数据结构（八）：区间信息

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31

# 块状数组
from math import isqrt
def apply(a,l,r,x):
    if [l,r]==[a*sq,a*sq+sq-1]:
        f[a] += x
        return
    for i in range(a*sq,min(n,a*sq+sq)):
        A[i] += f[a]
        if l<=i<=r:
            A[i] += x
    B[a] = set(A[a*sq:a*sq+sq])
    f[a] = 0

def update(l,r,x):
    a,b = l//sq,r//sq
    if a==b:
        apply(a,l,r,x)
    else:
        apply(a,l,a*sq+sq-1,x)
        apply(b,b*sq,r,x)
        for i in range(a+1,b):
            f[i] += x

A = []
n = len(A)
sq = isqrt(n)
m = (n-1)//sq+1
B = [0]*m
f = [0]*m
for a in range(m):
    B[a] = set(A[a*sq:a*sq+sq])

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15

# st 表
class ST:
    def __init__(self,A,func=max):
        f = A[:]
        self.st = st = [f]
        j, N = 1, len(f)
        while 2*j<=N:
            f = [func(f[i],f[i+j]) for i in range(N-2*j+1)]
            st.append(f)
            j <<= 1
        self.func = func
            
    def query(self,l,r):
        j = (r-l+1).bit_length()-1
        return self.func(self.st[j][l],self.st[j][r-(1<<j)+1])

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

#树状数组
class BIT:
    def __init__(self, n):
        self.n = n
        self.t = [0]*n
        self.L = n.bit_length()

    def update(self, i, x):
        while i<self.n:
            self.t[i] += x
            i += i&-i

    def get(self, i):
        res = 0
        while i:
            res += self.t[i]
            i &= i-1
        return res
    
    def query(self,l,r):
        return self.get(r)-self.get(l-1)
    
    def kth(self, k):
        x = 0
        for i in range(self.L-1,-1,-1):
            y = x|1<<i
            if y<self.n and self.t[y]<k:
                x = y
                k -= self.t[y]
        return x+1

A = []
n = len(A)
bit = BIT(n+1)
for i,x in enumerate(A,1):
	bit.update(i,x)

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99

#  线段树 单点修改
class Seg:
    def __init__(self,n):  
        self.N = N = 1<<n.bit_length()         
        self.t = [0]*N*2

    def merge(self,a,b):                # 区间信息怎么合并的   
        return a+b

    def build(self,A):
        self.t[self.N:self.N+len(A)] = A
        for o in range(self.N-1,0,-1):
            self.pull(o)

    def pull(self,o):
        self.t[o] = self.merge(self.t[o*2],self.t[o*2+1])

    def modify(self,a,x):
        a += self.N
        self.t[a] = x
        while a>>1:
            a >>= 1
            self.pull(a)

    def query(self,a,b):
        a,b = a+self.N-1,b+self.N+1
        lres,rres = 0,0
        while a^b^1:  
            if not a&1: lres = self.merge(lres,self.t[a^1])  
            if b&1: rres = self.merge(self.t[b^1],rres)  
            a,b = a>>1,b>>1  
        return self.merge(lres,rres)  
    
    # 递归式修改
    def _modify(self,a,x,o=1,l=0,r=None):
        r = self.N-1 if r is None else r
        if l==r:
            self.t[o] = x
            return
        m = (l+r)//2
        if a<=m: self._modify(a,x,o*2,l,m)
        else: self._modify(a,x,o*2+1,m+1,r)
        self.pull(o)
    
    # 递归式查询
    def _query(self,a,b,o=1,l=0,r=None):
        r = self.N-1 if r is None else r
        if a<=l and r<=b:
            return self.t[o]
        res = self.t[0]
        m = (l+r)//2
        if a<=m: res = self._query(a,b,o*2,l,m)
        if m<b: res = self.merge(res,self._query(a,b,o*2+1,m+1,r))
        return res
    
    def find_first(self,a,func):
        a += self.N
        lres = 0
        while True:
            while not a&1:
                a >>= 1
            tmp = self.merge(lres,self.t[a])
            if func(tmp):
                while a<self.N:
                    a <<= 1
                    tmp2 = self.merge(lres,self.t[a])
                    if not func(tmp2):
                        lres = tmp2
                        a += 1
                return a-self.N
            if a&(a+1)==0:
                return self.N
            lres = tmp
            a += 1

    def find_last(self,b,func):
        b += self.N
        rres = 0
        while True:
            while b>1 and b&1:
                b >>= 1
            tmp = self.merge(self.t[b],rres)
            if func(tmp):
                while b<self.N:
                    b = 2*b+1
                    tmp2 = self.merge(self.t[b],rres)
                    if not func(tmp2):
                        rres = tmp2
                        b -= 1
                return b-self.N
            if b&(b-1)==0:
                return -1
            rres = tmp
            b -= 1

A = []
n = len(A)
seg = Seg(n)
seg.build(A)

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130

# lazy 线段树 区间修改
class Seg:
    def __init__(self,n):
        self.L = n.bit_length()
        self.N = N = 1<<self.L
        self.t = [0]*N*2
        self.f = [0]*N*2
        
    def apply(self,o,x):            
        self.t[o] += x
        self.f[o] += x

    def merge(self,a,b):           
        return max(a,b)
    
    def build(self,A):
        self.t[self.N:self.N+len(A)] = A
        for o in range(self.N-1,0,-1):
            self.pull(o)

    def pull(self,o):
        self.t[o] = self.merge(self.t[o*2],self.t[o*2+1])

    def push(self,o):
        if self.f[o] != self.f[0]:
            self.apply(o*2,self.f[o])
            self.apply(o*2+1,self.f[o])
            self.f[o] = self.f[0]

    def modify(self,a,b,x):
        a,b = a+self.N-1,b+self.N+1  
        for i in range(self.L,0,-1):  
            self.push(a>>i)  
            self.push(b>>i)  
        while a^b^1:  
            if not a&1: self.apply(a^1,x)  
            if b&1: self.apply(b^1,x)  
            a,b = a>>1,b>>1  
            self.pull(a)  
            self.pull(b)  
        while a>>1:
            a >>= 1
            self.pull(a)

    def query(self,a,b):
        a,b = a+self.N-1,b+self.N+1  
        for i in range(self.L,0,-1):  
            self.push(a>>i)  
            self.push(b>>i)  
        lres,rres = 0,0
        while a^b^1:  
            if not a&1: lres = self.merge(lres,self.t[a^1])
            if b&1: rres = self.merge(self.t[b^1],rres)
            a,b = a>>1,b>>1  
        return self.merge(lres,rres)
    
    # 递归式修改
    def _modify(self,a,b,x,o=1,l=0,r=None):
        r = self.N-1 if r is None else r
        if a<=l and r<=b:
            self.apply(o,x)
            return
        self.push(o)
        m = (l+r)//2
        if a<=m: self._modify(a,b,x,o*2,l,m)
        if m<b: self._modify(a,b,x,o*2+1,m+1,r)
        self.pull(o)
    
    # 递归式查询
    def _query(self,a,b,o=1,l=0,r=None):
        r = self.N-1 if r is None else r
        if a<=l and r<=b:
            return self.t[o]
        self.push(o)
        res = self.t[0]
        m = (l+r)//2
        if a<=m: res = self._query(a,b,o*2,l,m)
        if m<b: res = self.merge(res,self._query(a,b,o*2+1,m+1,r))
        return res
    
    def find_first(self,a,func):
        a += self.N
        lres = 0
        for i in range(self.L,0,-1):  
            self.push(a>>i) 
        while True:
            while not a&1:
                a >>= 1
            tmp = self.merge(lres,self.t[a])
            if func(tmp):
                while a<self.N:
                    self.push(a)
                    a <<= 1
                    tmp2 = self.merge(lres,self.t[a])
                    if not func(tmp2):
                        lres = tmp2
                        a += 1
                return a-self.N
            if a&(a+1)==0:
                return self.N
            lres = tmp
            a += 1

    def find_last(self,b,func):
        b += self.N
        rres = 0
        for i in range(self.L,0,-1):  
            self.push(b>>i) 
        while True:
            while b>1 and b&1:
                b >>= 1
            tmp = self.merge(self.t[b],rres)
            if func(tmp):
                while b<self.N:
                    self.push(b)
                    b = 2*b+1
                    tmp2 = self.merge(self.t[b],rres)
                    if not func(tmp2):
                        rres = tmp2
                        b -= 1
                return b-self.N
            if b&(b-1)==0:
                return -1
            rres = tmp
            b -= 1

A = []
n = len(A)
seg = Seg(n)
seg.build(A)