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feat: add solutions to lcp problem: No.24 #2292

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feat: add solutions to lcp problem: No.24 #2292

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feat: add solutions to lcp problem: No.24
yanglbme Feb 1, 2024
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fix: code
yanglbme Feb 1, 2024
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235 changes: 235 additions & 0 deletions lcp/LCP 24. 数字游戏/README.md
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Expand Up @@ -55,4 +55,239 @@

## 解法

### 方法一：优先队列（大小根堆）

我们不妨假设最终的数组元素为 $x, x+1, x+2, \cdots, x+n-1$，那么操作次数为 $|nums[0] - x| + |nums[1] - (x+1)| + \cdots + |nums[n-1] - (x+n-1)|$。我们不妨变换一下式子，得到 $|nums[0] - x| + |nums[1] - 1 - x| + \cdots + |nums[n-1] - (n-1) - x|$。

如果我们将 $nums[i] - i$ 作为第 $i$ 个元素 $nums[i]$，那么上述式子就变成了求 $|nums[0] - x| + |nums[1] - x| + \cdots + |nums[n-1] - x|$ 的最小值。这等价于求：把数组 $nums$ 操作成相同数字的最小操作次数，即把数组 $nums$ 操作成中位数的最小操作数。

我们可以用大小根堆，动态维护前缀数组的中位数。

具体地，我们创建大根堆、小根堆，其中：小根堆 $q1$ 存放较大的一半元素，大根堆 $q2$ 存放较小的一半元素。

添加元素时，先放入小根堆，然后将小根堆对顶元素弹出并放入大根堆（使得大根堆个数多 $1$）；若大小根堆元素个数差超过 $1$，则将大根堆元素弹出放入小根堆。过程中，用两个变量 $s1, s2$ 分别记录两个堆的元素和。

取中位数时，若大根堆元素较多，取大根堆堆顶，否则取两堆顶元素和的平均值。求出中位数 $x$ 后，当前最小操作次数为 $s1 - x \times |q1| + x \times |q2| - s2$。注意，我们需要对最小操作次数取模。

时间复杂度 $O(n \times \log n)$，空间复杂度 $O(n)$。其中 $n$ 为数组长度。

<!-- tabs:start -->

```python
class MedianFinder:
def __init__(self):
self.q1 = []
self.q2 = []
self.s1 = 0
self.s2 = 0

def addNum(self, num: int) -> None:
heappush(self.q1, num)
self.s1 += num
num = heappop(self.q1)
heappush(self.q2, -num)
self.s1 -= num
self.s2 += num
if len(self.q2) - len(self.q1) > 1:
num = -heappop(self.q2)
heappush(self.q1, num)
self.s1 += num
self.s2 -= num

def findMedian(self) -> int:
if len(self.q2) > len(self.q1):
return -self.q2[0]
return (self.q1[0] - self.q2[0]) // 2

def cal(self) -> int:
x = self.findMedian()
return (self.s1 - x * len(self.q1) + x * len(self.q2) - self.s2) % (10**9 + 7)


class Solution:
def numsGame(self, nums: List[int]) -> List[int]:
n = len(nums)
ans = [0] * n
finder = MedianFinder()
for i, x in enumerate(nums):
finder.addNum(x - i)
ans[i] = finder.cal()
return ans
```

```java
class MedianFinder {
private PriorityQueue<Integer> q1 = new PriorityQueue<>();
private PriorityQueue<Integer> q2 = new PriorityQueue<>(Collections.reverseOrder());
private final int mod = (int) 1e9 + 7;
private long s1;
private long s2;

public MedianFinder() {
}

public void addNum(int num) {
q1.offer(num);
s1 += num;
num = q1.poll();
q2.offer(num);
s1 -= num;
s2 += num;
if (q2.size() - q1.size() > 1) {
num = q2.poll();
q1.offer(num);
s1 += num;
s2 -= num;
}
}

public long findMedian() {
if (q2.size() > q1.size()) {
return q2.peek();
}
return (q1.peek() + q2.peek()) / 2;
}

public int cal() {
long x = findMedian();
return (int) ((s1 - x * q1.size() + x * q2.size() - s2) % mod);
}
}

class Solution {
public int[] numsGame(int[] nums) {
int n = nums.length;
int[] ans = new int[n];
MedianFinder finder = new MedianFinder();
for (int i = 0; i < n; ++i) {
finder.addNum(nums[i] - i);
ans[i] = finder.cal();
}
return ans;
}
}
```

```cpp
class MedianFinder {
public:
MedianFinder() {
}

void addNum(int num) {
q1.push(num);
s1 += num;
num = q1.top();
q2.push(num);
q1.pop();
s2 += num;
s1 -= num;
if (q2.size() - q1.size() > 1) {
num = q2.top();
q1.push(num);
q2.pop();
s1 += num;
s2 -= num;
}
}

int findMedian() {
if (q2.size() > q1.size()) {
return q2.top();
}
return (q1.top() + q2.top()) / 2;
}

int cal() {
long long x = findMedian();
return (s1 - x * q1.size() + x * q2.size() - s2) % mod;
}

private:
priority_queue<int, vector<int>, greater<int>> q1;
priority_queue<int> q2;
long long s1 = 0;
long long s2 = 0;
const int mod = 1e9 + 7;
};

class Solution {
public:
vector<int> numsGame(vector<int>& nums) {
int n = nums.size();
vector<int> ans(n);
MedianFinder finder;
for (int i = 0; i < n; ++i) {
finder.addNum(nums[i] - i);
ans[i] = finder.cal();
}
return ans;
}
};
```

```go
func numsGame(nums []int) []int {
n := len(nums)
ans := make([]int, n)
finder := newMedianFinder()
for i, x := range nums {
finder.AddNum(x - i)
ans[i] = finder.Cal()
}
return ans
}

type MedianFinder struct {
q1 hp
q2 hp
s1 int
s2 int
}

func newMedianFinder() *MedianFinder {
return &MedianFinder{hp{}, hp{}, 0, 0}
}

func (this *MedianFinder) AddNum(num int) {
heap.Push(&this.q1, num)
this.s1 += num
num = heap.Pop(&this.q1).(int)
heap.Push(&this.q2, -num)
this.s1 -= num
this.s2 += num
if this.q2.Len()-this.q1.Len() > 1 {
num = -heap.Pop(&this.q2).(int)
heap.Push(&this.q1, num)
this.s1 += num
this.s2 -= num
}
}

func (this *MedianFinder) FindMedian() int {
if this.q2.Len() > this.q1.Len() {
return -this.q2.IntSlice[0]
}
return (this.q1.IntSlice[0] - this.q2.IntSlice[0]) / 2
}

func (this *MedianFinder) Cal() int {
x := this.FindMedian()
return (this.s1 - x*this.q1.Len() + x*this.q2.Len() - this.s2) % 1000000007
}

type hp struct{ sort.IntSlice }

func (h hp) Less(i, j int) bool { return h.IntSlice[i] < h.IntSlice[j] }
func (h *hp) Push(v any) { h.IntSlice = append(h.IntSlice, v.(int)) }
func (h *hp) Pop() any {
a := h.IntSlice
v := a[len(a)-1]
h.IntSlice = a[:len(a)-1]
return v
}
```

<!-- tabs:end -->

<!-- end -->
55 changes: 55 additions & 0 deletions lcp/LCP 24. 数字游戏/Solution.cpp
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Original file line number Diff line number Diff line change
@@ -0,0 +1,55 @@
class MedianFinder {
public:
MedianFinder() {
}

void addNum(int num) {
q1.push(num);
s1 += num;
num = q1.top();
q2.push(num);
q1.pop();
s2 += num;
s1 -= num;
if (q2.size() - q1.size() > 1) {
num = q2.top();
q1.push(num);
q2.pop();
s1 += num;
s2 -= num;
}
}

int findMedian() {
if (q2.size() > q1.size()) {
return q2.top();
}
return (q1.top() + q2.top()) / 2;
}

int cal() {
long long x = findMedian();
return (s1 - x * q1.size() + x * q2.size() - s2) % mod;
}

private:
priority_queue<int, vector<int>, greater<int>> q1;
priority_queue<int> q2;
long long s1 = 0;
long long s2 = 0;
const int mod = 1e9 + 7;
};

class Solution {
public:
vector<int> numsGame(vector<int>& nums) {
int n = nums.size();
vector<int> ans(n);
MedianFinder finder;
for (int i = 0; i < n; ++i) {
finder.addNum(nums[i] - i);
ans[i] = finder.cal();
}
return ans;
}
};
59 changes: 59 additions & 0 deletions lcp/LCP 24. 数字游戏/Solution.go
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Original file line number Diff line number Diff line change
@@ -0,0 +1,59 @@
func numsGame(nums []int) []int {
n := len(nums)
ans := make([]int, n)
finder := newMedianFinder()
for i, x := range nums {
finder.AddNum(x - i)
ans[i] = finder.Cal()
}
return ans
}

type MedianFinder struct {
q1 hp
q2 hp
s1 int
s2 int
}

func newMedianFinder() *MedianFinder {
return &MedianFinder{hp{}, hp{}, 0, 0}
}

func (this *MedianFinder) AddNum(num int) {
heap.Push(&this.q1, num)
this.s1 += num
num = heap.Pop(&this.q1).(int)
heap.Push(&this.q2, -num)
this.s1 -= num
this.s2 += num
if this.q2.Len()-this.q1.Len() > 1 {
num = -heap.Pop(&this.q2).(int)
heap.Push(&this.q1, num)
this.s1 += num
this.s2 -= num
}
}

func (this *MedianFinder) FindMedian() int {
if this.q2.Len() > this.q1.Len() {
return -this.q2.IntSlice[0]
}
return (this.q1.IntSlice[0] - this.q2.IntSlice[0]) / 2
}

func (this *MedianFinder) Cal() int {
x := this.FindMedian()
return (this.s1 - x*this.q1.Len() + x*this.q2.Len() - this.s2) % 1000000007
}

type hp struct{ sort.IntSlice }

func (h hp) Less(i, j int) bool { return h.IntSlice[i] < h.IntSlice[j] }
func (h *hp) Push(v any) { h.IntSlice = append(h.IntSlice, v.(int)) }
func (h *hp) Pop() any {
a := h.IntSlice
v := a[len(a)-1]
h.IntSlice = a[:len(a)-1]
return v
}
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