查找算法

发表于

算法 第三章 查找

符号表

一种存储键值对的数据结构，支持: put 、get 操作。

无序链表中的顺序查找

实现：

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public class SequentialSearchST<Key, Value> {
    private int n;           // number of key-value pairs
    private Node first;      // the linked list of key-value pairs
    // a helper linked list data type
    private class Node {
        private Key key;
        private Value val;
        private Node next;
        public Node(Key key, Value val, Node next)  {
            this.key  = key;
            this.val  = val;
            this.next = next;
        }
    }
    /**
     * Initializes an empty symbol table.
     */
    public SequentialSearchST() {
    }
    /**
     * 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 n;
    }
    /**
     * Returns true if this symbol table is empty.
     *
     * @return {@code true} if this symbol table is empty;
     *         {@code false} otherwise
     */
    public boolean isEmpty() {
        return size() == 0;
    }
    /**
     * Returns true if this symbol table contains the specified key.
     *
     * @param  key the key
     * @return {@code true} if this symbol table contains {@code key};
     *         {@code false} otherwise
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public boolean contains(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to contains() is null");
        return get(key) != null;
    }
    /**
     * Returns the value associated with the given key in this symbol table.
     *
     * @param  key the key
     * @return the value associated with the given key if the key is in the symbol table
     *     and {@code null} if the key is not in the symbol table
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Value get(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to get() is null"); 
        for (Node x = first; x != null; x = x.next) {
            if (key.equals(x.key))
                return x.val;
        }
        return null;
    }
    /**
     * 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 {@code null}.
     *
     * @param  key the key
     * @param  val the value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void put(Key key, Value val) {
        if (key == null) throw new IllegalArgumentException("first argument to put() is null"); 
        if (val == null) {
            delete(key);
            return;
        }
        for (Node x = first; x != null; x = x.next) {
            if (key.equals(x.key)) {
                x.val = val;
                return;
            }
        }
        first = new Node(key, val, first);
        n++;
    }
    /**
     * 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 IllegalArgumentException if {@code key} is {@code null}
     */
    public void delete(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to delete() is null"); 
        first = delete(first, key);
    }
    // delete key in linked list beginning at Node x
    // warning: function call stack too large if table is large
    private Node delete(Node x, Key key) {
        if (x == null) return null;
        if (key.equals(x.key)) {
            n--;
            return x.next;
        }
        x.next = delete(x.next, key);
        return x;
    }
    /**
     * Returns all keys in the symbol table as an {@code Iterable}.
     * To iterate over all of the keys in the symbol table named {@code st},
     * use the foreach notation: {@code for (Key key : st.keys())}.
     *
     * @return all keys in the symbol table
     */
    public Iterable<Key> keys()  {
        Queue<Key> queue = new Queue<Key>();
        for (Node x = first; x != null; x = x.next)
            queue.enqueue(x.key);
        return queue;
    }
        /**
     * Unit tests the {@code SequentialSearchST} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) {
        SequentialSearchST<String, Integer> st = new SequentialSearchST<String, Integer>();
        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));
    }
}

有序数组的二分查找

代码实现：

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public class BinarySearchST<Key extends Comparable<Key>, Value> {
    private static final int INIT_CAPACITY = 2;
    private Key[] keys;
    private Value[] vals;
    private int n = 0;
    /**
     * Initializes an empty symbol table.
     */
    public BinarySearchST() {
        this(INIT_CAPACITY);
    }
    /**
     * Initializes an empty symbol table with the specified initial capacity.
     * @param capacity the maximum capacity
     */
    public BinarySearchST(int capacity) { 
        keys = (Key[]) new Comparable[capacity]; 
        vals = (Value[]) new Object[capacity]; 
    }   
    // resize the underlying arrays
    private void resize(int capacity) {
        assert capacity >= n;
        Key[]   tempk = (Key[])   new Comparable[capacity];
        Value[] tempv = (Value[]) new Object[capacity];
        for (int i = 0; i < n; i++) {
            tempk[i] = keys[i];
            tempv[i] = vals[i];
        }
        vals = tempv;
        keys = tempk;
    }
    /**
     * 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 n;
    }
    /**
     * Returns true if this symbol table is empty.
     *
     * @return {@code true} if this symbol table is empty;
     *         {@code false} otherwise
     */
    public boolean isEmpty() {
        return size() == 0;
    }
    /**
     * Does this symbol table contain the given key?
     *
     * @param  key the key
     * @return {@code true} if this symbol table contains {@code key} and
     *         {@code false} otherwise
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public boolean contains(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to contains() is null");
        return get(key) != null;
    }
    /**
     * Returns the value associated with the given key in this symbol table.
     *
     * @param  key the key
     * @return the value associated with the given key if the key is in the symbol table
     *         and {@code null} if the key is not in the symbol table
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Value get(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to get() is null"); 
        if (isEmpty()) return null;
        int i = rank(key); 
        if (i < n && keys[i].compareTo(key) == 0) return vals[i];
        return null;
    } 
    /**
     * Returns the number of keys in this symbol table strictly less than {@code key}.
     *
     * @param  key the key
     * @return the number of keys in the symbol table strictly less than {@code key}
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public int rank(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to rank() is null"); 
        int lo = 0, hi = n-1; 
        while (lo <= hi) { 
            int mid = lo + (hi - lo) / 2; 
            int cmp = key.compareTo(keys[mid]);
            if      (cmp < 0) hi = mid - 1; 
            else if (cmp > 0) lo = mid + 1; 
            else return mid; 
        } 
        return lo;
    } 
    /**
     * 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 {@code null}.
     *
     * @param  key the key
     * @param  val the value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void put(Key key, Value val)  {
        if (key == null) throw new IllegalArgumentException("first argument to put() is null"); 
        if (val == null) {
            delete(key);
            return;
        }
        int i = rank(key);
        // key is already in table
        if (i < n && keys[i].compareTo(key) == 0) {
            vals[i] = val;
            return;
        }
        // insert new key-value pair
        if (n == keys.length) resize(2*keys.length);
        for (int j = n; j > i; j--)  {
            keys[j] = keys[j-1];
            vals[j] = vals[j-1];
        }
        keys[i] = key;
        vals[i] = val;
        n++;
        assert check();
    } 
    /**
     * Removes the specified key and associated value from this symbol table
     * (if the key is in the symbol table).
     *
     * @param  key the key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void delete(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to delete() is null"); 
        if (isEmpty()) return;
        // compute rank
        int i = rank(key);
        // key not in table
        if (i == n || keys[i].compareTo(key) != 0) {
            return;
        }
        for (int j = i; j < n-1; j++)  {
            keys[j] = keys[j+1];
            vals[j] = vals[j+1];
        }
        n--;
        keys[n] = null;  // to avoid loitering
        vals[n] = null;
        // resize if 1/4 full
        if (n > 0 && n == keys.length/4) resize(keys.length/2);
        assert check();
    } 
    /**
     * Removes the smallest key and associated value from this symbol table.
     *
     * @throws NoSuchElementException if the symbol table is empty
     */
    public void deleteMin() {
        if (isEmpty()) throw new NoSuchElementException("Symbol table underflow error");
        delete(min());
    }
    /**
     * Removes the largest key and associated value from this symbol table.
     *
     * @throws NoSuchElementException if the symbol table is empty
     */
    public void deleteMax() {
        if (isEmpty()) throw new NoSuchElementException("Symbol table underflow error");
        delete(max());
    }
   /***************************************************************************
    *  Ordered symbol table methods.
    ***************************************************************************/
   /**
     * Returns the smallest key in this symbol table.
     *
     * @return the smallest key in this symbol table
     * @throws NoSuchElementException if this symbol table is empty
     */
    public Key min() {
        if (isEmpty()) throw new NoSuchElementException("called min() with empty symbol table");
        return keys[0]; 
    }
    /**
     * Returns the largest key in this symbol table.
     *
     * @return the largest key in this symbol table
     * @throws NoSuchElementException if this symbol table is empty
     */
    public Key max() {
        if (isEmpty()) throw new NoSuchElementException("called max() with empty symbol table");
        return keys[n-1];
    }
    /**
     * Return the kth smallest key in this symbol table.
     *
     * @param  k the order statistic
     * @return the {@code k}th smallest key in this symbol table
     * @throws IllegalArgumentException unless {@code k} is between 0 and
     *        <em>n</em>–1
     */
    public Key select(int k) {
        if (k < 0 || k >= size()) {
            throw new IllegalArgumentException("called select() with invalid argument: " + k);
        }
        return keys[k];
    }
    /**
     * Returns the largest key in this symbol table less than or equal to {@code key}.
     *
     * @param  key the key
     * @return the largest key in this symbol table less than or equal to {@code key}
     * @throws NoSuchElementException if there is no such key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Key floor(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to floor() is null"); 
        int i = rank(key);
        if (i < n && key.compareTo(keys[i]) == 0) return keys[i];
        if (i == 0) return null;
        else return keys[i-1];
    }
    /**
     * Returns the smallest key in this symbol table greater than or equal to {@code key}.
     *
     * @param  key the key
     * @return the smallest key in this symbol table greater than or equal to {@code key}
     * @throws NoSuchElementException if there is no such key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Key ceiling(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to ceiling() is null"); 
        int i = rank(key);
        if (i == n) return null; 
        else return keys[i];
    }
    /**
     * Returns the number of keys in this symbol table in the specified range.
     *
     * @param lo minimum endpoint
     * @param hi maximum endpoint
     * @return the number of keys in this symbol table between {@code lo} 
     *         (inclusive) and {@code hi} (inclusive)
     * @throws IllegalArgumentException if either {@code lo} or {@code hi}
     *         is {@code null}
     */
    public int size(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to size() is null"); 
        if (hi == null) throw new IllegalArgumentException("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);
    }
    /**
     * Returns all keys in this symbol table as an {@code Iterable}.
     * To iterate over all of the keys in the symbol table named {@code st},
     * use the foreach notation: {@code for (Key key : st.keys())}.
     *
     * @return all keys in this symbol table
     */
    public Iterable<Key> keys() {
        return keys(min(), max());
    }
    /**
     * Returns all keys in this symbol table in the given range,
     * as an {@code Iterable}.
     *
     * @param lo minimum endpoint
     * @param hi maximum endpoint
     * @return all keys in this symbol table between {@code lo} 
     *         (inclusive) and {@code hi} (inclusive)
     * @throws IllegalArgumentException if either {@code lo} or {@code hi}
     *         is {@code null}
     */
    public Iterable<Key> keys(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to keys() is null"); 
        if (hi == null) throw new IllegalArgumentException("second argument to keys() is null"); 
        Queue<Key> queue = new Queue<Key>(); 
        if (lo.compareTo(hi) > 0) return queue;
        for (int i = rank(lo); i < rank(hi); i++) 
            queue.enqueue(keys[i]);
        if (contains(hi)) queue.enqueue(keys[rank(hi)]);
        return queue; 
    }
   /***************************************************************************
    *  Check internal invariants.
    ***************************************************************************/
    private boolean check() {
        return isSorted() && rankCheck();
    }
    // are the items in the array in ascending order?
    private boolean isSorted() {
        for (int i = 1; i < size(); i++)
            if (keys[i].compareTo(keys[i-1]) < 0) return false;
        return true;
    }
    // check that rank(select(i)) = i
    private boolean rankCheck() {
        for (int i = 0; i < size(); i++)
            if (i != rank(select(i))) return false;
        for (int i = 0; i < size(); i++)
            if (keys[i].compareTo(select(rank(keys[i]))) != 0) return false;
        return true;
    }
    /**
     * Unit tests the {@code BinarySearchST} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) { 
        BinarySearchST<String, Integer> st = new BinarySearchST<String, Integer>();
        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));
    }
}

二分查找：

  • 递归

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    public int rank(Key key, int lo, int hi) {
        if (hi < lo) return lo;
        int mid = lo + (hi - lo) / 2;
        int cmp = key.compareTo(keys[mid]);
        if (cmp < 0) retunr rank(key, lo, mid -1);
        else if (cmp > 0) {return rank(key, mid+1, hi);}
        else return mid;
    }

  • 迭代

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    public int rank(Key key) {
        int lo = 0; hi = N - 1;
        while(lo <= hi) {
             int mid = lo + (hi - lo) / 2;
             int cmp = key.compareTo(keys[mid]);
             if (cmp < 0) hi = mid - 1;
             else if (cmp > 0) lo = mid + 1;
             else return mid;
        }
        return lo;
    }

    ​

二叉查找树 （BST）

定义：一棵二叉查找树（BST） 是一棵二叉树，其中每个结点都含有一个 Comparable 的键（以及相关联的值）且每个结点的键都大于其左子树中的任意结点的键而小于右子树的任意结点的键。

代码实现：

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public class BST<Key extends Comparable<Key>, Value> {
    private Node root;             // root of BST
    private class Node {
        private Key key;           // sorted by key
        private Value val;         // associated data
        private Node left, right;  // left and right subtrees
        private int size;          // number of nodes in subtree
        public Node(Key key, Value val, int size) {
            this.key = key;
            this.val = val;
            this.size = size;
        }
    }
    /**
     * Initializes an empty symbol table.
     */
    public BST() {
    }
    /**
     * Returns true if this symbol table is empty.
     * @return {@code true} if this symbol table is empty; {@code false} otherwise
     */
    public boolean isEmpty() {
        return size() == 0;
    }
    /**
     * 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);
    }
    // return number of key-value pairs in BST rooted at x
    private int size(Node x) {
        if (x == null) return 0;
        else return x.size;
    }
    /**
     * Does this symbol table contain the given key?
     *
     * @param  key the key
     * @return {@code true} if this symbol table contains {@code key} and
     *         {@code false} otherwise
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public boolean contains(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to contains() is null");
        return get(key) != null;
    }
    /**
     * 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 {@code null} if the key is not in the symbol table
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Value get(Key key) {
        return get(root, key);
    }
    private Value get(Node x, Key key) {
        if (key == null) throw new IllegalArgumentException("calls get() with a null key");
        if (x == null) return null;
        int cmp = key.compareTo(x.key);
        if      (cmp < 0) return get(x.left, key);
        else if (cmp > 0) return get(x.right, key);
        else              return x.val;
    }
    /**
     * 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 {@code null}.
     *
     * @param  key the key
     * @param  val the value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void put(Key key, Value val) {
        if (key == null) throw new IllegalArgumentException("calls put() with a null key");
        if (val == null) {
            delete(key);
            return;
        }
        root = put(root, key, val);
        assert check();
    }
    private Node put(Node x, Key key, Value val) {
        if (x == null) return new Node(key, val, 1);
        int cmp = key.compareTo(x.key);
        if      (cmp < 0) x.left  = put(x.left,  key, val);
        else if (cmp > 0) x.right = put(x.right, key, val);
        else              x.val   = val;
        x.size = 1 + size(x.left) + size(x.right);
        return x;
    }
    /**
     * 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("Symbol table underflow");
        root = deleteMin(root);
        assert check();
    }
    private Node deleteMin(Node x) {
        if (x.left == null) return x.right;
        x.left = deleteMin(x.left);
        x.size = size(x.left) + size(x.right) + 1;
        return x;
    }
    /**
     * 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("Symbol table underflow");
        root = deleteMax(root);
        assert check();
    }
    private Node deleteMax(Node x) {
        if (x.right == null) return x.left;
        x.right = deleteMax(x.right);
        x.size = size(x.left) + size(x.right) + 1;
        return x;
    }
    /**
     * 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 IllegalArgumentException if {@code key} is {@code null}
     */
    public void delete(Key key) {
        if (key == null) throw new IllegalArgumentException("calls delete() with a null key");
        root = delete(root, key);
        assert check();
    }
    private Node delete(Node x, Key key) {
        if (x == null) return null;
        int cmp = key.compareTo(x.key);
        if      (cmp < 0) x.left  = delete(x.left,  key);
        else if (cmp > 0) x.right = delete(x.right, key);
        else { 
            if (x.right == null) return x.left;
            if (x.left  == null) return x.right;
            Node t = x;
            x = min(t.right);
            x.right = deleteMin(t.right);
            x.left = t.left;
        } 
        x.size = size(x.left) + size(x.right) + 1;
        return x;
    } 
    /**
     * 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("calls min() with empty symbol table");
        return min(root).key;
    } 
    private Node min(Node x) { 
        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("calls max() with empty symbol table");
        return max(root).key;
    } 
    private Node max(Node x) {
        if (x.right == null) return x; 
        else                 return max(x.right); 
    } 
    /**
     * Returns the largest key in the symbol table less than or equal to {@code key}.
     *
     * @param  key the key
     * @return the largest key in the symbol table less than or equal to {@code key}
     * @throws NoSuchElementException if there is no such key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Key floor(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to floor() is null");
        if (isEmpty()) throw new NoSuchElementException("calls floor() with empty symbol table");
        Node x = floor(root, key);
        if (x == null) return null;
        else return x.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; 
    } 
    public Key floor2(Key key) {
        return floor2(root, key, null);
    }
    private Key floor2(Node x, Key key, Key best) {
        if (x == null) return best;
        int cmp = key.compareTo(x.key);
        if      (cmp  < 0) return floor2(x.left, key, best);
        else if (cmp  > 0) return floor2(x.right, key, x.key);
        else               return x.key;
    } 
    /**
     * Returns the smallest key in the symbol table greater than or equal to {@code key}.
     *
     * @param  key the key
     * @return the smallest key in the symbol table greater than or equal to {@code key}
     * @throws NoSuchElementException if there is no such key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Key ceiling(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to ceiling() is null");
        if (isEmpty()) throw new NoSuchElementException("calls ceiling() with empty symbol table");
        Node x = ceiling(root, key);
        if (x == null) return null;
        else return x.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) { 
            Node t = ceiling(x.left, key); 
            if (t != null) return t;
            else return x; 
        } 
        return ceiling(x.right, key); 
    } 
    /**
     * Return the key in the symbol table whose rank is {@code k}.
     * This is the (k+1)st smallest key in the symbol table.
     *
     * @param  k the order statistic
     * @return the key in the symbol table of rank {@code k}
     * @throws IllegalArgumentException unless {@code k} is between 0 and
     *        <em>n</em>–1
     */
    public Key select(int k) {
        if (k < 0 || k >= size()) {
            throw new IllegalArgumentException("argument to select() is invalid: " + k);
        }
        Node x = select(root, k);
        return x.key;
    }
    // Return key of rank k. 
    private Node select(Node x, int k) {
        if (x == null) return null; 
        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 {@code key}.
     *
     * @param  key the key
     * @return the number of keys in the symbol table strictly less than {@code key}
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public int rank(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to rank() is null");
        return rank(key, root);
    } 
    // Number of keys in the subtree less than key.
    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); 
    } 
    /**
     * Returns all keys in the symbol table as an {@code Iterable}.
     * To iterate over all of the keys in the symbol table named {@code st},
     * use the foreach notation: {@code for (Key key : st.keys())}.
     *
     * @return all keys in the symbol table
     */
    public Iterable<Key> keys() {
        if (isEmpty()) return new Queue<Key>();
        return keys(min(), max());
    }
    /**
     * Returns all keys in the symbol table in the given range,
     * as an {@code Iterable}.
     *
     * @param  lo minimum endpoint
     * @param  hi maximum endpoint
     * @return all keys in the symbol table between {@code lo} 
     *         (inclusive) and {@code hi} (inclusive)
     * @throws IllegalArgumentException if either {@code lo} or {@code hi}
     *         is {@code null}
     */
    public Iterable<Key> keys(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to keys() is null");
        if (hi == null) throw new IllegalArgumentException("second argument to keys() is null");
        Queue<Key> queue = new Queue<Key>();
        keys(root, queue, lo, hi);
        return queue;
    } 
    private void keys(Node x, Queue<Key> 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.
     *
     * @param  lo minimum endpoint
     * @param  hi maximum endpoint
     * @return the number of keys in the symbol table between {@code lo} 
     *         (inclusive) and {@code hi} (inclusive)
     * @throws IllegalArgumentException if either {@code lo} or {@code hi}
     *         is {@code null}
     */
    public int size(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to size() is null");
        if (hi == null) throw new IllegalArgumentException("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);
    }
    /**
     * 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));
    }
    /**
     * Returns the keys in the BST in level order (for debugging).
     *
     * @return the keys in the BST in level order traversal
     */
    public Iterable<Key> levelOrder() {
        Queue<Key> keys = new Queue<Key>();
        Queue<Node> queue = new Queue<Node>();
        queue.enqueue(root);
        while (!queue.isEmpty()) {
            Node x = queue.dequeue();
            if (x == null) continue;
            keys.enqueue(x.key);
            queue.enqueue(x.left);
            queue.enqueue(x.right);
        }
        return keys;
    }
  /*************************************************************************
    *  Check integrity of BST 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");
        return isBST() && isSizeConsistent() && isRankConsistent();
    }
    // 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.size != 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;
    }
    /**
     * Unit tests the {@code BST} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) { 
        BST<String, Integer> st = new BST<String, Integer>();
        for (int i = 0; !StdIn.isEmpty(); i++) {
            String key = StdIn.readString();
            st.put(key, i);
        }
        for (String s : st.levelOrder())
            StdOut.println(s + " " + st.get(s));
        StdOut.println();
        for (String s : st.keys())
            StdOut.println(s + " " + st.get(s));
    }
}

  • 二叉树中的删除操作：在删除结点 X 后用它的后继结点填补它的位置。因为 x 有一个右子结点，因此它的后继结点就是其右子树的最小结点。

    • 将指向即将被删除的结点的链接保存为 t；

    • 将 x 指向它的后继结点 min(t.right) ;

    • 将 x 的右链接指向 deleteMin(t.right) , 也就是在删除后所有结点仍然都大于 x.key 的子二叉查找树；

    • 将 x 的左链接（本为空）设为 t.left (其下所有的键都小于被删除的结点和他的后继结点)；

      实现：

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    private Node delete(Node x, Key key) {
        if (x == null) return null;
      
        int cmp = key.compareTo(x.key);
        if      (cmp < 0) x.left  = delete(x.left,  key);
        else if (cmp > 0) x.right = delete(x.right, key);
        else { 
            if (x.right == null) return x.left;
            if (x.left  == null) return x.right;
            Node t = x;
            x = min(t.right);
            x.right = deleteMin(t.right);
            x.left = t.left;
        } 
        x.size = size(x.left) + size(x.right) + 1;
        return x;
    }

  • 二叉树的范围查找

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    public Iterable<Key> keys(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to keys() is null");
        if (hi == null) throw new IllegalArgumentException("second argument to keys() is null");
      
        Queue<Key> queue = new Queue<Key>();
        keys(root, queue, lo, hi);
        return queue;
    } 
      
    private void keys(Node x, Queue<Key> 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); 
    }

    ​

红黑树

2-3 查找树

定义：

一棵 2 -3 查找树或为一棵空树，或由一下结点组成：

  • 2 - 结点 ，含有一个键（及其对应的值）和两个链接，左链接指向的 2-3 树中的键都小于该结点，右链接指向的 2- 3树中的键都大于该结点。
  • 3 - 结点， 含有两个键（及其对应的值）和三个链接，左链接指向的 2-3树中的键都小于该结点，中链接指向的 2-3 树中的键都位于该结点的两个键之间，右链接指向的 2-3 树中的键都大于该结点。
  • 一棵完美平衡的 2-3 查找树中的所有空链接到根结点的距离都是相同的。
插入操作
  • 向 2- 结点中插入新建
    • 使该2-结点变为 3- 结点
  • 向一棵只含有一个 3-结点的树中插入新建
    • 创建一个 4- 结点，同时将 4- 结点分解为 2-3 树
  • 向一个父结点为2-结点的 3- 结点中插入新建
    • 将 3- 结点替换为 4- 结点，将 4 - 结点分解为两个 2- 结点并将中键移动到至父结点。
  • 向一个父结点为 3- 结点的 3- 结点中插入新建
    • 如果从插入结点到根结点的路径上全都是3- 结点，那么我们的根结点就是一个 4- 结点，我们将执行分解根结点，将树高加 1；

红黑树（红黑二叉查找树）

定义：

红链接：将两个2-结点连接起来构成一个3 - 结点；

黑链接：2 -3 树中的普通链接；

  • 定义：
    • 红链接均为左链接；
    • 没有任何一个结点同时和两条红链接相连；
    • 该树是完美黑色平衡的，即任意空链接到根结点的路径上的黑链接数量相同。
  • 满足这样定义的红黑树和相应的 2-3 树是一一对应的。
  • 优点：二叉查找树的简洁高效的查找算法和2-3树中高效的平衡插入算法 完美的结合。

结点表示：

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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 size;          // subtree count
        public Node(Key key, Value val, boolean color, int size) {
            this.key = key;
            this.val = val;
            this.color = color;
            this.size = size;
        }
    }
    /**
     * Initializes an empty symbol table.
     */
    public RedBlackBST() {
    }
旋转

旋转操作可以保持红黑树的两个重要性质：有序性和完美平衡性。

  • 左旋转 ： 如果右子结点是红色的而左子结点是黑色的，进行左旋转。

    code 实现：

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    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;
    }

    ​

  • 右旋转：如果左子结点是红色的且它的左子结点也是红色的，进行右旋转。

    code prompt

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    // 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;
    }
插入

插入新建时，使用旋转操作可以保证2 -3树和红黑树的一一对应关系。

  • 向单个 2- 结点插入新建

    • 左插入，新增一个红色结点即可，将这个 2 - 结点变为一个 3- 结点；
    • 右插入，将会产生一条红色的右链接，运用rotateRight 方法将其旋转为红色左链接并修正根结点的链接。

  • 向树底部的2- 结点插入新建，方法同上

  • 向一颗双键树（一个 3- 结点）中插入新建

    • 新键大于原树中的两个键
      • 用红链接和新结点相连，父结点左右链接都是红链接，同时将红链接变为黑链接（颜色变换）。
    • 新键小于原树中的两个键
      • 用红链接和新结点相连，此时存在连续两条红链接，运用右旋转，旋转后变为红色右链接，运用颜色变换。
    • 新键介于原树中的两个键之间
      • 用红链接和新结点相连，运用左旋转方式，变为红色左链接，进一步运用右旋转方法，变为红色右链接，再利用颜色变换。

  • 颜色变换：如果左右结点均为红色，进行颜色转换。

    Code Prompt：

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    // 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;
    }

  • 根结点总是黑色

    颜色变换会使根结点变为红色，红色的根结点说明根结点是一个 3- 结点的一部分，因此我们在每次插入后都会将根结点设为黑色，同时将树的黑链接高度加1；

  • 向树底部的 3- 结点插入新建， 是以上方法的集合使用；

  • 将红链接在树中向上传递

    要在一个 3- 结点下插入新键， 先创建一个临时的 4- 结点， 将其分解并将红链接由中间键传递给它的父结点。重复上述步骤，就能将红链接在树中向上传递，直至遇到一个2 - 结点或者根结点。

代码实现：

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public class RedBlackBST<Key extends Comparable<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 size;          // subtree count
        public Node(Key key, Value val, boolean color, int size) {
            this.key = key;
            this.val = val;
            this.color = color;
            this.size = size;
        }
    }
    /**
     * 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.size;
    } 
    /**
     * 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 {@code true} if this symbol table is empty and {@code 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 {@code null} if the key is not in the symbol table
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Value get(Key key) {
        if (key == null) throw new IllegalArgumentException("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 {@code true} if this symbol table contains {@code key} and
     *     {@code false} otherwise
     * @throws IllegalArgumentException if {@code key} is {@code 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 {@code null}.
     *
     * @param key the key
     * @param val the value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void put(Key key, Value val) {
        if (key == null) throw new IllegalArgumentException("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.size = 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 IllegalArgumentException if {@code key} is {@code null}
     */
    public void delete(Key key) { 
        if (key == null) throw new IllegalArgumentException("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.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;
    }
   /***************************************************************************
    *  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("calls 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("calls 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 {@code key}.
     * @param key the key
     * @return the largest key in the symbol table less than or equal to {@code key}
     * @throws NoSuchElementException if there is no such key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Key floor(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to floor() is null");
        if (isEmpty()) throw new NoSuchElementException("calls 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 {@code key}.
     * @param key the key
     * @return the smallest key in the symbol table greater than or equal to {@code key}
     * @throws NoSuchElementException if there is no such key
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Key ceiling(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to ceiling() is null");
        if (isEmpty()) throw new NoSuchElementException("calls 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 key in the symbol table whose rank is {@code k}.
     * This is the (k+1)st smallest key in the symbol table. 
     *
     * @param  k the order statistic
     * @return the key in the symbol table of rank {@code k}
     * @throws IllegalArgumentException unless {@code k} is between 0 and
     *        <em>n</em>–1
     */
    public Key select(int k) {
        if (k < 0 || k >= size()) {
            throw new IllegalArgumentException("argument to select() is invalid: " + k);
        }
        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 {@code key}.
     * @param key the key
     * @return the number of keys in the symbol table strictly less than {@code key}
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public int rank(Key key) {
        if (key == null) throw new IllegalArgumentException("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 {@code Iterable}.
     * To iterate over all of the keys in the symbol table named {@code st},
     * use the foreach notation: {@code for (Key key : st.keys())}.
     * @return all keys in the symbol table as an {@code Iterable}
     */
    public Iterable<Key> keys() {
        if (isEmpty()) return new Queue<Key>();
        return keys(min(), max());
    }
    /**
     * Returns all keys in the symbol table in the given range,
     * as an {@code Iterable}.
     *
     * @param  lo minimum endpoint
     * @param  hi maximum endpoint
     * @return all keys in the sybol table between {@code lo} 
     *    (inclusive) and {@code hi} (inclusive) as an {@code Iterable}
     * @throws IllegalArgumentException if either {@code lo} or {@code hi}
     *    is {@code null}
     */
    public Iterable<Key> keys(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to keys() is null");
        if (hi == null) throw new IllegalArgumentException("second argument to keys() is null");
        Queue<Key> queue = new Queue<Key>();
        // 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<Key> 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.
     *
     * @param  lo minimum endpoint
     * @param  hi maximum endpoint
     * @return the number of keys in the sybol table between {@code lo} 
     *    (inclusive) and {@code hi} (inclusive)
     * @throws IllegalArgumentException if either {@code lo} or {@code hi}
     *    is {@code null}
     */
    public int size(Key lo, Key hi) {
        if (lo == null) throw new IllegalArgumentException("first argument to size() is null");
        if (hi == null) throw new IllegalArgumentException("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.size != 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 {@code RedBlackBST} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) { 
        RedBlackBST<String, Integer> st = new RedBlackBST<String, Integer>();
        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();
    }
}
删除
  • 删除
  • 删除最小键
红黑树的性质

所有基于红黑树的符号表实现都能保证操作的运行时间为对数级别。

一棵大小为 N 的红黑树的高度不会超过 2lgN

一棵大小为 N 的红黑树中，根结点到任意结点的平均路径长度为～1.00lgN。

散列表

散列函数

  • 除留余数法

  • Java 中 hasCode() 的两种实现：

    • 数组的索引， 产生一个 0 到 M-1 （M=32） 的整数

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      private int hash(Key x) {
          return ((x.hasCode) & 0x7fffffff ) % M;
      }

    • 自定义 hasCode() 方法

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      public int hashCode() {
           int hash = 1;
           hash = 31*hash + who.hashCode();
           hash = 31*hash + when.hashCode();
           hash = 31*hash + ((Double) amount).hashCode();
           return hash;
           // return Objects.hash(who, when, amount);
       }

基于拉链法的散列表

拉链法

定义：将大小为M 的数组中的每一个元素指向一条链表，链表中的每个结点都存储了散列值为该元素的索引的键值对。基本思想是选择足够大的 M，使得链表尽可能短以提高查找效率。

用 M 条链表来保存 N 个键，无论键在各个链表中如何分布，链表 的平均长度肯定是 N / M;

代码实现：

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public class SeparateChainingHashST<Key, Value> {
    private static final int INIT_CAPACITY = 4;
    private int n;                                // number of key-value pairs
    private int m;                                // hash table size
    private SequentialSearchST<Key, Value>[] st;  // array of linked-list symbol tables
    /**
     * Initializes an empty symbol table.
     */
    public SeparateChainingHashST() {
        this(INIT_CAPACITY);
    } 
    /**
     * Initializes an empty symbol table with {@code m} chains.
     * @param m the initial number of chains
     */
    public SeparateChainingHashST(int m) {
        this.m = m;
        st = (SequentialSearchST<Key, Value>[]) new SequentialSearchST[m];
        for (int i = 0; i < m; i++)
            st[i] = new SequentialSearchST<Key, Value>();
    } 
    // resize the hash table to have the given number of chains,
    // rehashing all of the keys
    private void resize(int chains) {
        SeparateChainingHashST<Key, Value> temp = new SeparateChainingHashST<Key, Value>(chains);
        for (int i = 0; i < m; i++) {
            for (Key key : st[i].keys()) {
                temp.put(key, st[i].get(key));
            }
        }
        this.m  = temp.m;
        this.n  = temp.n;
        this.st = temp.st;
    }
    // hash value between 0 and m-1
    private int hash(Key key) {
        return (key.hashCode() & 0x7fffffff) % m;
    } 
    /**
     * 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 n;
    } 
    /**
     * Returns true if this symbol table is empty.
     *
     * @return {@code true} if this symbol table is empty;
     *         {@code false} otherwise
     */
    public boolean isEmpty() {
        return size() == 0;
    }
    /**
     * Returns true if this symbol table contains the specified key.
     *
     * @param  key the key
     * @return {@code true} if this symbol table contains {@code key};
     *         {@code false} otherwise
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public boolean contains(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to contains() is null");
        return get(key) != null;
    } 
    /**
     * Returns the value associated with the specified key in this symbol table.
     *
     * @param  key the key
     * @return the value associated with {@code key} in the symbol table;
     *         {@code null} if no such value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Value get(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to get() is null");
        int i = hash(key);
        return st[i].get(key);
    } 
    /**
     * 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 {@code null}.
     *
     * @param  key the key
     * @param  val the value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void put(Key key, Value val) {
        if (key == null) throw new IllegalArgumentException("first argument to put() is null");
        if (val == null) {
            delete(key);
            return;
        }
        // double table size if average length of list >= 10
        if (n >= 10*m) resize(2*m);
        int i = hash(key);
        if (!st[i].contains(key)) n++;
        st[i].put(key, val);
    } 
    /**
     * 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 IllegalArgumentException if {@code key} is {@code null}
     */
    public void delete(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to delete() is null");
        int i = hash(key);
        if (st[i].contains(key)) n--;
        st[i].delete(key);
        // halve table size if average length of list <= 2
        if (m > INIT_CAPACITY && n <= 2*m) resize(m/2);
    } 
    // return keys in symbol table as an Iterable
    public Iterable<Key> keys() {
        Queue<Key> queue = new Queue<Key>();
        for (int i = 0; i < m; i++) {
            for (Key key : st[i].keys())
                queue.enqueue(key);
        }
        return queue;
    } 
    /**
     * Unit tests the {@code SeparateChainingHashST} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) { 
        SeparateChainingHashST<String, Integer> st = new SeparateChainingHashST<String, Integer>();
        for (int i = 0; !StdIn.isEmpty(); i++) {
            String key = StdIn.readString();
            st.put(key, i);
        }
        // print keys
        for (String s : st.keys()) 
            StdOut.println(s + " " + st.get(s)); 
    }
}

基于线性探测的散列表

开放地址散列表

定义：用大小为 M 的数组保存 N个键值对，其中 M> N 。依靠数组中的空位解决碰撞冲突。

核心思想：与其将内存用作链表，不如将它们作为在散列表的空元素。

线性探测法： 当碰撞发生时，直接检查散列表中的下一个位置。
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public class LinearProbingHashST<Key, Value> {
    private static final int INIT_CAPACITY = 4;
    private int n;           // number of key-value pairs in the symbol table
    private int m;           // size of linear probing table
    private Key[] keys;      // the keys
    private Value[] vals;    // the values
    /**
     * Initializes an empty symbol table.
     */
    public LinearProbingHashST() {
        this(INIT_CAPACITY);
    }
    /**
     * Initializes an empty symbol table with the specified initial capacity.
     *
     * @param capacity the initial capacity
     */
    public LinearProbingHashST(int capacity) {
        m = capacity;
        n = 0;
        keys = (Key[])   new Object[m];
        vals = (Value[]) new Object[m];
    }
    /**
     * 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 n;
    }
    /**
     * Returns true if this symbol table is empty.
     *
     * @return {@code true} if this symbol table is empty;
     *         {@code false} otherwise
     */
    public boolean isEmpty() {
        return size() == 0;
    }
    /**
     * Returns true if this symbol table contains the specified key.
     *
     * @param  key the key
     * @return {@code true} if this symbol table contains {@code key};
     *         {@code false} otherwise
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public boolean contains(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to contains() is null");
        return get(key) != null;
    }
    // hash function for keys - returns value between 0 and M-1
    private int hash(Key key) {
        return (key.hashCode() & 0x7fffffff) % m;
    }
    // resizes the hash table to the given capacity by re-hashing all of the keys
    private void resize(int capacity) {
        LinearProbingHashST<Key, Value> temp = new LinearProbingHashST<Key, Value>(capacity);
        for (int i = 0; i < m; i++) {
            if (keys[i] != null) {
                temp.put(keys[i], vals[i]);
            }
        }
        keys = temp.keys;
        vals = temp.vals;
        m    = temp.m;
    }
    /**
     * 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 {@code null}.
     *
     * @param  key the key
     * @param  val the value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public void put(Key key, Value val) {
        if (key == null) throw new IllegalArgumentException("first argument to put() is null");
        if (val == null) {
            delete(key);
            return;
        }
        // double table size if 50% full
        if (n >= m/2) resize(2*m);
        int i;
        for (i = hash(key); keys[i] != null; i = (i + 1) % m) {
            if (keys[i].equals(key)) {
                vals[i] = val;
                return;
            }
        }
        keys[i] = key;
        vals[i] = val;
        n++;
    }
    /**
     * Returns the value associated with the specified key.
     * @param key the key
     * @return the value associated with {@code key};
     *         {@code null} if no such value
     * @throws IllegalArgumentException if {@code key} is {@code null}
     */
    public Value get(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to get() is null");
        for (int i = hash(key); keys[i] != null; i = (i + 1) % m)
            if (keys[i].equals(key))
                return vals[i];
        return null;
    }
    /**
     * 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 IllegalArgumentException if {@code key} is {@code null}
     */
    public void delete(Key key) {
        if (key == null) throw new IllegalArgumentException("argument to delete() is null");
        if (!contains(key)) return;
        // find position i of key
        int i = hash(key);
        while (!key.equals(keys[i])) {
            i = (i + 1) % m;
        }
        // delete key and associated value
        keys[i] = null;
        vals[i] = null;
        // rehash all keys in same cluster
        i = (i + 1) % m;
        while (keys[i] != null) {
            // delete keys[i] an vals[i] and reinsert
            Key   keyToRehash = keys[i];
            Value valToRehash = vals[i];
            keys[i] = null;
            vals[i] = null;
            n--;
            put(keyToRehash, valToRehash);
            i = (i + 1) % m;
        }
        n--;
        // halves size of array if it's 12.5% full or less
        if (n > 0 && n <= m/8) resize(m/2);
        assert check();
    }
    /**
     * Returns all keys in this symbol table as an {@code Iterable}.
     * To iterate over all of the keys in the symbol table named {@code st},
     * use the foreach notation: {@code for (Key key : st.keys())}.
     *
     * @return all keys in this symbol table
     */
    public Iterable<Key> keys() {
        Queue<Key> queue = new Queue<Key>();
        for (int i = 0; i < m; i++)
            if (keys[i] != null) queue.enqueue(keys[i]);
        return queue;
    }
    // integrity check - don't check after each put() because
    // integrity not maintained during a delete()
    private boolean check() {
        // check that hash table is at most 50% full
        if (m < 2*n) {
            System.err.println("Hash table size m = " + m + "; array size n = " + n);
            return false;
        }
        // check that each key in table can be found by get()
        for (int i = 0; i < m; i++) {
            if (keys[i] == null) continue;
            else if (get(keys[i]) != vals[i]) {
                System.err.println("get[" + keys[i] + "] = " + get(keys[i]) + "; vals[i] = " + vals[i]);
                return false;
            }
        }
        return true;
    }
    /**
     * Unit tests the {@code LinearProbingHashST} data type.
     *
     * @param args the command-line arguments
     */
    public static void main(String[] args) { 
        LinearProbingHashST<String, Integer> st = new LinearProbingHashST<String, Integer>();
        for (int i = 0; !StdIn.isEmpty(); i++) {
            String key = StdIn.readString();
            st.put(key, i);
        }
        // print keys
        for (String s : st.keys()) 
            StdOut.println(s + " " + st.get(s)); 
    }
}
删除操作

将要删除的元素位置至为空，同时将数组中被删除键的右侧的所有键重新插入散列表。

散列表的使用率 N/M，它不能大于1；此值最好不要超过 1/ 2；