The Bakery (Codeforces 833B)
Tags: DP, divide and conquer

  Some time ago Slastyona the Sweetmaid decided to open her own bakery! She bought required ingredients and a wonder-oven which can bake several types of cakes, and opened the bakery.Soon the expenses started to overcome the income, so Slastyona decided to study the sweets market. She learned it's profitable to pack cakes in boxes, and that the more distinct cake types a box contains (let's denote this number as the value of the box), the higher price it has.She needs to change the production technology! The problem is that the oven chooses the cake types on its own and Slastyona can't affect it. However, she knows the types and order of n cakes the oven is going to bake today. Slastyona has to pack exactly k boxes with cakes today, and she has to put in each box several (at least one) cakes the oven produced one right after another (in other words, she has to put in a box a continuous segment of cakes).Slastyona wants to maximize the total value of all boxes with cakes. Help her determine this maximum possible total value.

Input: The first line contains two integers n and k (1 ≤ n ≤ 35000, 1 ≤ k ≤ min(n, 50)) – the number of cakes and the number of boxes, respectively.The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ n) – the types of cakes in the order the oven bakes them.

Output: Print the only integer – the maximum total value of all boxes with cakes.

Note: In the first example Slastyona has only one box. She has to put all cakes in it, so that there are two types of cakes in the box, so the value is equal to 2.In the second example it is profitable to put the first two cakes in the first box, and all the rest in the second. There are two distinct types in the first box, and three in the second box then, so the total value is 5.

Sample Inputs:
4 1
1 2 2 1

7 2
1 3 3 1 4 4 4

8 3
7 7 8 7 7 8 1 7

Sample Outputs:
2

5

6

Solution (Python):
import sys
input = sys.stdin.readline

class LazySegTree:
    """"""
    __slots__ = ['n', 'log', 'size', 'd', 'lz', 'e', 'op', 'mapping', 'composition', 'identity']

    def _update(self, k):
        self.d[k] = self.op(self.d[2 * k], self.d[2 * k + 1])

    def _all_apply(self, k, f):
        self.d[k] = self.mapping(f, self.d[k])
        if k < self.size:
            self.lz[k] = self.composition(f, self.lz[k])

    def _push(self, k):
        self._all_apply(2 * k, self.lz[k])
        self._all_apply(2 * k + 1, self.lz[k])
        self.lz[k] = self.identity

    def __init__(self, V, OP, E, MAPPING, COMPOSITION, ID):
        self.n = len(V)
        self.log = (self.n - 1).bit_length()
        self.size = 1 << self.log
        self.d = [E for i in range(2 * self.size)]
        self.lz = [ID for i in range(self.size)]
        self.e = E
        self.op = OP
        self.mapping = MAPPING
        self.composition = COMPOSITION
        self.identity = ID
        for i in range(self.n):
            self.d[self.size + i] = V[i]
        for i in range(self.size - 1, 0, -1):
            self._update(i)

    def set(self, p, x):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        self.d[p] = x
        for i in range(1, self.log + 1):
            self._update(p >> i)

    def get(self, p):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        return self.d[p]

    def prod(self, l, r):
        assert 0 <= l and l <= r and (r <= self.n)
        if l == r:
            return self.e
        l += self.size
        r += self.size
        for i in range(self.log, 0, -1):
            if l >> i << i != l:
                self._push(l >> i)
            if r >> i << i != r:
                self._push(r >> i)
        (sml, smr) = (self.e, self.e)
        while l < r:
            if l & 1:
                sml = self.op(sml, self.d[l])
                l += 1
            if r & 1:
                r -= 1
                smr = self.op(self.d[r], smr)
            l >>= 1
            r >>= 1
        return self.op(sml, smr)

    def all_prod(self):
        return self.d[1]

    def apply_point(self, p, f):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        self.d[p] = self.mapping(f, self.d[p])
        for i in range(1, self.log + 1):
            self._update(p >> i)

    def apply(self, l, r, f):
        assert 0 <= l and l <= r and (r <= self.n)
        if l == r:
            return
        l += self.size
        r += self.size
        for i in range(self.log, 0, -1):
            if l >> i << i != l:
                self._push(l >> i)
            if r >> i << i != r:
                self._push(r - 1 >> i)
        (l2, r2) = (l, r)
        while l < r:
            if l & 1:
                self._all_apply(l, f)
                l += 1
            if r & 1:
                r -= 1
                self._all_apply(r, f)
            l >>= 1
            r >>= 1
        (l, r) = (l2, r2)
        for i in range(1, self.log + 1):
            if l >> i << i != l:
                self._update(l >> i)
            if r >> i << i != r:
                self._update(r - 1 >> i)

    def max_right(self, l, g):
        assert 0 <= l and l <= self.n
        assert g(self.e)
        if l == self.n:
            return self.n
        l += self.size
        for i in range(self.log, 0, -1):
            self._push(l >> i)
        sm = self.e
        while 1:
            while i % 2 == 0:
                l >>= 1
            if not g(self.op(sm, self.d[l])):
                while l < self.size:
                    self._push(l)
                    l = 2 * l
                    if g(self.op(sm, self.d[l])):
                        sm = self.op(sm, self.d[l])
                        l += 1
                return l - self.size
            sm = self.op(sm, self.d[l])
            l += 1
            if l & -l == l:
                break
        return self.n

    def min_left(self, r, g):
        assert 0 <= r and r <= self.n
        assert g(self.e)
        if r == 0:
            return 0
        r += self.size
        for i in range(self.log, 0, -1):
            self._push(r - 1 >> i)
        sm = self.e
        while 1:
            r -= 1
            while r > 1 and r % 2:
                r >>= 1
            if not g(self.op(self.d[r], sm)):
                while r < self.size:
                    self._push(r)
                    r = 2 * r + 1
                    if g(self.op(self.d[r], sm)):
                        sm = self.op(self.d[r], sm)
                        r -= 1
                return r + 1 - self.size
            sm = self.op(self.d[r], sm)
            if r & -r == r:
                break
        return 0
(n, k) = map(int, input().split())
a = list(map(int, input().split()))

def add(x, y):
    return x + y
last = [-1] * (n + 1)
pre = [0] * n
for i in range(n):
    pre[i] = last[a[i]]
    last[a[i]] = i
dp = [0] * n
for _ in range(k):
    seg = LazySegTree([0] + dp[:-1], max, 0, add, add, 0)
    for i in range(n):
        seg.apply(pre[i] + 1, i + 1, 1)
        dp[i] = seg.prod(0, i + 1)
print(dp[-1])

Solution (Python):
import sys
input = sys.stdin.readline
_max = lambda a, b: a if a > b else b
_min = lambda a, b: b if a > b else a
_add = lambda a, b: a + b
mod = 1000000007
''

class LazySegTree:
    """"""
    __slots__ = ['n', 'log', 'size', 'd', 'lz', 'e', 'op', 'mapping', 'composition', 'identity']

    def _update(self, k):
        self.d[k] = self.op(self.d[2 * k], self.d[2 * k + 1])

    def _all_apply(self, k, f):
        self.d[k] = self.mapping(f, self.d[k])
        if k < self.size:
            self.lz[k] = self.composition(f, self.lz[k])

    def _push(self, k):
        self._all_apply(2 * k, self.lz[k])
        self._all_apply(2 * k + 1, self.lz[k])
        self.lz[k] = self.identity

    def __init__(self, V, OP, E, MAPPING, COMPOSITION, ID):
        self.n = len(V)
        self.log = (self.n - 1).bit_length()
        self.size = 1 << self.log
        self.d = [E for i in range(2 * self.size)]
        self.lz = [ID for i in range(self.size)]
        self.e = E
        self.op = OP
        self.mapping = MAPPING
        self.composition = COMPOSITION
        self.identity = ID
        for i in range(self.n):
            self.d[self.size + i] = V[i]
        for i in range(self.size - 1, 0, -1):
            self._update(i)

    def set(self, p, x):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        self.d[p] = x
        for i in range(1, self.log + 1):
            self._update(p >> i)

    def get(self, p):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        return self.d[p]

    def prod(self, l, r):
        assert 0 <= l and l <= r and (r <= self.n)
        if l == r:
            return self.e
        l += self.size
        r += self.size
        for i in range(self.log, 0, -1):
            if l >> i << i != l:
                self._push(l >> i)
            if r >> i << i != r:
                self._push(r >> i)
        (sml, smr) = (self.e, self.e)
        while l < r:
            if l & 1:
                sml = self.op(sml, self.d[l])
                l += 1
            if r & 1:
                r -= 1
                smr = self.op(self.d[r], smr)
            l >>= 1
            r >>= 1
        return self.op(sml, smr)

    def all_prod(self):
        return self.d[1]

    def apply_point(self, p, f):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        self.d[p] = self.mapping(f, self.d[p])
        for i in range(1, self.log + 1):
            self._update(p >> i)

    def apply(self, l, r, f):
        assert 0 <= l and l <= r and (r <= self.n)
        if l == r:
            return
        l += self.size
        r += self.size
        for i in range(self.log, 0, -1):
            if l >> i << i != l:
                self._push(l >> i)
            if r >> i << i != r:
                self._push(r - 1 >> i)
        (l2, r2) = (l, r)
        while l < r:
            if l & 1:
                self._all_apply(l, f)
                l += 1
            if r & 1:
                r -= 1
                self._all_apply(r, f)
            l >>= 1
            r >>= 1
        (l, r) = (l2, r2)
        for i in range(1, self.log + 1):
            if l >> i << i != l:
                self._update(l >> i)
            if r >> i << i != r:
                self._update(r - 1 >> i)

    def max_right(self, l, g):
        assert 0 <= l and l <= self.n
        assert g(self.e)
        if l == self.n:
            return self.n
        l += self.size
        for i in range(self.log, 0, -1):
            self._push(l >> i)
        sm = self.e
        while 1:
            while i % 2 == 0:
                l >>= 1
            if not g(self.op(sm, self.d[l])):
                while l < self.size:
                    self._push(l)
                    l = 2 * l
                    if g(self.op(sm, self.d[l])):
                        sm = self.op(sm, self.d[l])
                        l += 1
                return l - self.size
            sm = self.op(sm, self.d[l])
            l += 1
            if l & -l == l:
                break
        return self.n

    def min_left(self, r, g):
        assert 0 <= r and r <= self.n
        assert g(self.e)
        if r == 0:
            return 0
        r += self.size
        for i in range(self.log, 0, -1):
            self._push(r - 1 >> i)
        sm = self.e
        while 1:
            r -= 1
            while r > 1 and r % 2:
                r >>= 1
            if not g(self.op(self.d[r], sm)):
                while r < self.size:
                    self._push(r)
                    r = 2 * r + 1
                    if g(self.op(self.d[r], sm)):
                        sm = self.op(self.d[r], sm)
                        r -= 1
                return r + 1 - self.size
            sm = self.op(self.d[r], sm)
            if r & -r == r:
                break
        return 0
(n, k) = map(int, input().split())
a = list(map(int, input().split()))
last = [-1] * (n + 1)
pre = [0] * n
for i in range(n):
    pre[i] = last[a[i]]
    last[a[i]] = i
dp = [0] * n
for _ in range(k):
    seg = LazySegTree([0] + dp[:-1], max, 0, _add, _add, 0)
    for i in range(n):
        seg.apply(pre[i] + 1, i + 1, 1)
        dp[i] = seg.prod(0, i + 1)
print(dp[-1])

Solution (Python):
import random

class LazySegTree:
    """"""
    __slots__ = ['n', 'log', 'size', 'd', 'lz', 'e', 'op', 'mapping', 'composition', 'identity']

    def _update(self, k):
        self.d[k] = self.op(self.d[2 * k], self.d[2 * k + 1])

    def _all_apply(self, k, f):
        self.d[k] = self.mapping(f, self.d[k])
        if k < self.size:
            self.lz[k] = self.composition(f, self.lz[k])

    def _push(self, k):
        self._all_apply(2 * k, self.lz[k])
        self._all_apply(2 * k + 1, self.lz[k])
        self.lz[k] = self.identity

    def __init__(self, V, OP, E, MAPPING, COMPOSITION, ID):
        self.n = len(V)
        self.log = (self.n - 1).bit_length()
        self.size = 1 << self.log
        self.d = [E for i in range(2 * self.size)]
        self.lz = [ID for i in range(self.size)]
        self.e = E
        self.op = OP
        self.mapping = MAPPING
        self.composition = COMPOSITION
        self.identity = ID
        for i in range(self.n):
            self.d[self.size + i] = V[i]
        for i in range(self.size - 1, 0, -1):
            self._update(i)

    def set(self, p, x):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        self.d[p] = x
        for i in range(1, self.log + 1):
            self._update(p >> i)

    def get(self, p):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        return self.d[p]

    def prod(self, l, r):
        assert 0 <= l and l <= r and (r <= self.n)
        if l == r:
            return self.e
        l += self.size
        r += self.size
        for i in range(self.log, 0, -1):
            if l >> i << i != l:
                self._push(l >> i)
            if r >> i << i != r:
                self._push(r >> i)
        (sml, smr) = (self.e, self.e)
        while l < r:
            if l & 1:
                sml = self.op(sml, self.d[l])
                l += 1
            if r & 1:
                r -= 1
                smr = self.op(self.d[r], smr)
            l >>= 1
            r >>= 1
        return self.op(sml, smr)

    def all_prod(self):
        return self.d[1]

    def apply_point(self, p, f):
        assert 0 <= p and p < self.n
        p += self.size
        for i in range(self.log, 0, -1):
            self._push(p >> i)
        self.d[p] = self.mapping(f, self.d[p])
        for i in range(1, self.log + 1):
            self._update(p >> i)

    def apply(self, l, r, f):
        assert 0 <= l and l <= r and (r <= self.n)
        if l == r:
            return
        l += self.size
        r += self.size
        for i in range(self.log, 0, -1):
            if l >> i << i != l:
                self._push(l >> i)
            if r >> i << i != r:
                self._push(r - 1 >> i)
        (l2, r2) = (l, r)
        while l < r:
            if l & 1:
                self._all_apply(l, f)
                l += 1
            if r & 1:
                r -= 1
                self._all_apply(r, f)
            l >>= 1
            r >>= 1
        (l, r) = (l2, r2)
        for i in range(1, self.log + 1):
            if l >> i << i != l:
                self._update(l >> i)
            if r >> i << i != r:
                self._update(r - 1 >> i)

    def max_right(self, l, g):
        assert 0 <= l and l <= self.n
        assert g(self.e)
        if l == self.n:
            return self.n
        l += self.size
        for i in range(self.log, 0, -1):
            self._push(l >> i)
        sm = self.e
        while 1:
            while i % 2 == 0:
                l >>= 1
            if not g(self.op(sm, self.d[l])):
                while l < self.size:
                    self._push(l)
                    l = 2 * l
                    if g(self.op(sm, self.d[l])):
                        sm = self.op(sm, self.d[l])
                        l += 1
                return l - self.size
            sm = self.op(sm, self.d[l])
            l += 1
            if l & -l == l:
                break
        return self.n

    def min_left(self, r, g):
        assert 0 <= r and r <= self.n
        assert g(self.e)
        if r == 0:
            return 0
        r += self.size
        for i in range(self.log, 0, -1):
            self._push(r - 1 >> i)
        sm = self.e
        while 1:
            r -= 1
            while r > 1 and r % 2:
                r >>= 1
            if not g(self.op(self.d[r], sm)):
                while r < self.size:
                    self._push(r)
                    r = 2 * r + 1
                    if g(self.op(self.d[r], sm)):
                        sm = self.op(self.d[r], sm)
                        r -= 1
                return r + 1 - self.size
            sm = self.op(self.d[r], sm)
            if r & -r == r:
                break
        return 0
(n, k) = map(int, input().split())
nums = list(map(int, input().split()))
pos = [-1] * (n + 1)
prev = [-1] * n
for i in range(n):
    prev[i] = pos[nums[i]]
    pos[nums[i]] = i

def add(x, y):
    return x + y
dp = [0] * n
for _ in range(k):
    seg = LazySegTree([0] + dp[:-1], max, 0, add, add, 0)
    for i in range(n):
        seg.apply(prev[i] + 1, i + 1, 1)
        dp[i] = seg.prod(0, i + 1)
print(dp[-1])