Question

JAVA: Add to RedBlackBST class a method height() that computes the height of the tree (notice the height includes red and black edges and the height of a tree is the maximum depth of any node in the tree). Develop a recursive method which takes O(n), where n is the number of keys in the tree and space proportional to the height. You need to argue both the correctness of your method and its time complexity. Write a client that inserts the integer keys from 1 to 15 -in any order- and use your new method to print the height of the tree. This was not clear for many experts. So how i took this question, was to write a new method height that computes the height of the red black tree and takes linear time (big oh of n).. Prove that it takes O(n) and test it with a client with integers 1 - 15 in any order ***************this is the Class******************** public class RedBlackBST, Value> { 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 Key key; // key private Value val; // associated data private Node left, right; // links to left and right subtrees private boolean color; // color of parent link private int N; // subtree count public Node(Key key, Value val, boolean color, int N) { this.key = key; this.val = val; this.color = color; this.N = N; } } /**  * Initializes an empty symbol table.  */ public RedBlackBST() { } /***************************************************************************  * Node helper methods.  ***************************************************************************/ // 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.N; } /**  * Returns the number of key-value pairs in this symbol table.  * @return the number of key-value pairs in this symbol table  */ public int size() { return size(root); } /**  * Is this symbol table empty?  * @return true if this symbol table is empty and false otherwise  */ public boolean isEmpty() { return root == null; } /***************************************************************************  * Standard BST search.  ***************************************************************************/ /**  * Returns the value associated with the given key.  * @param key the key  * @return the value associated with the given key if the key is in the symbol table  * and null if the key is not in the symbol table  * @throws NullPointerException if key is null  */ public Value get(Key key) { if (key == null) throw new NullPointerException("argument to get() is null"); return get(root, key); } // value associated with the given key in subtree rooted at x; null if no such key private Value get(Node x, Key key) { while (x != null) { int cmp = key.compareTo(x.key); if (cmp < 0) x = x.left; else if (cmp > 0) x = x.right; else return x.val; } return null; } /**  * Does this symbol table contain the given key?  * @param key the key  * @return true if this symbol table contains key and  * false otherwise  * @throws NullPointerException if key is null  */ public boolean contains(Key key) { return get(key) != null; } /***************************************************************************  * Red-black tree insertion.  ***************************************************************************/ /**  * Inserts the specified key-value pair into the symbol table, overwriting the old   * value with the new value if the symbol table already contains the specified key.  * Deletes the specified key (and its associated value) from this symbol table  * if the specified value is null.  *  * @param key the key  * @param val the value  * @throws NullPointerException if key is null  */ public void put(Key key, Value val) { if (key == null) throw new NullPointerException("first argument to put() is null"); if (val == null) { delete(key); return; } root = put(root, key, val); root.color = BLACK; // assert check(); } // insert the key-value pair in the subtree rooted at h private Node put(Node h, Key key, Value val) { if (h == null) return new Node(key, val, RED, 1); int cmp = key.compareTo(h.key); if (cmp < 0) h.left = put(h.left, key, val); else if (cmp > 0) h.right = put(h.right, key, val); else h.val = 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.N = size(h.left) + size(h.right) + 1; return h; } /***************************************************************************  * Red-black tree deletion.  ***************************************************************************/ /**  * Removes the smallest key and associated value from the symbol table.  * @throws NoSuchElementException if the symbol table is empty  */ public void deleteMin() { if (isEmpty()) throw new NoSuchElementException("BST underflow"); // if both children of root are black, set root to red if (!isRed(root.left) && !isRed(root.right)) root.color = RED; root = deleteMin(root); if (!isEmpty()) root.color = BLACK; // assert check(); } // delete the key-value pair with the minimum key rooted at h private Node deleteMin(Node h) { if (h.left == null) return null; if (!isRed(h.left) && !isRed(h.left.left)) h = moveRedLeft(h); h.left = deleteMin(h.left); return balance(h); } /**  * Removes the largest key and associated value from the symbol table.  * @throws NoSuchElementException if the symbol table is empty  */ public void deleteMax() { if (isEmpty()) throw new NoSuchElementException("BST underflow"); // if both children of root are black, set root to red if (!isRed(root.left) && !isRed(root.right)) root.color = RED; root = deleteMax(root); if (!isEmpty()) root.color = BLACK; // assert check(); } // delete the key-value pair with the maximum key rooted at h private Node deleteMax(Node h) { if (isRed(h.left)) h = rotateRight(h); if (h.right == null) return null; if (!isRed(h.right) && !isRed(h.right.left)) h = moveRedRight(h); h.right = deleteMax(h.right); return balance(h); } /**  * Removes the specified key and its associated value from this symbol table   * (if the key is in this symbol table).   *  * @param key the key  * @throws NullPointerException if key is null  */ public void delete(Key key) { if (key == null) throw new NullPointerException("argument to delete() is null"); if (!contains(key)) return; // if both children of root are black, set root to red if (!isRed(root.left) && !isRed(root.right)) root.color = RED; root = delete(root, key); if (!isEmpty()) root.color = BLACK; // assert check(); } // delete the key-value pair with the given key rooted at h private Node delete(Node h, Key key) { // assert get(h, key) != null; if (key.compareTo(h.key) < 0) { if (!isRed(h.left) && !isRed(h.left.left)) h = moveRedLeft(h); h.left = delete(h.left, key); } else { if (isRed(h.left)) h = rotateRight(h); if (key.compareTo(h.key) == 0 && (h.right == null)) return null; if (!isRed(h.right) && !isRed(h.right.left)) h = moveRedRight(h); if (key.compareTo(h.key) == 0) { Node x = min(h.right); h.key = x.key; h.val = x.val; // h.val = get(h.right, min(h.right).key); // h.key = min(h.right).key; h.right = deleteMin(h.right); } else h.right = delete(h.right, key); } return balance(h); } /***************************************************************************  * Red-black tree helper functions.  ***************************************************************************/ // 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.N = h.N; h.N = 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.N = h.N; h.N = 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.N = size(h.left) + size(h.right) + 1; return h; } /***************************************************************************  * Utility functions.  ***************************************************************************/ /**  * 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)); } /***************************************************************************  * Ordered symbol table methods.  ***************************************************************************/ /**  * Returns the smallest key in the symbol table.  * @return the smallest key in the symbol table  * @throws NoSuchElementException if the symbol table is empty  */ public Key min() { if (isEmpty()) throw new NoSuchElementException("called min() with empty symbol table"); return min(root).key; } // the smallest key in subtree rooted at x; null if no such key private Node min(Node x) { // assert x != null; if (x.left == null) return x; else return min(x.left); } /**  * Returns the largest key in the symbol table.  * @return the largest key in the symbol table  * @throws NoSuchElementException if the symbol table is empty  */ public Key max() { if (isEmpty()) throw new NoSuchElementException("called max() with empty symbol table"); return max(root).key; } // the largest key in the subtree rooted at x; null if no such key private Node max(Node x) { // assert x != null; if (x.right == null) return x; else return max(x.right); } /**  * Returns the largest key in the symbol table less than or equal to key.  * @param key the key  * @return the largest key in the symbol table less than or equal to key  * @throws NoSuchElementException if there is no such key  * @throws NullPointerException if key is null  */ public Key floor(Key key) { if (key == null) throw new NullPointerException("argument to floor() is null"); if (isEmpty()) throw new NoSuchElementException("called floor() with empty symbol table"); Node x = floor(root, key); if (x == null) return null; else return x.key; } // the largest key in the subtree rooted at x less than or equal to the given key private Node floor(Node x, Key key) { if (x == null) return null; int cmp = key.compareTo(x.key); if (cmp == 0) return x; if (cmp < 0) return floor(x.left, key); Node t = floor(x.right, key); if (t != null) return t; else return x; } /**  * Returns the smallest key in the symbol table greater than or equal to key.  * @param key the key  * @return the smallest key in the symbol table greater than or equal to key  * @throws NoSuchElementException if there is no such key  * @throws NullPointerException if key is null  */ public Key ceiling(Key key) { if (key == null) throw new NullPointerException("argument to ceiling() is null"); if (isEmpty()) throw new NoSuchElementException("called ceiling() with empty symbol table"); Node x = ceiling(root, key); if (x == null) return null; else return x.key; } // the smallest key in the subtree rooted at x greater than or equal to the given key private Node ceiling(Node x, Key key) { if (x == null) return null; int cmp = key.compareTo(x.key); if (cmp == 0) return x; if (cmp > 0) return ceiling(x.right, key); Node t = ceiling(x.left, key); if (t != null) return t; else return x; } /**  * Return the kth smallest key in the symbol table.  * @param k the order statistic  * @return the kth smallest key in the symbol table  * @throws IllegalArgumentException unless k is between 0 and  * N  1  */ public Key select(int k) { if (k < 0 || k >= size()) throw new IllegalArgumentException(); Node x = select(root, k); return x.key; } // the key of rank k in the subtree rooted at x private Node select(Node x, int k) { // assert x != null; // assert k >= 0 && k < size(x); int t = size(x.left); if (t > k) return select(x.left, k); else if (t < k) return select(x.right, k-t-1); else return x; } /**  * Return the number of keys in the symbol table strictly less than key.  * @param key the key  * @return the number of keys in the symbol table strictly less than key  * @throws NullPointerException if key is null  */ public int rank(Key key) { if (key == null) throw new NullPointerException("argument to rank() is null"); return rank(key, root); } // number of keys less than key in the subtree rooted at x private int rank(Key key, Node x) { if (x == null) return 0; int cmp = key.compareTo(x.key); if (cmp < 0) return rank(key, x.left); else if (cmp > 0) return 1 + size(x.left) + rank(key, x.right); else return size(x.left); } /***************************************************************************  * Range count and range search.  ***************************************************************************/ /**  * Returns all keys in the symbol table as an Iterable.  * To iterate over all of the keys in the symbol table named st,  * use the foreach notation: for (Key key : st.keys()).  * @return all keys in the sybol table as an Iterable  */ public Iterable keys() { if (isEmpty()) return new Queue(); return keys(min(), max()); } /**  * Returns all keys in the symbol table in the given range,  * as an Iterable.  * @return all keys in the sybol table between lo   * (inclusive) and hi (exclusive) as an Iterable  * @throws NullPointerException if either lo or hi  * is null  */ public Iterable keys(Key lo, Key hi) { if (lo == null) throw new NullPointerException("first argument to keys() is null"); if (hi == null) throw new NullPointerException("second argument to keys() is null"); Queue queue = new Queue(); // if (isEmpty() || lo.compareTo(hi) > 0) return queue; keys(root, queue, lo, hi); return queue; } // add the keys between lo and hi in the subtree rooted at x // to the queue private void keys(Node x, Queue queue, Key lo, Key hi) { if (x == null) return; int cmplo = lo.compareTo(x.key); int cmphi = hi.compareTo(x.key); if (cmplo < 0) keys(x.left, queue, lo, hi); if (cmplo <= 0 && cmphi >= 0) queue.enqueue(x.key); if (cmphi > 0) keys(x.right, queue, lo, hi); } /**  * Returns the number of keys in the symbol table in the given range.  * @return the number of keys in the sybol table between lo   * (inclusive) and hi (exclusive)  * @throws NullPointerException if either lo or hi  * is null  */ public int size(Key lo, Key hi) { if (lo == null) throw new NullPointerException("first argument to size() is null"); if (hi == null) throw new NullPointerException("second argument to size() is null"); if (lo.compareTo(hi) > 0) return 0; if (contains(hi)) return rank(hi) - rank(lo) + 1; else return rank(hi) - rank(lo); } /***************************************************************************  * Check integrity of red-black tree data structure.  ***************************************************************************/ private boolean check() { if (!isBST()) StdOut.println("Not in symmetric order"); if (!isSizeConsistent()) StdOut.println("Subtree counts not consistent"); if (!isRankConsistent()) StdOut.println("Ranks not consistent"); if (!is23()) StdOut.println("Not a 2-3 tree"); if (!isBalanced()) StdOut.println("Not balanced"); return isBST() && isSizeConsistent() && isRankConsistent() && is23() && isBalanced(); } // does this binary tree satisfy symmetric order? // Note: this test also ensures that data structure is a binary tree since order is strict private boolean isBST() { return isBST(root, null, null); } // is the tree rooted at x a BST with all keys strictly between min and max // (if min or max is null, treat as empty constraint) // Credit: Bob Dondero's elegant solution private boolean isBST(Node x, Key min, Key max) { if (x == null) return true; if (min != null && x.key.compareTo(min) <= 0) return false; if (max != null && x.key.compareTo(max) >= 0) return false; return isBST(x.left, min, x.key) && isBST(x.right, x.key, max); } // are the size fields correct? private boolean isSizeConsistent() { return isSizeConsistent(root); } private boolean isSizeConsistent(Node x) { if (x == null) return true; if (x.N != size(x.left) + size(x.right) + 1) return false; return isSizeConsistent(x.left) && isSizeConsistent(x.right); } // check that ranks are consistent private boolean isRankConsistent() { for (int i = 0; i < size(); i++) if (i != rank(select(i))) return false; for (Key key : keys()) if (key.compareTo(select(rank(key))) != 0) return false; return true; } // 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); } /**  * Unit tests the RedBlackBST data type.  */ public static void main(String[] args) { RedBlackBST st = new RedBlackBST(); for (int i = 0; !StdIn.isEmpty(); i++) { String key = StdIn.readString(); st.put(key, i); } for (String s : st.keys()) StdOut.println(s + " " + st.get(s)); StdOut.println(); } }