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number-of-beautiful-integers-in-the-range.cpp
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number-of-beautiful-integers-in-the-range.cpp
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// Time: O(n^2 * k), n = len(str(high))
// Space: O(n^2 * k)
// memoization (faster but more space)
class Solution {
public:
int numberOfBeautifulIntegers(int low, int high, int k) {
enum {TIGHT, UNTIGHT, UNBOUND};
const auto& to_digits = [](int x) {
vector<int> result;
for (; x; x /= 10) {
result.emplace_back(x % 10);
}
reverse(begin(result), end(result));
return !empty(result) ? result : vector<int>{0};
};
const auto& f = [&](int x) {
const auto& digits = to_digits(x);
vector<vector<vector<vector<int>>>> lookup(size(digits), vector<vector<vector<int>>>(3, vector<vector<int>>(2 * size(digits) + 1, vector<int>(k, -1))));
function<int (int, int, int, int)> memoization = [&](int i, int state, int diff, int total) {
if (i == size(digits)) {
return static_cast<int>(state != UNBOUND && diff == size(digits) && total == 0);
}
if (lookup[i][state][diff][total] == -1) {
int result = static_cast<int>(i != 0 && diff == size(digits) && total == 0); // count if the beautiful integer x s.t. len(str(x)) < len(digits)
for (int d = (i == 0) ? 1 : 0; d <= 9; ++d) {
int new_state = state;
if (state == TIGHT && d != digits[i]) {
new_state = d < digits[i] ? UNTIGHT : UNBOUND;
}
const int new_diff = diff + (d % 2 == 0 ? 1 : -1);
const int new_total = (total * 10 + d) % k;
result += memoization(i + 1, new_state, new_diff, new_total);
}
lookup[i][state][diff][total] = result;
}
return lookup[i][state][diff][total];
};
return memoization(0, TIGHT, size(digits), 0);
};
return f(high) - f(low - 1);
}
};
// Time: O(n^2 * k), n = len(str(high))
// Space: O(n * k)
// dp (slower but less space)
class Solution2 {
public:
int numberOfBeautifulIntegers(int low, int high, int k) {
enum {TIGHT, UNTIGHT, UNBOUND};
const auto& to_digits = [](int x) {
vector<int> result;
for (; x; x /= 10) {
result.emplace_back(x % 10);
}
reverse(begin(result), end(result));
return !empty(result) ? result : vector<int>{0};
};
const auto& f = [&](int x) {
const auto& digits = to_digits(x);
vector<vector<vector<int64_t>>> dp(3, vector<vector<int64_t>>(2 * size(digits) + 1, vector<int64_t>(k)));
for (int tight = 0; tight < 2; ++tight) {
for (const auto& state : {TIGHT, UNTIGHT}) {
dp[state][size(digits)][0] = 1;;
}
}
for (int i = size(digits) - 1; i >= 0; --i) {
vector<vector<vector<int64_t>>> new_dp(3, vector<vector<int64_t>>(2 * size(digits) + 1, vector<int64_t>(k)));
for (const auto& state : {TIGHT, UNTIGHT, UNBOUND}) {
new_dp[state][size(digits)][0] = static_cast<int>(i != 0); // count if the beautiful integer x s.t. len(str(x)) < len(digits)
for (int d = (i == 0) ? 1 : 0; d <= 9; ++d) {
int new_state = state;
if (state == TIGHT && d != digits[i]) {
new_state = d < digits[i] ? UNTIGHT : UNBOUND;
}
for (int diff = 0; diff < 2 * size(digits) + 1; ++diff) {
const int new_diff = diff + (d % 2 == 0 ? 1 : -1);
if (!(0 <= new_diff && new_diff < 2 * size(digits) + 1)) {
continue;
}
for (int total = 0; total < k; ++total) {
const int new_total = (total * 10 + d) % k;
new_dp[state][diff][total] += dp[new_state][new_diff][new_total];
}
}
}
}
dp = move(new_dp);
}
return dp[TIGHT][size(digits)][0];
};
return f(high) - f(low - 1);
}
};
// Time: O(n^2 * k), n = len(str(high))
// Space: O(n^2 * k)
// memoization (faster but more space)
class Solution3 {
public:
int numberOfBeautifulIntegers(int low, int high, int k) {
const auto& to_digits = [](int x) {
vector<int> result;
for (; x; x /= 10) {
result.emplace_back(x % 10);
}
reverse(begin(result), end(result));
return !empty(result) ? result : vector<int>{0};
};
const auto& f = [&](int x) {
const auto& digits = to_digits(x);
vector<vector<vector<vector<vector<int>>>>> lookup(size(digits), vector<vector<vector<vector<int>>>>(2, vector<vector<vector<int>>>(2, vector<vector<int>>(2 * size(digits) + 1, vector<int>(k, -1)))));
function<int (int, int, int, int, int)> memoization = [&](int i, int zero, int tight, int diff, int total) {
if (i == size(digits)) {
return static_cast<int>(zero == 0 && diff == size(digits) && total == 0);
}
if (lookup[i][zero][tight][diff][total] == -1) {
int result = 0;
for (int d = 0; d <= (tight ? digits[i] : 9); ++d) {
const int new_zero = static_cast<int>(zero && d == 0);
const int new_tight = static_cast<int>(tight && d == digits[i]);
const int new_diff = diff + (new_zero == 0 ? (d % 2 == 0 ? 1 : -1): 0);
const int new_total = (total * 10 + d) % k;
result += memoization(i + 1, new_zero, new_tight, new_diff, new_total);
}
lookup[i][zero][tight][diff][total] = result;
}
return lookup[i][zero][tight][diff][total];
};
return memoization(0, 1, 1, size(digits), 0);
};
return f(high) - f(low - 1);
}
};
// Time: O(n^2 * k), n = len(str(high))
// Space: O(n * k)
// dp (slower but less space)
class Solution4 {
public:
int numberOfBeautifulIntegers(int low, int high, int k) {
const auto& to_digits = [](int x) {
vector<int> result;
for (; x; x /= 10) {
result.emplace_back(x % 10);
}
reverse(begin(result), end(result));
return !empty(result) ? result : vector<int>{0};
};
const auto& f = [&](int x) {
const auto& digits = to_digits(x);
vector<vector<vector<vector<int64_t>>>> dp(2, vector<vector<vector<int64_t>>>(2, vector<vector<int64_t>>(2 * size(digits) + 1, vector<int64_t>(k))));
for (int tight = 0; tight < 2; ++tight) {
dp[0][tight][size(digits)][0] = 1;
}
for (int i = size(digits) - 1; i >= 0; --i) {
vector<vector<vector<vector<int64_t>>>> new_dp(2, vector<vector<vector<int64_t>>>(2, vector<vector<int64_t>>(2 * size(digits) + 1, vector<int64_t>(k))));
for (int zero = 0; zero < 2; ++zero) {
for (int tight = 0; tight < 2; ++tight) {
for (int d = 0; d <= (tight ? digits[i] : 9); ++d) {
const int new_zero = static_cast<int>(zero && d == 0);
const int new_tight = static_cast<int>(tight && d == digits[i]);
for (int diff = 0; diff < 2 * size(digits) + 1; ++diff) {
const int new_diff = diff + (new_zero == 0 ? (d % 2 == 0 ? 1 : -1): 0);
if (!(0 <= new_diff && new_diff < 2 * size(digits) + 1)) {
continue;
}
for (int total = 0; total < k; ++total) {
const int new_total = (total * 10 + d) % k;
new_dp[zero][tight][diff][total] += dp[new_zero][new_tight][new_diff][new_total];
}
}
}
}
}
dp = move(new_dp);
}
return dp[1][1][size(digits)][0];
};
return f(high) - f(low - 1);
}
};