说明

        如果需要用到这些知识却没有掌握，则会让人感到沮丧，也可能导致面试被拒。无论是花几天时间“突击”，还是利用零碎的时间持续学习，在数据结构上下点功夫都是值得的。那么Python 中有哪些数据结构呢？列表、字典、集合，还有……栈？Python 有栈吗？本系列文章将给出详细拼图。
 

9章：Advanced Linked Lists

之前曾经介绍过单链表，一个链表节点只有data和next字段，本章介绍高级的链表。

Doubly Linked List，双链表，每个节点多了个prev指向前一个节点。双链表可以用来编写文本编辑器的buffer。

class DListNode:
    def __init__(self, data):
        self.data = data
        self.prev = None
        self.next = None


def revTraversa(tail):
    curNode = tail
    while cruNode is not None:
        print(curNode.data)
        curNode = curNode.prev


def search_sorted_doubly_linked_list(head, tail, probe, target):
    """ probing technique探查法，改进直接遍历，不过最坏时间复杂度仍是O(n)
    searching a sorted doubly linked list using the probing technique
    Args:
        head (DListNode obj)
        tail (DListNode obj)
        probe (DListNode or None)
        target (DListNode.data): data to search
    """
    if head is None:    # make sure list is not empty
        return False
    if probe is None:    # if probe is null, initialize it to first node
        probe = head

    # if the target comes before the probe node, we traverse backward, otherwise
    # traverse forward
    if target < probe.data:
        while probe is not None and target <= probe.data:
            if target == probe.dta:
                return True
            else:
                probe = probe.prev
    else:
        while probe is not None and target >= probe.data:
            if target == probe.data:
                return True
            else:
                probe = probe.next
    return False


def insert_node_into_ordered_doubly_linekd_list(value):
    """ 最好画个图看，链表操作很容易绕晕，注意赋值顺序"""
    newnode = DListNode(value)

    if head is None:    # empty list
        head = newnode
        tail = head

    elif value < head.data:    # insert before head
        newnode.next = head
        head.prev = newnode
        head = newnode

    elif value > tail.data:    # insert after tail
        newnode.prev = tail
        tail.next = newnode
        tail = newnode

    else:    # insert into middle
        node = head
        while node is not None and node.data < value:
            node = node.next
        newnode.next = node
        newnode.prev = node.prev
        node.prev.next = newnode
        node.prev = newnode

循环链表

def travrseCircularList(listRef):
    curNode = listRef
    done = listRef is None
    while not None:
        curNode = curNode.next
        print(curNode.data)
        done = curNode is listRef   # 回到遍历起始点


def searchCircularList(listRef, target):
    curNode = listRef
    done = listRef is None
    while not done:
        curNode = curNode.next
        if curNode.data == target:
            return True
        else:
            done = curNode is listRef or curNode.data > target
    return False


def add_newnode_into_ordered_circular_linked_list(listRef, value):
    """ 插入并维持顺序
    1.插入空链表；2.插入头部；3.插入尾部；4.按顺序插入中间
    """
    newnode = ListNode(value)
    if listRef is None:    # empty list
        listRef = newnode
        newnode.next = newnode

    elif value < listRef.next.data:    # insert in front
        newnode.next = listRef.next
        listRef.next = newnode

    elif value > listRef.data:    # insert in back
        newnode.next = listRef.next
        listRef.next = newnode
        listRef = newnode

    else:    # insert in the middle
        preNode = None
        curNode = listRef
        done = listRef is None
        while not done:
            preNode = curNode
            preNode = curNode.next
            done = curNode is listRef or curNode.data > value

        newnode.next = curNode
        preNode.next = newnode

利用循环双端链表我们可以实现一个经典的缓存失效算法，lru：

# -*- coding: utf-8 -*-

class Node(object):
    def __init__(self, prev=None, next=None, key=None, value=None):
        self.prev, self.next, self.key, self.value = prev, next, key, value


class CircularDoubleLinkedList(object):
    def __init__(self):
        node = Node()
        node.prev, node.next = node, node
        self.rootnode = node

    def headnode(self):
        return self.rootnode.next

    def tailnode(self):
        return self.rootnode.prev

    def remove(self, node):
        if node is self.rootnode:
            return
        else:
            node.prev.next = node.next
            node.next.prev = node.prev

    def append(self, node):
        tailnode = self.tailnode()
        tailnode.next = node
        node.next = self.rootnode
        self.rootnode.prev = node


class LRUCache(object):
    def __init__(self, maxsize=16):
        self.maxsize = maxsize
        self.cache = {}
        self.access = CircularDoubleLinkedList()
        self.isfull = len(self.cache) >= self.maxsize

    def __call__(self, func):
        def wrapper(n):
            cachenode = self.cache.get(n)
            if cachenode is not None:  # hit
                self.access.remove(cachenode)
                self.access.append(cachenode)
                return cachenode.value
            else:  # miss
                value = func(n)
                if not self.isfull:
                    tailnode = self.access.tailnode()
                    newnode = Node(tailnode, self.access.rootnode, n, value)
                    self.access.append(newnode)
                    self.cache[n] = newnode

                    self.isfull = len(self.cache) >= self.maxsize
                    return value
                else:  # full
                    lru_node = self.access.headnode()
                    del self.cache[lru_node.key]
                    self.access.remove(lru_node)
                    tailnode = self.access.tailnode()
                    newnode = Node(tailnode, self.access.rootnode, n, value)
                    self.access.append(newnode)
                    self.cache[n] = newnode
                return value
        return wrapper


@LRUCache()
def fib(n):
    if n <= 2:
        return 1
    else:
        return fib(n - 1) + fib(n - 2)


for i in range(1, 35):
    print(fib(i))

---

10章：Recursion

Recursion is a process for solving problems by subdividing a larger problem into smaller cases of the problem itself and then solving the smaller, more trivial parts.

递归函数：调用自己的函数

# 递归函数：调用自己的函数，看一个最简单的递归函数，倒序打印一个数
def printRev(n):
    if n > 0:
        print(n)
        printRev(n-1)


printRev(3)    # 从10输出到1


# 稍微改一下，print放在最后就得到了正序打印的函数
def printInOrder(n):
    if n > 0:
        printInOrder(n-1)
        print(n)    # 之所以最小的先打印是因为函数一直递归到n==1时候的最深栈，此时不再
                    # 递归，开始执行print语句，这时候n==1，之后每跳出一层栈，打印更大的值

printInOrder(3)    # 正序输出

Properties of Recursion: 使用stack解决的问题都能用递归解决

• A recursive solution must contain a base case; 递归出口，代表最小子问题(n == 0退出打印)
• A recursive solution must contain a recursive case; 可以分解的子问题
• A recursive solution must make progress toward the base case. 递减n使得n像递归出口靠近

Tail Recursion: occurs when a function includes a single recursive call as the last statement of the function. In this case, a stack is not needed to store values to te used upon the return of the recursive call and thus a solution can be implemented using a iterative loop instead.

# Recursive Binary Search

def recBinarySearch(target, theSeq, first, last):
    # 你可以写写单元测试来验证这个函数的正确性
    if first > last:    # 递归出口1
        return False
    else:
        mid = (first + last) // 2
        if theSeq[mid] == target:
            return True    # 递归出口2
        elif theSeq[mid] > target:
            return recBinarySearch(target, theSeq, first, mid - 1)
        else:
            return recBinarySearch(target, theSeq, mid + 1, last)

---

11章：Hash Tables

基于比较的搜索（线性搜索，有序数组的二分搜索）最好的时间复杂度只能达到O(logn)，利用hash可以实现O(1)查找，python内置dict的实现方式就是hash，你会发现dict的key必须要是实现了 __hash__ 和 __eq__ 方法的。

Hashing: hashing is the process of mapping a search a key to a limited range of array indeices with the goal of providing direct access to the keys.

hash方法有个hash函数用来给key计算一个hash值，作为数组下标，放到该下标对应的槽中。当不同key根据hash函数计算得到的下标相同时，就出现了冲突。解决冲突有很多方式，比如让每个槽成为链表，每次冲突以后放到该槽链表的尾部，但是查询时间就会退化，不再是O(1)。还有一种探查方式，当key的槽冲突时候，就会根据一种计算方式去寻找下一个空的槽存放，探查方式有线性探查，二次方探查法等，cpython解释器使用的是二次方探查法。还有一个问题就是当python使用的槽数量大于预分配的2/3时候，会重新分配内存并拷贝以前的数据，所以有时候dict的add操作代价还是比较高的，牺牲空间但是可以始终保证O(1)的查询效率。如果有大量的数据，建议还是使用bloomfilter或者redis提供的HyperLogLog。

如果你感兴趣，可以看看这篇文章，介绍c解释器如何实现的python dict对象：Python dictionary implementation。我们使用Python来实现一个类似的hash结构。

import ctypes

class Array:  # 第二章曾经定义过的ADT，这里当做HashMap的槽数组使用
    def __init__(self, size):
        assert size > 0, 'array size must be > 0'
        self._size = size
        PyArrayType = ctypes.py_object * size
        self._elements = PyArrayType()
        self.clear(None)

    def __len__(self):
        return self._size

    def __getitem__(self, index):
        assert index >= 0 and index < len(self), 'out of range'
        return self._elements[index]

    def __setitem__(self, index, value):
        assert index >= 0 and index < len(self), 'out of range'
        self._elements[index] = value

    def clear(self, value):
        """ 设置每个元素为value """
        for i in range(len(self)):
            self._elements[i] = value

    def __iter__(self):
        return _ArrayIterator(self._elements)


class _ArrayIterator:
    def __init__(self, items):
        self._items = items
        self._idx = 0

    def __iter__(self):
        return self

    def __next__(self):
        if self._idx < len(self._items):
            val = self._items[self._idx]
            self._idx += 1
            return val
        else:
            raise StopIteration


class HashMap:
    """ HashMap ADT实现，类似于python内置的dict
    一个槽有三种状态：
    1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过，查找时只要找到UNUSEd就不用再继续探查了
    2.使用过但是remove了，此时是 HashMap.EMPTY，该探查点后边的元素扔可能是有key
    3.槽正在使用 _MapEntry节点
    """

    class _MapEntry:    # 槽里存储的数据
        def __init__(self, key, value):
            self.key = key
            self.value = value

    UNUSED = None    # 没被使用过的槽，作为该类变量的一个单例，下边都是is 判断
    EMPTY = _MapEntry(None, None)     # 使用过但是被删除的槽

    def __init__(self):
        self._table = Array(7)    # 初始化7个槽
        self._count = 0
        # 超过2/3空间被使用就重新分配，load factor = 2/3
        self._maxCount = len(self._table) - len(self._table) // 3

    def __len__(self):
        return self._count

    def __contains__(self, key):
        slot = self._findSlot(key, False)
        return slot is not None

    def add(self, key, value):
        if key in self:    # 覆盖原有value
            slot = self._findSlot(key, False)
            self._table[slot].value = value
            return False
        else:
            slot = self._findSlot(key, True)
            self._table[slot] = HashMap._MapEntry(key, value)
            self._count += 1
            if self._count == self._maxCount:    # 超过2/3使用就rehash
                self._rehash()
            return True

    def valueOf(self, key):
        slot = self._findSlot(key, False)
        assert slot is not None, 'Invalid map key'
        return self._table[slot].value

    def remove(self, key):
        """ remove操作把槽置为EMPTY"""
        assert key in self, 'Key error %s' % key
        slot = self._findSlot(key, forInsert=False)
        value = self._table[slot].value
        self._count -= 1
        self._table[slot] = HashMap.EMPTY
        return value

    def __iter__(self):
        return _HashMapIteraotr(self._table)

    def _slot_can_insert(self, slot):
        return (self._table[slot] is HashMap.EMPTY or
                self._table[slot] is HashMap.UNUSED)

    def _findSlot(self, key, forInsert=False):
        """ 注意原书有错误，代码根本不能运行，这里我自己改写的
        Args:
            forInsert (bool): if the search is for an insertion
        Returns:
            slot or None
        """
        slot = self._hash1(key)
        step = self._hash2(key)
        _len = len(self._table)

        if not forInsert:    # 查找是否存在key
            while self._table[slot] is not HashMap.UNUSED:
                # 如果一个槽是UNUSED，直接跳出
                if self._table[slot] is HashMap.EMPTY:
                    slot = (slot + step) % _len
                    continue
                elif self._table[slot].key == key:
                    return slot
                slot = (slot + step) % _len
            return None

        else:    # 为了插入key
            while not self._slot_can_insert(slot):    # 循环直到找到一个可以插入的槽
                slot = (slot + step) % _len
            return slot

    def _rehash(self):    # 当前使用槽数量大于2/3时候重新创建新的table
        origTable = self._table
        newSize = len(self._table) * 2 + 1    # 原来的2*n+1倍
        self._table = Array(newSize)

        self._count = 0
        self._maxCount = newSize - newSize // 3

        # 将原来的key value添加到新的table
        for entry in origTable:
            if entry is not HashMap.UNUSED and entry is not HashMap.EMPTY:
                slot = self._findSlot(entry.key, True)
                self._table[slot] = entry
                self._count += 1

    def _hash1(self, key):
        """ 计算key的hash值"""
        return abs(hash(key)) % len(self._table)

    def _hash2(self, key):
        """ key冲突时候用来计算新槽的位置"""
        return 1 + abs(hash(key)) % (len(self._table)-2)


class _HashMapIteraotr:
    def __init__(self, array):
        self._array = array
        self._idx = 0

    def __iter__(self):
        return self

    def __next__(self):
        if self._idx < len(self._array):
            if self._array[self._idx] is not None and self._array[self._idx].key is not None:
                key = self._array[self._idx].key
                self._idx += 1
                return key
            else:
                self._idx += 1
        else:
            raise StopIteration


def print_h(h):
    for idx, i in enumerate(h):
        print(idx, i)
    print('\n')


def test_HashMap():
    """ 一些简单的单元测试，不过测试用例覆盖不是很全面 """
    h = HashMap()
    assert len(h) == 0
    h.add('a', 'a')
    assert h.valueOf('a') == 'a'
    assert len(h) == 1

    a_v = h.remove('a')
    assert a_v == 'a'
    assert len(h) == 0

    h.add('a', 'a')
    h.add('b', 'b')
    assert len(h) == 2
    assert h.valueOf('b') == 'b'
    b_v = h.remove('b')
    assert b_v == 'b'
    assert len(h) == 1
    h.remove('a')
    assert len(h) == 0

    n = 10
    for i in range(n):
        h.add(str(i), i)
    assert len(h) == n
    print_h(h)
    for i in range(n):
        assert str(i) in h
    for i in range(n):
        h.remove(str(i))
    assert len(h) == 0

---

12章: Advanced Sorting

第5章介绍了基本的排序算法，本章介绍高级排序算法。

归并排序(mergesort): 分治法

def merge_sorted_list(listA, listB):
    """ 归并两个有序数组，O(max(m, n)) ,m和n是数组长度"""
    print('merge left right list', listA, listB, end='')
    new_list = list()
    a = b = 0
    while a < len(listA) and b < len(listB):
        if listA[a] < listB[b]:
            new_list.append(listA[a])
            a += 1
        else:
            new_list.append(listB[b])
            b += 1

    while a < len(listA):
        new_list.append(listA[a])
        a += 1

    while b < len(listB):
        new_list.append(listB[b])
        b += 1

    print(' ->', new_list)
    return new_list


def mergesort(theList):
    """ O(nlogn)，log层调用，每层n次操作
    mergesort: divided and conquer 分治
    1. 把原数组分解成越来越小的子数组
    2. 合并子数组来创建一个有序数组
    """
    print(theList)    # 我把关键步骤打出来了，你可以运行下看看整个过程
    if len(theList) <= 1:    # 递归出口
        return theList
    else:
        mid = len(theList) // 2

        # 递归分解左右两边数组
        left_half = mergesort(theList[:mid])
        right_half = mergesort(theList[mid:])

        # 合并两边的有序子数组
        newList = merge_sorted_list(left_half, right_half)
        return newList

""" 这是我调用一次打出来的排序过程
[10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
[10, 9, 8, 7, 6]
[10, 9]
[10]
[9]
merge left right list [10] [9] -> [9, 10]
[8, 7, 6]
[8]
[7, 6]
[7]
[6]
merge left right list [7] [6] -> [6, 7]
merge left right list [8] [6, 7] -> [6, 7, 8]
merge left right list [9, 10] [6, 7, 8] -> [6, 7, 8, 9, 10]
[5, 4, 3, 2, 1]
[5, 4]
[5]
[4]
merge left right list [5] [4] -> [4, 5]
[3, 2, 1]
[3]
[2, 1]
[2]
[1]
merge left right list [2] [1] -> [1, 2]
merge left right list [3] [1, 2] -> [1, 2, 3]
merge left right list [4, 5] [1, 2, 3] -> [1, 2, 3, 4, 5]
"""

快速排序

def quicksort(theSeq, first, last):    # average: O(nlog(n))
    """
    quicksort :也是分而治之，但是和归并排序不同的是，采用选定主元（pivot）而不是从中间
    进行数组划分
    1. 第一步选定pivot用来划分数组，pivot左边元素都比它小，右边元素都大于等于它
    2. 对划分的左右两边数组递归，直到递归出口（数组元素数目小于2）
    3. 对pivot和左右划分的数组合并成一个有序数组
    """
    if first < last:
        pos = partitionSeq(theSeq, first, last)
        # 对划分的子数组递归操作
        quicksort(theSeq, first, pos - 1)
        quicksort(theSeq, pos + 1, last)


def partitionSeq(theSeq, first, last):
    """ 快排中的划分操作，把比pivot小的挪到左边，比pivot大的挪到右边"""
    pivot = theSeq[first]
    print('before partitionSeq', theSeq)

    left = first + 1
    right = last

    while True:
        # 找到第一个比pivot大的
        while left <= right and theSeq[left] < pivot:
            left += 1

        # 从右边开始找到比pivot小的
        while right >= left and theSeq[right] >= pivot:
            right -= 1

        if right < left:
            break
        else:
            theSeq[left], theSeq[right] = theSeq[right], theSeq[left]

    # 把pivot放到合适的位置
    theSeq[first], theSeq[right] = theSeq[right], theSeq[first]

    print('after partitionSeq {}: {}\t'.format(theSeq, pivot))
    return right    # 返回pivot的位置


def test_partitionSeq():
    l = [0,1,2,3,4]
    assert partitionSeq(l, 0, len(l)-1) == 0
    l = [4,3,2,1,0]
    assert partitionSeq(l, 0, len(l)-1) == 4
    l = [2,3,0,1,4]
    assert partitionSeq(l, 0, len(l)-1) == 2

test_partitionSeq()


def test_quicksort():
    def _is_sorted(seq):
        for i in range(len(seq)-1):
            if seq[i] > seq[i+1]:
                return False
        return True

    from random import randint
    for i in range(100):
        _len = randint(1, 100)
        to_sort = []
        for i in range(_len):
            to_sort.append(randint(0, 100))
        quicksort(to_sort, 0, len(to_sort)-1)    # 注意这里用了原地排序，直接更改了数组
        print(to_sort)
        assert _is_sorted(to_sort)

test_quicksort()

利用快排中的partitionSeq操作，我们还能实现另一个算法，nth_element，快速查找一个无序数组中的第k大元素

def nth_element(seq, beg, end, k):
    if beg == end:
        return seq[beg]
    pivot_index = partitionSeq(seq, beg, end)
    if pivot_index == k:
        return seq[k]
    elif pivot_index > k:
        return nth_element(seq, beg, pivot_index-1, k)
    else:
        return nth_element(seq, pivot_index+1, end, k)

def test_nth_element():
    from random import shuffle
    n = 10
    l = list(range(n))
    shuffle(l)
    print(l)
    for i in range(len(l)):
        assert nth_element(l, 0, len(l)-1, i) == i

test_nth_element()