JAVA: Add to RedBlackBST class a method height() that computes the height of the
ID: 3679289 • Letter: J
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 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(); } }
Explanation / Answer
import java.util.NoSuchElementException;
import secondweek.Queue;
import secondweek.StdIn;
import secondweek.StdOut;
public class RedBlackBST<Key, 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 0;
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();
}
}
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