第十九章 斐波那契堆

ps：这章写的挺烂的，感觉没头没尾的提出一个结构和算法，然后说明它具有这些性质。

19.1 斐波那契堆结构

def：一个斐波那契堆是一系列最有最小堆的有根树集合。也就是说每棵树均遵循最小堆性质：每个节点的关键字大于或等于它的父节点的关键字，节点结构定义如下：

class NODE():    def __init__(self, key):        self.key = key        self.left = self        self.right = self        self.parent = None        self.child = None        self.degree = 0        self.mark = Falseclass FIB_HEAP():    def __init__(self):        self.num = 0        self.min = None        self.rootList = None

我们使用摊还分析的势函数分析方法来分析其复杂度：

我们可以实现如下的算法来保证上表中的复杂度能够成立：

import mathclass NODE():    def __init__(self, key):        self.key = key        self.left = self        self.right = self        self.parent = None        self.child = None        self.degree = 0        self.mark = Falseclass FIB_HEAP():    def __init__(self):        self.num = 0        self.min = None        self.rootList = Noneclass DL_LIST():    def __init__(self):        self.nil = NODE(None)        self.nil.right = self.nil        self.nil.left = self.nil        def DL_LIST_SEARCH(L, k):    x = L.nil.right    while x != L.nil and x.key != k:        x = x.right    return xdef DL_LIST_INSERT(L, x):    x.right = L.nil.right    L.nil.right.left = x    L.nil.right = x    x.left = L.nildef DL_LIST_INSERT_BEFORE(L, x, y):    y.left = x.left    y.right = x    x.left.right = y    x.left = y    def DL_LIST_DELETE(L, x):    x.left.right = x.right    x.right.left = x.leftdef DL_LIST_UNION(L1, L2):    head1 = L1.nil.right    tail1 = L1.nil.left    head2 = L2.nil.right    tail2 = L2.nil.left    tail1.right = head2    head2.left = tail1    tail2.right = L1.nil    L1.nil.left = tail2    L2.nil.left = None    L2.nil.right = None    def DL_LIST_PRINT(L):    y = L.nil.right    while y != L.nil:        print(y.key, y.mark, y.degree)        if y.child != None:            print("---------child of ", y.key)            DL_LIST_PRINT(y.child)            print("----------child end")        y = y.rightdef FIB_HEAP_INSERT(H, x):    x.parent = None    x.child = None    x.degree = 0    x.mark = False    if H.min == None:        H.root = DL_LIST()        DL_LIST_INSERT(H.root, x)        H.min = x    else:        DL_LIST_INSERT_BEFORE(H.root, H.min, x)        if x.key < H.min.key:            H.min = x    H.num = H.num + 1def FIB_HEAP_UNION(H1, H2):    H = FIB_HEAP()    H.min = H1.min    DL_LIST_UNION(H.root, H2.root)    if (H1.min == None) or (H2.min != None and H2.min.key < H1.min.key):        H.min = H2.min    H.num = H1.num + H2.num        H1.root = None    H1.min = None    H2.root = None    H2.min = None    return Hdef FIB_HEAP_EXTRACT_MIN(H):    z = H.min    if z != None:        if z.child != None:            x = z.child.nil.right            while x != z.child.nil:                x.parent = None                y = x.right                DL_LIST_INSERT_BEFORE(H.root, H.min, x)                x = y                DL_LIST_DELETE(H.root, z)        if H.root.nil.right == H.root.nil  and H.root.nil.left == H.root.nil:            H.min = None        else:            H.min = z.right            CONSOLIDATE(H)        H.num = H.num -1    return zdef D(num):    return int(math.log(num, 2))+1def CONSOLIDATE(H):    A = [None for i in range(D(H.num)+1)]    w = H.min    last = w.left    flag = True    while flag:        x = w        w = w.right        if x == H.root.nil:            continue        elif x == last:            flag = False        d = x.degree        while A[d] != None:            y = A[d]            if x.key > y.key:                temp = x                x = y                y = temp            FIB_HEAP_LINK(H, y, x)            A[d] = None            d = d + 1        A[d] = x    H.min = None    for i in range(0, D(H.num)+1):        if A[i] != None:            if H.min == None:                H.root = DL_LIST()                DL_LIST_INSERT(H.root, A[i])                H.min = A[i]            else:                DL_LIST_INSERT_BEFORE(H.root, H.min, A[i])                if A[i].key < H.min.key:                    H.min = A[i]            def FIB_HEAP_LINK(H, y, x):    DL_LIST_DELETE(H.root, y)    if x.child == None:        x.child = DL_LIST()    DL_LIST_INSERT(x.child, y)    x.degree = x.degree + 1    y.parent = x    y.mark = Falsedef FIB_HEAP_DECREASE_KEY(H, x, k):    if k > x.key:        raise("new key is greater than current key")    x.key = k    y = x.parent    if y != None and x.key < y.key:        CUT(H, x, y)        CASCADING_CUT(H, y)    if x.key < H.min.key:        H.min = xdef CUT(H, x, y):    DL_LIST_DELETE(y.child, x)    y.degree = y.degree - 1    DL_LIST_INSERT_BEFORE(H.root, H.min, x)    x.parent = None    x.mark = Falsedef CASCADING_CUT(H, y):    z = y.parent    if z != None:        if y.mark == False:            y.mark = True        else:            CUT(H, y, z)            CASCADING_CUT(H, z)def FIB_HEAP_DELETE(H, x):    FIB_HEAP_DECREASE_KEY(H, x, -float("inf"))    FIB_HEAP_EXTRACT_MIN(H)if __name__ == "__main__":    H = FIB_HEAP()    node24 = NODE(24)    FIB_HEAP_INSERT(H, node24)    node17 = NODE(17)    FIB_HEAP_INSERT(H, node17)    node3 = NODE(3)    FIB_HEAP_INSERT(H, node3)    node23 = NODE(23)    FIB_HEAP_INSERT(H, node23)        node7 = NODE(7)    FIB_HEAP_INSERT(H, node7)        node39 = NODE(39)    FIB_HEAP_INSERT(H, node39)        node18 = NODE(18)    FIB_HEAP_INSERT(H, node18)    node52 = NODE(52)    FIB_HEAP_INSERT(H, node52)        node38 = NODE(38)    FIB_HEAP_INSERT(H, node38)        node41 = NODE(41)    FIB_HEAP_INSERT(H, node41)        node30 = NODE(30)    FIB_HEAP_INSERT(H, node30)        node35 = NODE(35)    FIB_HEAP_INSERT(H, node35)        node26 = NODE(26)    FIB_HEAP_INSERT(H, node26)                node46 = NODE(46)    FIB_HEAP_INSERT(H, node46)    FIB_HEAP_LINK(H, node39, node18)    FIB_HEAP_LINK(H, node41, node38)    FIB_HEAP_LINK(H, node35, node26)    node39.mark = True        FIB_HEAP_LINK(H, node38, node3)    FIB_HEAP_LINK(H, node52, node3)    FIB_HEAP_LINK(H, node18, node3)    FIB_HEAP_LINK(H, node30, node17)    FIB_HEAP_LINK(H, node46, node24)    FIB_HEAP_LINK(H, node26, node24)    node18.mark = True    node26.mark = True    print("init ********************")    DL_LIST_PRINT(H.root)    print("insert ******************")    node21 = NODE(21)    FIB_HEAP_INSERT(H, node21)    DL_LIST_PRINT(H.root)    print("extract******************")    FIB_HEAP_EXTRACT_MIN(H)    DL_LIST_PRINT(H.root)    print("decrease*****************")    FIB_HEAP_DECREASE_KEY(H, node46, 15)    DL_LIST_PRINT(H.root)    print("decrease-----------------")    FIB_HEAP_DECREASE_KEY(H, node35, 5)    DL_LIST_PRINT(H.root)    print("delete*******************")    FIB_HEAP_DELETE(H, node18)    DL_LIST_PRINT(H.root)

19.4 最大度数的界

习题解答