges? private boolean isBalanced() { int black = 0; // number of black links on path from root to min Node x = root; while (x != null) { if (!isRed(x)) black++; x = x.left; } return isBalanced(root, black); } // does every path from the root to a leaf have the given number of black links? private boolean isBalanced(Node x, int black) { if (x == null) return black == 0; if (!isRed(x)) black--; return isBalanced(x.left, black) && isBalanced(x.right, black); } public void clear() { root = null; } }