数组序列(Array-Based Sequences)

1. 二维数组转置

grid = [['.', '.', '.', '.', '.', '.'],
['.', 'O', 'O', '.', '.', '.'],
['O', 'O', 'O', 'O', '.', '.'],
['O', 'O', 'O', 'O', 'O', '.'],
['.', 'O', 'O', 'O', 'O', 'O'],
['O', 'O', 'O', 'O', 'O', '.'],
['O', 'O', 'O', 'O', '.', '.'],
['.', 'O', 'O', '.', '.', '.'],
['.', '.', '.', '.', '.', '.']]

#转置四种种方法
grid_T =map(list,zip(*grid))
#grid_T = list(zip(*grid))
#grid_T = list(list(i) for i in zip(*grid))
#grid_T = [[row[i] for row in grid] for i in range(len(grid[0]))]

for line in grid_T:
    dots = ''
    for dot in line:
        dots += dot
    print(dots,end='\n')

2.有序二维数组查找

题目描述：在一个二维数组中（每个一维数组的长度相同），每一行都按照从左到右递增的顺序排序，每一列都按照从上到下递增的顺序排序。请完成一个函数，输入这样的一个二维数组和一个整数，判断数组中是否含有该整数

class Solution:
    def Find_lower_left(self,target,array):
        """        从左下向右上检测        """
        cols = len(array[0])-1
        rows = len(array)-1
        i = rows
        j=0
        while j<=cols and i>=0:
            if target>array[i][j]:
                j +=1
            elif target<array[i][j]:
                i -=1
            else:
                return True
        return False

    def Find_upper_right(self,target,array):
        """        从右上向左下检测       """
        rows = len(array)-1
        cols = len(array[0])-1
        i = 0
        j = cols
        while i<=rows and j >=0:
            if target>array[i][j]:
                i +=1
            elif target< array[i][j]:
                j -=1
            else:
                return True
        return False


if __name__=="__main__":
    solution = Solution()
    target = 11
    array = [[1,3,5],[7,9,11],[13,15,17]]
    print(solution.Find_lower_left(target,array))
    print(solution.Find_upper_right(target,array))

链表(Linked Lists)

1.单项列表实现反向打印

#-*- coding:utf-8 -*-
class ListNode:
    def __init__(self, x):
        self.val = x
        self.next = None

class Solution:
    """
    反向打印链表
    """
    def printListFromTailToHead(self, listNode):
        # write code here
        pre_header = listNode
        result = []
        while pre_header != None:
            #result.append(pre_header.val)
            #  return result[::-1] or result.reverse()
            result.insert(0,pre_header.val)
            pre_header = pre_header.next
        return result

    def printListFromTailToHead_1(self, listNode):
        #   递归的方法
        if listNode is not None:
            return list(self.printListFromTailToHead_1(listNode.next))+ list([listNode.val])
        else:
            return []

if __name__ == "__main__":
    solution = Solution()
    Node_1 = ListNode(1)
    Node_2 = ListNode(2)
    Node_3 = ListNode(3)
    Node_1.next = Node_2
    Node_2.next = Node_3

    print(solution.printListFromTailToHead(Node_1))

2.双向列表


class _DoubleLinkedBase:
    '''A base class providing a doubly linked list representation.'''
    class _Node:
        __slots__ = '_element','_prev','_next'

        def __init__(self,element,prev,next_):
            self._element = element
            self._prev = prev
            self._next = next_
    def __init__(self):
        # 这里header和trailer只是一个前后标识，并没有实际存储作用
        self._header = self._Node(None,None,None)
        self._trailer = self._Node(None,None,None)
        self._header._next = self._trailer
        self._trailer._prev = self._header
        self._size = 0
    def __len__(self):
        return self._size
    def is_empty(self):
        return self._size == 0
    def _insert_between(self,e,predecessor,successor):
        newest = self._Node(e,predecessor,successor)
        predecessor._next = newest
        successor._prev = newest
        self._size += 1
        return newest
    def _delete_node(self,node):
        predecessor = node._prev
        successor = node._next
        predecessor._next = successor
        successor._prev = predecessor
        self._size -= 1
        element = node._element
        node._prev = node._next = node._element = None
        return element

3.循环列表

class CircularQueue:
    '''使用循环链表实现队列的存储。'''
    class _Node:
        __slots__ = '_element','_next'

        def __init__(self,element,next_):
            self._element = element
            self._next = next_

    def __init__(self):
        self._tail = None
        self._size = 0

    def __len__(self):
        return self._size

    def is_empty(self):
        return self._size == 0

    def first(self):
        if self.is_empty():
            raise IndexError
        head = self._tail._next
        return head._element

    def dequeue(self):
        if self.is_empty():
            raise IndexError
        oldhead = self._tail._next
        if self._size==1:
            self._tail = None
        else:
            self._tail._next = oldhead._next
        self._size -= 1
        return oldhead._element

    def enqueue(self,e):
        newest = self._Node(e,None)
        if self.is_empty():
            newest._next = newest
        else:
            newest._next = self._tail._next
            self._tail._next = newest
        self._tail = newest
        self._size +=1

    def rotat(self):
        '''将前部元素旋转到队列的后面'''
        if self._size > 0:
            self._tail = self._tail._next

4.位置列表

我们定义了一个 PositionalList ADT以及一个更简单的 class Position 抽象数据类型来描述列表中的位置。位置实例是一个简单对象，只支持以下方法：p.element() 返回存储在位置p的元素

class PositionalList(_DoubleLinkedBase):
    '''元素的顺序链表容器允许位置访问'''
    class Position:
        '''表示单个元素位置的抽象'''
        def __init__(self,container,node):
            self._container = container
            self._node = node

        def element(self):
            return self._node._element

        def __eq__(self, other):
            return type(other) is type(self) and other._node is self._node

        def __ne__(self, other):
            return not(self == other)   # opposite of __eq__

    def _validate(self, p):
        if not isinstance(p, self.Position):
            raise TypeError(' p must be proper Position type')
        if p._container is not self:
            raise ValueError('p does not belong to this container')
        if p._node._next is None:
            raise ValueError(' p is no longer valid')
        return p._node

    def _make_position(self, node):
        '''添加position属性'''
        if node is self._header or node is self._trailer:
            return None
        else:
            return self.Position(self, node)

    def first(self):
        return self._make_position(self._header._next)

    def last(self):
        return self._make_position(self._trailer._prev)

    def before(self, p):
        node = self._validate(p)
        return self._make_position(node._prev)

    def after(self, p):
        node = self._validate(p)
        return self._make_position(node._next)

    def __iter__(self):
        cursor = self.first()
        while cursor is not None:
            yield cursor.element()
            cursor = self.after(cursor)

    def _insert_between(self,e,predecessor,successor):
        node = super()._insert_between(e,predecessor,successor)
        return self._make_position(node)

    def add_first(self, e):
        return self._insert_between(e, self._header,self._header._next)

    def add_last(self, e):
        return self._insert_between(e, self._trailer._prev,self._trailer)

    def add_before(self, p ,e):
        original = self._validate(p)
        return self._insert_between(e, original._prev, original)

    def add_after(self, p, e):
        original = self._validate(p)
        return self._insert_between(e, original, original._next)

    def delete(self, p):
        original = self._validate(p)
        return self._delete_node(original)

    def replace(self, p, e):
        original = self._validate(p)
        old_value = original._element
        original._element = e
        return old_value

栈(Stacks)

1.利用列表实现栈

class ArrayStack:
    '''LIFO Stack implementation using a Python list as underlying storage.'''
    def __init__(self):
        self.data = []
    def __len__(self):
        return len(self.data)
    def is_empty(self):
        return len(self.data) == 0
    def push(self, e):
        self.data.append(e)
    def top(self):
        if self.is_empty():
            raise Exception('Stack is empty')
        return self.data[-1]
    def pop(self):
        if self.is_empty():
            raise Exception('Stack is empty')
        return self.data.pop()

2.利用链表实现栈

class LinkedStack:
    class _Node:
        __slots__ = '_element','_next'      # __slots__变量限制该class能添加的属性
        def __init__(self,element,next):
            self._element = element
            self._next = next
    def __init__(self):
        self._head = None                   # 链表的第一个node
        self._size = 0
    def __len__(self):
        return self._size
    def is_empty(self):
        return self._size == 0
    def push(self, e):
        self._head = self._Node(e,self._head)
        self._size += 1
    def top(self):
        if self.is_empty():
            raise IndexError('Stack is empty')
        return self._head._element
    def pop(self):
        if self.is_empty():
            raise IndexError('Stack is empty')
        answer = self._head._element
        self._head = self._head._next
        self._size -= 1
        return answer

Link-Based vs. Array-Based Sequences

数组：

数组提供基于整数索引的元素的O(1)时间访问。相反，定位第k个元素在链表中需要O(k)时间从开始遍历列表，或者可能是O(n-k)时间，如果从双端向后遍历链表。
基于数组的表示通常使用比链接结构更少的内存,但有时会出现存储空间大于存储元素情况，造成内存浪费。
插入和删除操作非常昂贵(后继内存会相应移动)

链表优势：

基于链表的结构支持O(1)时间插入和删除在任意位置。而基于数组的列表类，在索引k处插入或弹出使用O(n-k+1)时间，因为所有后续元素会被循环移位
利用分散内存存储元素，没有浪费内存，插入和删除操作方便

3.字符串公式计算


def Evaluate(expr):
    ops = LinkedStack()
    vals = LinkedStack()
    for i in expr:
        if i == '(':
            pass
        elif i == '+':
            ops.push(i)
        elif i == '-':
            ops.push(i)
        elif i == '*':
            ops.push(i)
        elif i == '/':
            ops.push(i)
        elif i == '//':
            ops.push(i)
        elif i == ')': # 如果字符为）,弹出运算符合操作数，计算结果并压入栈
            op = ops.pop()
            v = vals.pop()
            if op == '+':
                v = vals.pop() + v
            if op == '-':
                v = vals.pop() - v
            if op == '*':
                v = vals.pop() * v
            if op == '/':
                v = vals.pop() / v
            if op == '//':
                v = vals.pop()//v
            vals.push(v)
        else:
            vals.push(int(i))
    return vals.pop()

if __name__ == '__main__':
   print( Evaluate('(1+(2*2))'))

4.判断字符串数学公式是否合法

def match_math(expr):
    '''Return True if all delimiters are properly match; False otherwise.'''
    lefty = '({['
    righty = ')}]'
    S = LinkedStack()
    for c in expr:
        if c in lefty:
            S.push(c)
        elif c in righty:
            if S.is_empty():
                return False
            if righty.index(c) != lefty.index(S.pop()):
                return False
    return S.is_empty()

if __name__ == '__main__':
   print( match_math('(1+(2*2))'))

5.文件内容顺序倒置

def reverse_file(file_name):
    '''Overwrite given file with its contents line-by-line reversed.'''
    S = LinkedStack()
    original = open(file_name)
    for line in original:
        S.push(line.rstrip('\n'))
    original.close()

    output = open(file_name,'w')
    while not S.is_empty():
        output.write(S.pop() + '\n')
    output.close()


if __name__ == '__main__':
    reverse_file('tst.txt')

队列(Queues)

1.python内部集成队列

import queue                     #单向
from collections import deque    #双向

2.数组实现队列

class ArrayQueue:
    DEFAULT_CAPACITY = 10                                   #固定大小，循环使用
    def __init__(self):
        self._data = [None]*ArrayQueue.DEFAULT_CAPACITY     #是一个具有固定容量的列表实例的引用。
        self._size = 0                                      #表示存储在队列中的元素的当前数量
        self._front = 0                                     #队列第一个元素在数据中的索引

    def __len__(self):
        return self._size

    def is_empty(self):
        return self._size == 0

    def first(self):
        '''Return (but do not remove) the element at the front of the queue'''
        if self.is_empty():
            raise IndexError('Queue is empty')
        return self._data[self._front]

    def dequeue(self):
        '''Remove and return the first element of the queue'''
        if self.is_empty():
            raise IndexError('Queue is empty')
        answer = self._data[self._front]
        self._data[self._front] = None
        self._front = (self._front +1)%len(self._data)
        self._size -= 1
        return answer

    def enqueue(self, e):
        if self._size == len(self._data):
            self._resize(2 * len(self._data))
        avail = (self._front + self._size)%len(self._data)      #在_data范围内循环使用
        self._data[avail] = e
        self._size += 1

    def _resize(self,cap):
        old = self._data
        self._data = [None] * cap
        walk = self._front
        for k in range(self._size):
            self._data[k] = old[walk]
            walk = (1+walk) % len(old)
        self._front = 0

3.链表实现队列


class LinkedQueue:
    class _Node:
        __slots__ = '_element','_next'

        def __init__(self, element, next):
            self._element = element
            self._next = next

    def __init__(self):
        self._head = None
        self._tail = None
        self._size = 0

    def __len__(self):
        return self._size

    def is_empty(self):
        return self._size == 0

    def first(self):
        if self.is_empty():
            raise IndexError('Queue is empty')
        return self._head._element

    def dequeue(self):
        if self.is_empty():
            raise IndexError('Queue is empty')
        answer = self._head._element
        self._head = self._head._next
        self._size -= 1
        if self.is_empty():
            self._tail = None
        return answer

    def enqueue(self, e):
        newest = self._Node(e,None)
        if self.is_empty():
            self._head = newest
        else:
            self._tail._next = newest
        self._tail = newest
        self._size += 1

树(Trees)

将链表插入的灵活性和有序数组查找的高效性结合的数据结构：

p.element( )：返回在节点p点存储的元素

T.root():返回根节点T,or None if T is empty.

T.is root(p): 如果是根节点则返回为真.

T.parent(p): 返回节点p的父节点,or None if p is the root of T.

T.num children(p): 返回节点p的子节点数量

T.children(p): 生成节点p的子结点迭代器.

T.is leaf(p): 判断是否为叶节点.

len(T): 计算T中存储的元素个数

T.is empty( ): 判断T是否为空.

T.positions( ):产生遍历树中所有节点的迭代器（前向，后向，中序，宽度优先）

iter(T): 产生遍历树中所有存储在T中元素的迭代器

这里实现上述基本方法的预定义类Tree

class Tree:
    '''Abstract base class representing a tree structure.'''
    class Position:

        def element(self):
            raise NotImplementedError(' must be implemented by subclass') #必须由子类实现

        def __eq__(self, other):
            raise NotImplementedError(' must be implemented by subclass')

        def __ne__(self, other):
            return not (self == other)

    def root(self):
        raise NotImplementedError(' must be implemented by subclass')

    def parent(self, p):
        raise NotImplementedError(' must be implemented by subclass')

    def num_children(self, p):
        raise NotImplementedError(' must be implemented by subclass')

    def children(self, p):
        raise NotImplementedError(' must be implemented by subclass')

    def __len__(self):
        raise NotImplementedError(' must be implemented by subclass')

    def __iter__(self):
        for p in self.positions():
            yield p.element()

    def is_root(self, p):
        return self.root() == p

    def is_leaf(self, p):
        return self.num_children(p) == 0

    def is_empty(self):
        return len(self) == 0

    def positions(self):
        #return self.preorder()
        #return self.inorder()
        #return self.postorder()
        #return self.breadthfirst()

1. 计算树的深度和高度

    def depth(self, p):
    """自上而下计算深度"""
        if self.is_root(p):
            return 0
        else:
            return 1+self.depth(self.parent(p))

#######################################################################

    def _height1(self):  # works, but O(n^2) worst-case time
        """Return the height of the tree."""
        return max(self.depth(p) for p in self.positions() if self.is_leaf(p))

    def _height2(self, p):  # time is linear in size of subtree
        """Return the height of the subtree rooted at Position p."""
        if self.is_leaf(p):
            return 0
        else:
            return 1 + max(self._height2(c) for c in self.children(p))

    def height(self, p):
        if p is None:
            p = self.root()
        return self._height2(p)

2.前序遍历

输入：树结构T，遍历以p为根的子树

Algorithm preorder(T, p):
    perform the “visit” action for position p
    for each child c in T.children(p) do
        preorder(T, c)        #递归遍历以C为根的子树

###############################################################

def preorder(self):
    if not self.is_empty():
        for p in self._subtree_preorder(self.root()):
            yield p

def _subtree_preorder(self, p):
"""迭代器"""
    yield p                                         #先对p进行访问
    for c in self.children(p):
        for other in self._subtree_preorder(c):
            yield other

3.中序遍历

输入：遍历以p为根的子树

Algorithm inorder(p):
    if p has a left child lc then
        inorder(lc)             #递归遍历P的左子树
    perform the “visit” action for position p
    if p has a right child rc then
        inorder(rc)             #递归遍历P的右子树

#############################################################

    def inorder(self):
        if not self.is_empty():
            for p in self._subtree_inorder(self.root()):
                yield p

    def _subtree_inorder(self, p):
        if self.left(p) is not None:
            for other in self._subtree_inorder(self.left(p)):
                yield other
        yield p
        if self.right(p) is not None:
            for other in self._subtree_inorder(self.right(p)):
                yield other

4.后序遍历

输入：树结构T，遍历以p为根的子树

Algorithm postorder(T, p):
    for each child c in T.children(p) do
        postorder(T, c)             #递归遍历以C为根的子树
    perform the “visit” action for position p

#########################################################
    def postorder(self):
        """Generate a postorder iteration of positions in the tree."""
        if not self.is_empty():
            for p in self._subtree_postorder(self.root()):  # start recursion
                yield p

    def _subtree_postorder(self, p):
        """Generate a postorder iteration of positions in subtree rooted at p."""
        for c in self.children(p):  # for each child c
            for other in self._subtree_postorder(c):  # do postorder of c's subtree
                yield other  # yielding each to our caller
        yield p  # visit p after its subtrees

5.广度优先遍历

广度优先遍历是用于游戏中常用的方法

输入：树结构T

Algorithm breadthfirst(T):
    Initialize queue Q to contain T.root( )
    while Q not empty do
        p = Q.dequeue( )                     #P是队列中最古老的条目
        perform the “visit” action for position p
        for each child c in T.children(p) do
            Q.enqueue(c)                     #将P的孩子添加到队列的末尾以供稍后访问

############################################################################

    def breadthfirst(self):
        if not self.is_empty():
            fringe = LinkedQueue()
            fringe.enqueue(self.root())
            while not fringe.is_empty():
                p = fringe.dequeue()
                yield p
                for c in self.children(p):
                    fringe.enqueue(c)

二叉树(Binary Trees)

令T为非空二叉树，n为节点数， $n_{E}$ 为外部节点(叶节点)， $n_{I}$ 为内部节点，h为高度：

$\begin{align*} & 1.h+1 \leq n \leq 2^{h+1}-1 \\ & 2.1\leq n_{E} \leq 2^{h} \\ & 3.h \leq n_{I}\leq2^{h}-1\\ & 4.log(n+1)-1\leq h \leq n-1 \end{align*}$

对于一棵满二叉树，外部节点或者说是叶子节点数是n，则内部节点数是n-1。

T.left(p): 返回左节点，or None if p has no left child.

T.right(p): 返回右节点,or None if p has no right child.

T.sibling(p): 返回同级的兄弟节点,or None if p has no sibling.

class BinaryTree(Tree):
    '''Abstract base class representing a binary tree structure.'''
    def left(self, p):
        raise NotImplementedError(' must be implemented by subclass')

    def right(self, p):
        raise NotImplementedError(' must be implemented by subclass')

    def sibling(self, p):
        '''返回兄弟结点位置'''
        parent = self.parent(p)
        if parent is None:
            return None
        else:
            if p == self.left(parent):
                return self.right(parent)
            else:
                return self.left(parent)

    def children(self, p):
        if self.left(p) is not None:
            yield self.left(p)
        if self.right(p) is not None:
            yield  self.right(p)

1.根据前向和中序遍历得到树结构


class TreeNode:
    def __init__(self,x):
        self.val = x
        self.right = None
        self.left = None

class Solution:
    def reConstructBinaryTree(self, pre, tin):
        """
        :param pre: 前向遍历
        :param tin: 中序遍历
        :return: 树结构
        """
        if not pre or not tin:
            return None
        root = TreeNode(pre.pop(0))
        index = tin.index(root.val)
        # 递归
        root.left = self.reConstructBinaryTree(pre,tin[:index])
        root.right = self.reConstructBinaryTree(pre,tin[index+1:])
        return root

if __name__=="__main__":
    solution = Solution()
    pre = [1,2,4,7,3,5,6,8]
    tin = [4,7,2,1,5,3,8,6]
    tree = solution.reConstructBinaryTree(pre,tin)
    print(tree)

利用链表结构实现二叉树

此处将上述两节中未实现函数实现，并添加如下函数

T.add root(e): 为空树T创建根节点, 并存储元素e,返回根节点; an error occurs if thetree is not empty.

T.add left(p, e): 创建新节点存储e, 作为左子结点连接到 p节点, and return the resulting position;an error occurs if p already has a left child.

T.add right(p, e): 创建新节点存储e, 作为右子结点连接到 p节点, and return the resulting position;an error occurs if p already has a right child.

T.replace(p, e):用元素e替换p节点所存储元素,and return the previously stored element

T.delete(p): 移除节点p,用子结点替换,并返回节点p所存储元素，an error occurs if p has two children.

T.attach(p, T1, T2): 将树T1、树T2分别作为左子树和右子树，合并到p节点。an error condition occurs if p is not a leaf.

class LinkedBinaryTree(BinaryTree):

    class _Node:
        __slots__ = '_element','_parent','_left','_right'
        def __init__(self,element,parent=None,left=None,right=None):
            self._element = element
            self._parent = parent
            self._left = left
            self._right = right

    class Position(BinaryTree.Position):

        def __init__(self, container, node):
            self._container = container
            self._node = node

        def element(self):
            return self._node._element

        def __eq__(self, other):
            return type(other) is type(self) and other._node is self._node

    def _validate(self, p):
        if not isinstance(p , self.Position):
            raise TypeError
        if p._container is not self:
            raise ValueError
        if p._node._parent is p._node:
            raise ValueError
        return p._node

    def _make_position(self, node):
        return self.Position(self, node) if node is not None else None

    def __init__(self):
        self._root = None
        self._size = 0

    def __len__(self):
        return self._size

    def root(self):
        return self._make_position(self._root)

    def parent(self, p):
        node = self._validate(p)
        return self._make_position(node._parent)

    def left(self, p):
        node = self._validate(p)
        return self._make_position(node._left)

    def right(self, p):
        node = self._validate(p)
        return self._make_position(node._right)

    def num_children(self, p):
        node = self._validate(p)
        count = 0
        if node._left is not None:
            count += 1
        if node._right is not None:
            count += 1
        return count

    def _add_root(self, e):
        if self._root is not None:
            raise ValueError
        self._size = 1
        self._root = self._Node(e)
        return self._make_position(self._root)

    def _add_left(self, p, e):
        node = self._validate(p)
        if node._left is not None:
            raise ValueError
        self._size += 1
        node._left = self._Node(e,node)
        return self._make_position(node._left)

    def _add_right(self, p, e):
        node = self._validate(p)
        if node._right is not None:
            raise ValueError
        self._size += 1
        node._right = self._Node(e,node)
        return self._make_position(node._right)

    def _replace(self, p, e):
        node = self._validate(p)
        old = node._element
        node._element = e
        return old

    def _delete(self, p):
        node = self._validate(p)
        if self.num_children(p) == 2:
            raise ValueError
        child = node._left if node._left else node._right
        if child is not None:
            child._parent = node._parent
        if node is self._root:
            self._root = child
        else:
            parent = node._parent
            if node is parent._left:
                parent._left = child
            else:
                parent._right = child
        self._size -= 1
        node._parent = node
        return node._element

    def _attach(self, p, t1, t2):
        node = self._validate(p)
        if not self.is_leaf(p):
          raise ValueError('position must be leaf')
        if not type(self) is type(t1) is type(t2):    # all 3 trees must be same type
          raise TypeError('Tree types must match')
        self._size += len(t1) + len(t2)
        if not t1.is_empty():         # attached t1 as left subtree of node
          t1._root._parent = node
          node._left = t1._root
          t1._root = None             # set t1 instance to empty
          t1._size = 0
        if not t2.is_empty():         # attached t2 as right subtree of node
          t2._root._parent = node
          node._right = t2._root
          t2._root = None             # set t2 instance to empty
          t2._size = 0

基于链表结构的二叉树的性能

优先队列(Priority Queue)

总是处理下一个优先级最高的事件，一个合适的数据结构应该支持两种操作：删除优先级最高元素和插入元素。

优先队列和队列(删除最老的元素)以及栈(删除最新元素)类似

P.add(k, v)	插入键值对(k,v)
P.min()	返回优先队列中键值最小的键值对; 如果为队列空报错
P.remove_min()	删除并返回优先队列中键值最小的键值对; 如果为队列空报错
P.is_empty( ):	是否为空
len(P):	返回键值对的个数

class PriorityQueueBase:

    class _Item:
        __slots__ = '_key', '_value'

        def __init__(self, k, v):
            self._key = k
            self._value = v

        def __lt__(self, other):                      # self < other
            return self._key < other._key

    def is_empty(self):
        return len(self) == 0

1. 利用数组实现（无序）

class UnsortedPriorityQueue(PriorityQueueBase):

    def _find_min(self):                                # 选择排序
        if self.is_empty():
            raise IndexError
        small = self._data.first()
        walk = self._data.after(small)
        while walk is not None:
            if walk.element() < small.element():
                small = walk
            walk = self._data.after(walk)
        return small

    def __init__(self):
        self._data = PositionalList()

    def __len__(self):
        return len(self._data)

    def add(self, key, value):
        self._data.add_first(self._Item(key , value))

    def min(self):
        p = self._find_min()
        item = p.element()
        return (item._key, item._value)

    def remove_min(self):
        p = self._find_min()
        item = self._data.delete(p)
        return (item._key, item._value)

2. 利用数组实现（有序）

class SortedPriorityQueue(PriorityQueueBase):

    def __init__(self):
        self._data = PositionalList()

    def __len__(self):
        return len(self._data)

    def add(self, key, value):
        newest = self._Item(key, value)
        walk = self._data.last()
        while walk is not None and newest < walk.element():     # 插入排序
            walk = self._data.before(walk)
        if walk is None:
            self._data.add_first(newest)
        else:
            self._data.add_after(walk,newest)

    def min(self):
        """Return but do not remove (k,v) tuple with minimum key.

        Raise Empty exception if empty.
        """
        if self.is_empty():
            raise IndexError('Priority queue is empty.')
        p = self._data.first()
        item = p.element()
        return (item._key, item._value)

    def remove_min(self):
        """Remove and return (k,v) tuple with minimum key.

        Raise Empty exception if empty.
        """
        if self.is_empty():
            raise IndexError('Priority queue is empty.')
        item = self._data.delete(self._data.first())
        return (item._key, item._value)

堆结构(Heap)

数据结构二叉堆能够更好的实现优先队列的基本操作。是一组能够用堆有序的完全二叉树排序的元素，并在数组中按照层级存储(不使用数组的第一个位置)

堆有序：当一颗二叉树的每个节点都大于等于它的两个子节点时。且根节点是二叉堆的最大节点

在堆中，位置k的父节点的位置为 $\left \lfloor k/2 \right \rfloor$ ,两个子节点的位置分别为2k和2k+1。
可以通过计算数组的索引在树中移动：从a[k]向上一层就令k等于k/2，向下一层则令k等于2k和2k+1.
一颗大小为N的完全二叉树的高度为 $\left \lfloor logN \right \rfloor$

插入元素后上浮(Up-Heap Bubbling)

当某个节点的优先级上升
如在堆底加入一个新的元素时

我们需要由下至上恢复堆的顺序，让新元素上浮到合适位置。

删除最大元素后下沉(Down-Heap Bubbling)

当某个节点的优先级下降
如将根节点替换为一个较小元素

我们需要由上至下恢复堆的顺序，让新元素下沉到合适位置。


class HeapPriorityQueue(PriorityQueueBase): # base class defines _Item
  """"""

  #------------------------------ nonpublic behaviors ------------------------------
  def _parent(self, j):
    return (j-1) // 2

  def _left(self, j):         #数组存储从index=0开始
    return 2*j + 1
  
  def _right(self, j):       
    return 2*j + 2

  def _has_left(self, j):
    return self._left(j) < len(self._data)     # index beyond end of list?
  
  def _has_right(self, j):
    return self._right(j) < len(self._data)    # index beyond end of list?
  
  def _swap(self, i, j):
    """Swap the elements at indices i and j of array."""
    self._data[i], self._data[j] = self._data[j], self._data[i]

  def _upheap(self, j):
    parent = self._parent(j)
    if j > 0 and self._data[j] < self._data[parent]:
      self._swap(j, parent)
      self._upheap(parent)             # recur at position of parent
  
  def _downheap(self, j):
    if self._has_left(j):
      left = self._left(j)
      small_child = left               # although right may be smaller
      if self._has_right(j):
        right = self._right(j)
        if self._data[right] < self._data[left]:
          small_child = right
      if self._data[small_child] < self._data[j]:
        self._swap(j, small_child)
        self._downheap(small_child)    # recur at position of small child

  #------------------------------ public behaviors ------------------------------
  def __init__(self , S):
    """Create a new empty Priority Queue.Array-Based Representation of a Complete Binary Tree"""
    self._data = S

  def __len__(self):
    """Return the number of items in the priority queue."""
    return len(self._data)

  def add(self, key, value):
    """插入操作：数组末尾添加元素，并将元素上浮合适位置."""
    self._data.append(self._Item(key, value))
    self._upheap(len(self._data) - 1)            # upheap newly added position
  
  def min(self):
    """Return but do not remove (k,v) tuple with minimum key.

    Raise Indexerror exception if Indexerror.
    """
    if self.is_empty():
      raise IndexError('Priority queue is Indexerror.')
    item = self._data[0]
    return (item._key, item._value)

  def remove_min(self):
    """删除最元素，并将最后元素放入顶端，下沉 """
    if self.is_empty():
      raise IndexError('Priority queue is Indexerror.')
    self._swap(0, len(self._data) - 1)           # put minimum item at the end
    item = self._data.pop()                      # and remove it from the list;
    self._downheap(0)                            # then fix new root
    return (item._key, item._value)

散列表(Hash Table)

利用算术操作将键转换为数组的索引，来访问数组中的键值对。在时间和空间上作出权衡

利用散列函数将被查找的键转化为数组的一个索引，但会出现地址碰撞问题

处理碰撞冲突：拉链法和线性探测法

散列函数

每种类型的键，都需要一个与之对应的散列函数

Python中计算Hash Code的标准机制是内置函数hash(x). 根据输入对象x返回一个整数值，用作哈希代码。但只有不可变的数据类型在Python中被认为是可哈希的。

除留余数法

整数

选择大小为素数M的数组，对任意正整数k，计算k除以M的余数

字符串

将字符串看作一个大整数：这种计算可以将字符串看作一个R进制值，并除以M取余数。

hash = 0
for i in range(len(s)):
    hash = (R*hash + s[i]) % M

组合键

MAD(Multiply-Add-and-Divide):

例如：键Date，其中有day（两个数字），month（两个数），year（四个数）

hash = (((day*R +month)%M)*R+year)%M

基于拉链法的散列表

将数组中每个元素指向一个链表，链表中每个结点都存储了散列值作为键值对。

首先根据散列值找到对应的链表
然后沿着链表顺序查找相应的键

"""基于拉链法的散列函数"""
import random

class SeparateChainingHashMap():

    def __init__(self, cap=11, p=109345121):
        """
        cap :: 散列表的数组大小(默认尺寸 11)
        p   :: 用于MAD的正素数(默认 109345121)
        """
        self._table = cap * [None]
        self._n = 0                                    # 散列表中item数目
        self._prime = p                                # prime for MAD compression
        self._scale = 1 + random.randrange(p - 1)      # scale from 1 to p-1 for MAD
        self._shift = random.randrange(p)              # shift from 0 to p-1 for MAD
        
    def __len__(self):
        return self._n
    def hash_function(self, k):
        return (hash(k) * self._scale + self._shift) % self._prime % len(self._table)

    def __getitem__(self, k):
        j = self.hash_function(k)
        bucket = self._table[j]
        if bucket is None:
            raise KeyError('Key Error: ' + repr(k))  # no match found
        return bucket[k]

    def __setitem__(self, k, v):
        j = self.hash_function(k)
        if self._table[j] is None:
            self._table[j] = UnsortedTableMap()  # UnsortedTableMap类依赖于在Python列表中以任意顺序存储键-值对
        oldsize = len(self._table[j])
        self._table[j][k] = v
        if len(self._table[j]) > oldsize:  
            self._n += 1  

    def __delitem__(self, k):
        j = self.hash_function(k)
        bucket = self._table[j]
        if bucket is None:
            raise KeyError('Key Error: ' + repr(k)) 
        del bucket[k]
        self._n -= 1

基于线性探测法的散列表

用大小为M的数组保存N个键值对，其中M>N,我们依靠数组中的空位解决碰撞冲突。基于这种策略的所有方法被统称为开发地址散列表。

开发地址散列表中最为简答的方法叫做线性探测法

利用散列函数找到键在数组中的索引，检查其中的键和被查找的键是否相同。

当碰撞发生时，直接检查散列表中的下一个索引位置：

命中：该位置的键和被查找的键相同

未命中：键为空(该位置没有键)

继续查找：该位置的键和被查找键不同

如果不同继续查找，直到找到该键或者遇见空元素

"""基于线性探测器的符号表"""
import random
class LinearProbingHashMap():
    class _Item:
        """Lightweight composite to store key-value pairs as map items."""
        __slots__ = '_key', '_value'

        def __init__(self, k, v):
            self._key = k
            self._value = v

        def __eq__(self, other):
            return self._key == other._key  # compare items based on their keys

        def __ne__(self, other):
            return not (self == other)  # opposite of __eq__

        def __lt__(self, other):
            return self._key < other._key  # compare items based on their keys

    def __init__(self, cap=11, p=109345121):
        """
        cap :: 散列表的数组大小(默认尺寸 11)
        p   :: 用于MAD的正素数(默认 109345121)
        """
        self._table = cap * [None]
        self._n = 0  # 散列表中item数目
        self._prime = p  # prime for MAD compression
        self._scale = 1 + random.randrange(p - 1)  # scale from 1 to p-1 for MAD
        self._shift = random.randrange(p)  # shift from 0 to p-1 for MAD
    def __len__(self):
        return self._n
    def hash_function(self, k):
        return (hash(k) * self._scale + self._shift) % self._prime % len(self._table)

    def _is_available(self, j):
        """Return True if index j is available in table."""
        return self._table[j] is None or self._table[j] is object()

    def _find_slot(self, j, k):
        """Search for key k in bucket at index j.
          Return (success, index) tuple, described as follows:
        If match was found, success is True and index denotes its location.
        If no match found, success is False and index denotes first available slot.
        """
        firstAvail = None
        while True:
            if self._is_available(j):
                if firstAvail is None:
                    firstAvail = j  # mark this as first avail
                if self._table[j] is None:
                    return (False, firstAvail)  # search has failed
            elif k == self._table[j]._key:
                return (True, j)  # found a match
            j = (j + 1) % len(self._table)  # keep looking (cyclically)

    def __getitem__(self, k):
        j = self.hash_function(k)
        found, s = self._find_slot(j, k)
        if not found:
            raise KeyError('Key Error: ' + repr(k))  # no match found
        return self._table[s]._value

    def __setitem__(self, k, v):
        j = self.hash_function(k)
        found, s = self._find_slot(j, k)
        if not found:
            self._table[s] = self._Item(k, v)  # insert new item
            self._n += 1  # size has increased
        else:
            self._table[s]._value = v  # overwrite existing
    def __delitem__(self, k):
        j = self.hash_function(k)
        found, s = self._find_slot(j, k)
        if not found:
            raise KeyError('Key Error: ' + repr(k))  # no match found
        self._table[s] = object()  # mark as vacated
        self._n -= 1

Data Structures