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minimum-sum-of-values-by-dividing-array.cpp
175 lines (161 loc) · 6.45 KB
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minimum-sum-of-values-by-dividing-array.cpp
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// Time: O(n * m * logr), r = max(nums)
// Space: O(n + logr)
// dp, mono deque, two pointers
class Solution {
public:
int minimumValueSum(vector<int>& nums, vector<int>& andValues) {
static const int INF = numeric_limits<int>::max();
const auto& bit_length = [](int x) {
return (x ? std::__lg(x) : -1) + 1;
};
const int L = bit_length(ranges::max(nums));
const auto& update = [&](auto& cnt, int x, int d) {
for (int i = 0; i < L; ++i) {
if (x & (1 << i)) {
cnt[i] += d;
}
}
};
const auto& mask = [&](const auto& cnt, int l) {
int result = 0;
for (int i = 0; i < L; ++i) {
if (cnt[i] == l) {
result |= 1 << i;
}
}
return result;
};
vector<int> dp(size(nums) + 1, INF);
dp[0] = 0;
for (int j = 0; j < size(andValues); ++j) {
vector<int> new_dp(size(dp), INF), cnt(L), l(size(dp));
deque<int> dq;
for (int right = j, left = right, idx = right; right < size(nums); ++right) {
update(cnt, nums[right], +1);
if (mask(cnt, right - left + 1) <= andValues[j]) {
for (; left <= right && mask(cnt, right - left + 1) <= andValues[j]; ++left) {
update(cnt, nums[left], -1);
}
--left;
update(cnt, nums[left], +1); // try to move to the last left s.t. mask(cnt, right - left + 1) == andValues[j]
}
if ((andValues[j] & nums[right]) == andValues[j]) {
l[right + 1] = l[right] + 1;
}
if (mask(cnt, right - left + 1) != andValues[j]) {
continue;
}
// new_dp[right+1] = min(dp[left-l[left]], dp[left-l[left]+1], ..., dp[left])+nums[right]
for (; idx <= left; ++idx) {
for (; !empty(dq) && dp[dq.back()] >= dp[idx]; dq.pop_back());
dq.emplace_back(idx);
}
for (; !empty(dq) && dq.front() < left - l[left]; dq.pop_front());
if (!empty(dq)) {
if (dp[dq.front()] != INF) {
new_dp[right + 1] = dp[dq.front()] + nums[right];
}
}
}
dp = move(new_dp);
}
return dp.back() != INF ? dp.back() : -1;
}
};
// Time: O(m * n * (logn + logr)), r = max(nums)
// Space: O(n + logr)
// dp, sparse table
class Solution2 {
public:
int minimumValueSum(vector<int>& nums, vector<int>& andValues) {
static const int INF = numeric_limits<int>::max();
vector<int> dp(size(nums) + 1, INF);
dp[0] = 0;
for (int j = 0; j < size(andValues); ++j) {
vector<int> new_dp(size(nums) + 1, INF);
vector<pair<int, int>> masks;
SparseTable st(dp, [&](int i, int j) { return min(i, j); });
for (int i = j; i < size(nums); ++i) {
masks.emplace_back(nums[i], i);
for (auto& [mask, _] : masks) {
mask &= nums[i];
}
masks.erase(unique(begin(masks), end(masks), [](const auto& a, const auto& b) {
return a.first == b.first;
}), end(masks));
for (int k = 0; k < size(masks); ++k) {
const auto [mask, left] = masks[k];
if (mask == andValues[j]) {
const int right = k + 1 != size(masks) ? masks[k + 1].second - 1 : i;
if (st.query(left, right) != INF) {
// any j in range(left, right+1) has same and(nums[j:i+1]) = mask
new_dp[i + 1] = min(new_dp[i + 1], st.query(left, right) + nums[i]);
}
break;
}
}
}
dp = move(new_dp);
}
return dp.back() == INF ? -1 : dp.back();
}
private:
// Reference: https://cp-algorithms.com/data_structures/sparse-table.html
class SparseTable {
public:
SparseTable(const vector<int>& arr, function<int (int, int)> fn)
: fn(fn) { // Time: O(nlogn) * O(fn) = O(nlogn), Space: O(nlogn)
const int n = size(arr);
const int k = __lg(n);
st.assign(k + 1, vector<int>(n));
st[0].assign(cbegin(arr), cend(arr));
for (int i = 1; i <= k; ++i) {
for (int j = 0; j + (1 << i) <= n; ++j) {
st[i][j] = fn(st[i - 1][j], st[i - 1][j + (1 << (i - 1))]);
}
}
}
int query(int L, int R) const {
const int i = __lg(R - L + 1);
return fn(st[i][L], st[i][R - (1 << i) + 1]); // Time: O(fn) = O(1)
}
private:
vector<vector<int>> st;
const function<int (int, int)>& fn;
};
};
// Time: O(n * m * logr), r = max(nums)
// Space: O(n * m * logr)
// memoization
class Solution3 {
public:
int minimumValueSum(vector<int>& nums, vector<int>& andValues) {
static const int INF = numeric_limits<int>::max();
const auto& bit_length = [](int x) {
return (x ? std::__lg(x) : -1) + 1;
};
const int FULL_MASK = (1 << bit_length(ranges::max(nums))) - 1;
vector<vector<unordered_map<int, int>>> lookup(size(nums), vector<unordered_map<int, int>>(size(andValues)));
const function<int (int, int, int)> memoization = [&](int i, int j, int mask) {
if (i == size(nums) && j == size(andValues)) {
return 0;
}
if (i == size(nums) || j == size(andValues) || mask < andValues[j]) {
return INF;
}
if (!lookup[i][j].count(mask)) {
int curr = memoization(i + 1, j, mask & nums[i]);
if ((mask & nums[i]) == andValues[j]) {
const int total = memoization(i + 1, j + 1, FULL_MASK);
if (total != INF) {
curr = min(curr, nums[i] + total);
}
}
lookup[i][j][mask] = curr;
}
return lookup[i][j][mask];
};
const int result = memoization(0, 0, FULL_MASK);
return result != INF ? result : -1;
}
};