算法模板
文章目录
- 算法模板
- 简介
- 数组
- 字符串
- 列表
- 数学
- 树
- 图
- 动态规划
简介
博主在LeetCode网站中学习算法的过程中使用到并总结的算法模板，在算法方面算是刚过初学者阶段，竞赛分数仅2000。
为了节省读者的宝贵时间，部分基础的算法与模板未列出。当然也并非全面。
文章及代码存在不正不明之处还望指正。
数组
- 生成数组测试数据
- 区间合并
- 前缀和 二维前缀和 差分数组
- 二分查找(各种开闭区间组合)
- 回溯 子集型(选或不选/选哪个) 组合型 排列
class ArrTemplates {
class Generator {
public void generateArr(int n, int max) {
Random r = new Random();
int[] arr = new int[n];
for (int i = 0; i < n; i++) {
arr[i] = r.nextInt(max);
}
System.out.println(Arrays.toString(arr));
}
public void generateArr(int m, int n, int max) {
Random r = new Random();
int[][] arr = new int[m][n];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
arr[i][j] = r.nextInt(max);
}
}
for (int[] ints : arr) {
System.out.println(Arrays.toString(ints));
}
}
}
/**
* 区间
*/
class Interval {
/**
* 区间合并
*/
List<int[]> intervalMerge(int[][] ranges) {
int n = ranges.length;
Arrays.sort(ranges, (a, b) -> a[0] == b[0] ? a[1] - b[1] : a[0] - b[0]);
List<int[]> rList = new ArrayList<>(n);
int[] r1 = ranges[0];
for (int i = 1; i < n; i++) {
int[] r2 = ranges[i];
if (r2[0] <= r1[1]) {
r1[1] = Math.max(r1[1], r2[1]);
} else {
rList.add(r1);
r1 = r2;
}
}
rList.add(r1);
return rList;
}
}
class PrefixSum {
/**
* 前缀和
*/
public int[] getPreFix(int[] arr) {
int n = arr.length;
int[] pre = new int[n + 1];
for (int i = 0; i < n; i++) {
pre[i + 1] = pre[i] + arr[i];
}
return pre;
}
/**
* 二维前缀和
*/
public int[][] getPrefix(int[][] matrix) {
int m = matrix.length, n = matrix[0].length;
int[][] pre = new int[m + 1][n + 1];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
pre[i + 1][j + 1] = pre[i + 1][j] + pre[i][j + 1] - pre[i][j] + matrix[i][j];
}
}
return pre;
}
/**
* 返回左上角在 (r1,c1) 右下角在 (r2,c2) 的子矩阵元素和
*/
public int query(int[][] pre, int r1, int c1, int r2, int c2) {
return pre[r2 + 1][c2 + 1] - pre[r2 + 1][c1] - pre[r1][c2 + 1] + pre[r1][c1];
}
//差分数组
public int[] getDiff(int n, int[] arr) {
int[] diff = new int[n];
diff[0] = arr[0];
for (int i = 1; i < n; i++) {
diff[i] = arr[i] - arr[i - 1];
}
// for (int i = 1; i < n; i++) {
// diff[i] += diff[i - 1]; // 直接在差分数组上复原数组 a
// }
return diff;
}
int[] getDiff(int n, int[][] arr) {
int[] diff = new int[n]; // 差分数组
for (int[] q : arr) {
int l = q[0], r = q[1], x = q[2];
diff[l] += x;
if (r + 1 < n) {
diff[r + 1] -= x;
}
}
for (int i = 1; i < n; i++) {
diff[i] += diff[i - 1]; // 直接在差分数组上复原数组 a
}
return diff;
}
}
class BinarySearch {
/**
* 二分查找
* 闭区间
* 循环不变量:左侧小于等于目标值,右侧大于目标值
* r-1<=target,l+1>target
*/
int binarySearch(int[] nums, int target) {
int l = 0, r = nums.length - 1;
while (l <= r) {
int m = l + (r - l) / 2;
if (nums[m] < target) {
l = m + 1;
} else {
r = m - 1;
}
}
return l;
}
/**
* 左闭右开区间[l,r)
* [l,mid) [mid+1,r)
*/
int binarySearch2(int[] nums, int target) {
int l = 0, r = nums.length;
while (l < r) {
int m = l + (r - l) / 2;
if (nums[m] < target) {
l = m + 1;
} else {
r = m;
}
}
return l;
}
/**
* 左开右闭区间(l,r]
* (l,mid-1] (mid,r]
*/
int binarySearch3(int[] nums, int target) {
int l = -1, r = nums.length - 1;
while (l < r) {
int m = l + (r - l) / 2;
if (nums[m] < target) {
l = m;
} else {
r = m - 1;
}
}
return l;
}
/**
* 开区间(l,r)
* (l,mid)(mid,r)
*/
int binarySearch4(int[] nums, int target) {
int l = -1, r = nums.length;
while (l + 1 < r) {
int m = l + (r - l) / 2;
if (nums[m] < target) {
l = m;
} else {
r = m;
}
}
return r;
}
}
class Backtrack {
List<List<Integer>> res = new ArrayList<>();
List<Integer> path = new ArrayList<>();
int n;
boolean[] visited = new boolean[n];
/**
* 子集型
* 选或不选 n*2^n
*/
void backtrack(int[] nums, int idx) {
//解
if (idx == n) {
res.add(new ArrayList<>(path));
}
//尝试 选或不选
//不选
backtrack(nums, idx + 1);
//选
path.add(nums[idx]);
backtrack(nums, idx + 1);
//回溯
path.remove(path.size() - 1);
}
/**
* 子集型
* 每次选一个
*/
void backtrack2(int[] nums, int idx) {
//解
res.add(new ArrayList<>(path));
if (idx == n) {
return;
}
for (int i = idx; i < n; i++) {
path.add(nums[i]);
backtrack2(nums, i + 1);
//回溯
path.remove(path.size() - 1);
}
}
/**
* 组合型
* 逆序排列
*/
void backtrack(int[] nums, int idx, int k) {
//剩余不足达到k个
if (idx < k - path.size()) {
return;
}
if (path.size() == k) {
res.add(new ArrayList<>(path));
return;
}
for (int i = idx; i >= 0; i--) {
path.add(nums[i]);
backtrack(nums, i - 1, k);
//回溯
path.remove(path.size() - 1);
}
}
/**
* 组合型
* 正序排列
*/
void backtrack2(int[] nums, int idx, int k) {
if (n - idx < k - path.size()) {
return;
}
if (path.size() == k) {
res.add(new ArrayList<>(path));
return;
}
for (int i = idx; i < n; i++) {
path.add(nums[i]);
backtrack2(nums, i + 1, k);
//回溯
path.remove(path.size() - 1);
}
}
/**
* 排列
* Tn*n!
*/
void backtrack3(int[] nums, int idx) {
if (idx == n) {
res.add(new ArrayList<>(path));
return;
}
for (int i = 0; i < n; i++) {
if (!visited[i]) {
visited[i] = true;
path.add(nums[i]);
backtrack3(nums, idx + 1);
visited[i] = false;
path.remove(path.size() - 1);
}
}
}
}
}
字符串
- kmp字符串匹配
- 子串
- 回文串
class StringTemplates {
/**
* kmp
*/
public int kmpSearchIndex(String source, String target) {
int n = source.length(), m = target.length();
if (m == 0) {
return 0;
}
// 创建部分匹配表
int[] kmp = new int[m];
for (int i = 1, j = 0; i < m; i++) {
// 字符不匹配时，j回溯到上一个匹配的字符，继续匹配
while (j > 0 && target.charAt(i) != target.charAt(j)) {
j = kmp[j - 1];
}
// 当haystack和needle的字符匹配时，将j的值加一
if (target.charAt(i) == target.charAt(j)) {
j++;
}
// 更新部分匹配表
kmp[i] = j;
}
// 在haystack中查找needle
for (int i = 0, j = 0; i < n; i++) {
// 字符不匹 j回溯到上一个匹配的字符
while (j > 0 && source.charAt(i) != target.charAt(j)) {
j = kmp[j - 1];
}
// 字符匹配时，已匹配字符数+1
if (source.charAt(i) == target.charAt(j)) {
j++;
}
//找到needle
if (j == m) {
return i - m + 1;
}
}
// 没有找到needle，则返回-1
return -1;
}
/**
* 获取长度为len的所有子串
*/
private String[] getSubstrings(String s, int len) {
int n = s.length();
String[] substrings = new String[n - len + 1];
for (int j = 0; j <= n - len; j++) {
substrings[j] = s.substring(j, j + len);
}
return substrings;
}
/**
* 是否回文串
*/
boolean isPalindrome(String str) {
int left = 0, right = str.length() - 1;
while (left < right) {
if (str.charAt(left) != str.charAt(right)) {
return false;
}
left++;
right--;
}
return true;
}
/**
* 是否回文串
*/
private boolean isPalindrome(char[] ss, int l, int r) {
if (l == r) {
return true;
}
while (l < r) {
if (ss[l++] != ss[r--]) {
return false;
}
}
return true;
}
/**
* 字符串任意子串是否回文串
*/
boolean[][] palindrome(char[] cs) {
int n = cs.length;
boolean[][] p = new boolean[n][n];
//1
// for (int i = 0; i < n; i++) {
// Arrays.fill(p[i], true);
// }
// for (int i = n - 1; i >= 0; i--) {
// for (int j = i + 1; j < n; ++j) {
// p[i][j] = cs[i] == cs[j] && p[i + 1][j - 1];
// }
// }
//2
for (int len = 1; len <= n; len++) {
//从i开始,统计长度为len的子串是否为回文串
for (int i = 0; i <= n - len; i++) {
int j = i + len - 1;
if (len == 1) {
p[i][j] = true;
} else if (len == 2) {
p[i][j] = cs[i] == cs[j];
} else {
// 大于两个字符时，判断首尾字符是否相等，并且去除首尾字符后的子串是否是回文串
p[i][j] = cs[i] == cs[j] && p[i + 1][j - 1];
}
}
}
return p;
}
}
列表
- 翻转
- 快慢指针
class ListTemplates {
/**
* 翻转链表
*/
public void reverse(ListNode head, ListNode tail) {
ListNode last = null;
ListNode curr = head;
ListNode end = tail.next;
while (curr != end) {
//下一个节点
ListNode next = curr.next;
//将当前节点的指针指向前一个节点
curr.next = last;
//将当前节点置位下一个节点的前置
last = curr;
//循环控制
curr = next;
}
}
/**
* 快慢指针
*/
boolean fast_slowPoints(ListNode head) {
ListNode slow = head;
ListNode fast = head;
while (slow != null && fast != null && fast.next != null) {
slow = slow.next;
fast = fast.next.next;
if (slow == fast) {
return true;
}
}
return false;
}
}
数学
- 最大/小值
- lcm
- gcd
- 快速幂
- lowbit
- 质数/素数(埃氏筛)
- 回文数
- 组合数
class MathTemplates {
/**
* min
*/
public int min(int a, int... b) {
for (int i : b) {
a = Math.min(a, i);
}
return a;
}
/**
* max
*/
public int max(int a, int... b) {
for (int i : b) {
a = Math.max(a, i);
}
return a;
}
/**
* 最小公倍数 Lowest Common Multiple
*/
private long lcm(long a, long b) {
return a * b / gcd(a, b);
}
/**
* 最大公约数 Greatest common divisor
*/
public long gcd(long x, long y) {
return y == 0 ? x : gcd(y, x % y);
}
/**
* 快速幂
*/
public double fastPow(double x, long n) {
double res = 1.0;
while (n > 0) {
if (n % 2 == 1) {
res *= x;
}
x *= x;
n /= 2;
}
return res;
}
/**
* 快速幂
*/
public long fastPow(long x, long n) {
long res = 1;
while (n > 0) {
if (n % 2 == 1) {
res *= x;
}
x *= x;
n /= 2;
}
return res;
}
public int lowbit(int i) {
//x & (~x + 1);
return i & -i;
}
/**
* 二进制1个数
* Integer.bitCount(i)
*/
public int bitCount(int i) {
i = i - ((i >>> 1) & 0x55555555);
i = (i & 0x33333333) + ((i >>> 2) & 0x33333333);
i = (i + (i >>> 4)) & 0x0f0f0f0f;
i = i + (i >>> 8);
i = i + (i >>> 16);
return i & 0x3f;
}
/**
* 是否质数
*/
private boolean isPrime(int n) {
for (int i = 2; i * i <= n; i++) {
if (n % i == 0) {
return false;
}
}
return true;
}
private boolean isPrime2(int n) {
// 小于等于1的整数不是素数
if (n <= 1) {
return false;
}
// 2和3是素数
if (n <= 3) {
return true;
}
// 如果整数能被2或3整除，不是素数
if (n % 2 == 0 || n % 3 == 0) {
return false;
}
// 除了2和3，素数都可以表示成6的倍数加1或减1的形式(在6的倍数两侧的数不是素数)
for (int i = 5; i * i <= n; i += 6) {
if (n % i == 0 || n % (i + 2) == 0) {
return false;
}
}
return true;
}
boolean[] isPrime;
/**
* 质数表-埃氏筛
* Tnlog(logn) Sn
*/
private void getPrimes(int n) {
isPrime = new boolean[n];
Arrays.fill(isPrime, true);
isPrime[1] = false;
for (int i = 2; i <= Math.sqrt(n); i++) {
if (isPrime[i]) {
// 将 i 的所有倍数标记为非质数(合数)
for (int j = i * i; j < n; j += i) {
isPrime[j] = false;
}
}
}
}
/**
* 回文数
*/
public boolean isPalindrome(int x) {
if (x < 0 || x % 10 == 0 && x != 0) {
return false;
}
int reverse = 0;
while (x > reverse) {
reverse = reverse * 10 + x % 10;
x /= 10;
}
return x == reverse || x == reverse / 10;
}
private final int N = 31;
private final int[][] c = new int[N][N];
{
for (int i = 0; i < N; i++) {
c[i][0] = c[i][i] = 1;
for (int j = 1; j < i; j++) {
//Cn,k = Cn-1,k-1 + Cn-1,k
c[i][j] = c[i - 1][j - 1] + c[i - 1][j];
}
}
}
/**
* 计算n个数里拿k个数的组合数
*/
public long comb(int n, int k) {
long ans = 1;
for (int i = 1; i <= k; i++) {
ans *= (n - k + i);
ans /= i;
}
return ans;
}
}
树
- 树状数组
class TreeTemplates {
/**
* 树状数组
*/
class BinaryIndexedTrees {
private int[] tree;
private int[] nums;
/**
* TOnlogn
*/
public BinaryIndexedTrees(int[] nums) {
//lowbit(0)计算为0;舍弃0下标;
this.tree = new int[nums.length + 1];
this.nums = nums;
for (int i = 0; i < nums.length; i++) {
add(i + 1, nums[i]);
}
}
/**
* TOlogn
*/
public void update(int index, int val) {
add(index + 1, val - nums[index]);
nums[index] = val;
}
/**
* TOlogn
*/
public int sumRange(int left, int right) {
return prefixSum(right + 1) - prefixSum(left);
}
/**
* 用于计算树状数组的 x节点的父节点
* x的父节点索引= x+=lowbit(x)
* lowbit(x) 未 非负整数x在二进制下的最低为1及其后面的0构成的数(x的除最后一位1外,其他位置0)
*/
private int lowBit(int x) {
//return x & (~x + 1);
return x & -x;
}
private void add(int index, int val) {
while (index < tree.length) {
tree[index] += val;
index += lowBit(index);
}
}
private int prefixSum(int index) {
int sum = 0;
while (index > 0) {
sum += tree[index];
index -= lowBit(index);
}
return sum;
}
}
}
图
- 无向图/带权图 领接矩阵/表
- 并查集
- Dijkstra
- Floyd
class GraphTemplates {
/**
* 无向图
*/
public List<Integer>[] edgesToGraph(int n, int[][] edges) {
List<Integer>[] graph = new List[n];
Arrays.setAll(graph, i -> new ArrayList<>());
for (int[] edge : edges) {
int x = edge[0];
int y = edge[1];
graph[x].add(y);
graph[y].add(x);
}
return graph;
}
/**
* 带权图
*/
public List<int[]>[] edgesToWGraph(int n, int[][] edges) {
List<int[]>[] wgraph = new List[n];
Arrays.setAll(wgraph, i -> new ArrayList<>());
for (int[] edge : edges) {
int x = edge[0];
int y = edge[1];
int w = edge[2];
wgraph[x].add(new int[]{y, w});
wgraph[y].add(new int[]{x, w});
}
return wgraph;
}
/**
* 带权领接矩阵
*/
public int[][] edgesToWGraph2(int n, int[][] edges) {
int INF = Integer.MAX_VALUE / 2;
int[][] graph = new int[n][n];
for (int i = 0; i < n; i++) {
Arrays.fill(graph[i], INF);
}
for (int[] edge : edges) {
graph[edge[0]][edge[1]] = edge[2];
}
return graph;
}
/**
* 并查集
*/
class UnionFind {
int[] fa;
public UnionFind(int n) {
fa = new int[n];
for (int i = 0; i < n; i++) {
fa[i] = i;
}
}
public int find(int x) {
if (fa[x] == x) {
return x;
}
return fa[x] = find(fa[x]);
}
boolean union(int x, int y) {
int fx = find(x), fy = find(y);
if (fx == fy) {
return false;
}
if (fx > fy) {
fa[fy] = fx;
} else {
fa[fx] = fy;
}
return true;
}
}
/**
* 长度统计 联通分量统计 高度压缩 并查集
*/
class UnionFind2 {
//父节点
int[] fa;
//高度
int[] rk;
//子集长度
int[] sz;
//连通分量数
int count;
public UnionFind2(int n) {
fa = new int[n];
rk = new int[n];
sz = new int[n];
for (int i = 0; i < n; i++) {
fa[i] = i;
sz[i] = 1;
}
count = n;
}
public int find(int x) {
if (fa[x] == x) {
return x;
}
return fa[x] = find(fa[x]);
}
boolean union(int x, int y) {
int fx = find(x), fy = find(y);
if (fx == fy) {
return false;
}
//如果 x的高度大于 y，则令 y的上级为 x
if (rk[fx] > rk[fy]) {
fa[fy] = fx;
sz[fx] += sz[fy];
} else {
//如果 x的高度和 y的高度相同，则令 y的高度加1
if (rk[fx] == rk[fy]) {
rk[fy]++;
}
fa[fx] = fy;
sz[fy] += sz[fx];
}
count--;
return true;
}
public boolean isSame(int x, int y) {
return find(x) == find(y);
}
public int size(int x) {
return sz[find(x)];
}
public int count() {
return count;
}
}
class Dijkstra {
/**
* dijkstra 统计单源最短路径长度
*/
public long dijkstraCalculateLenOfShortestPaths(int n, int[][] edges, int start, int end) {
List<int[]>[] wgraph = edgesToWGraph(n, edges);
//最短路径长度,最短路径次数
long[] dist = new long[n];
Arrays.fill(dist, Long.MAX_VALUE);
dist[start] = 0;
PriorityQueue<long[]> pq = new PriorityQueue<>((a, b) -> Long.compare(a[1], b[1]));
//节点,权重
pq.offer(new long[]{start, 0});
while (!pq.isEmpty()) {
long[] node = pq.poll();
int u = (int) node[0];
//s到达u的最短路径权重
long suw = node[1];
//此路径不是到达u的最短路径
if (suw > dist[u]) {
continue;
}
for (int[] edge : wgraph[u]) {
int v = edge[0], uvw = edge[1];
//s到达v的最短路径权重
long svw = dist[v];
if (suw + uvw < svw) {
dist[v] = suw + uvw;
pq.offer(new long[]{v, dist[v]});
}
}
}
return dist[end];
}
/**
* dijkstra 统计单源最短路径数量
*/
public int dijkstraCalculateCount0fShortestPaths(int n, int[][] edges, int start, int end) {
List<int[]>[] wgraph = edgesToWGraph(n, edges);
//最短路径长度,最短路径次数
long[] dist = new long[n];
Arrays.fill(dist, Long.MAX_VALUE);
dist[start] = 0;
int[] counts = new int[n];
counts[start] = 1;
PriorityQueue<long[]> pq = new PriorityQueue<>((a, b) -> Long.compare(a[1], b[1]));
//节点,权重
pq.offer(new long[]{start, 0});
while (!pq.isEmpty()) {
long[] node = pq.poll();
int u = (int) node[0];
//s到达u的最短路径权重
long suw = node[1];
//此路径不是到达u的最短路径
if (suw > dist[u]) {
continue;
}
for (int[] edge : wgraph[u]) {
int v = edge[0], uvw = edge[1];
//s到达v的最短路径权重
long svw = dist[v];
//s到达u的最短路径数量
int suc = counts[u];
if (suw + uvw < svw) {
dist[v] = suw + uvw;
counts[v] = suc;
pq.offer(new long[]{v, dist[v]});
} else if (suw + uvw == svw) {
counts[v] = (counts[v] + suc) % 1000000007;
}
}
}
return counts[end];
}
/**
* dijkstra 单元 统计最短路径长度与数量
*/
public long[][] dijkstraCalculateLenAndCount0fShortestPaths(int n, int[][] edges) {
List<int[]>[] wgraph = edgesToWGraph(n, edges);
//最短路径长度,最短路径次数
long[][] dist = new long[n][2];
Arrays.setAll(dist, i -> new long[]{Long.MAX_VALUE, 0});
dist[0] = new long[]{0, 1};
PriorityQueue<long[]> pq = new PriorityQueue<>((a, b) -> Long.compare(a[1], b[1]));
//节点,权重
pq.offer(new long[]{0, 0});
while (!pq.isEmpty()) {
long[] node = pq.poll();
int u = (int) node[0];
//s到达u的最短路径权重
long suw = node[1];
//此路径不是到达u的最短路径
if (suw > dist[u][0]) {
continue;
}
for (int[] edge : wgraph[u]) {
int v = edge[0], uvw = edge[1];
//s到达v的最短路径权重
long svw = dist[v][0];
//s到达u的最短路径数量
long suc = dist[u][1];
if (suw + uvw < svw) {
dist[v][0] = suw + uvw;
dist[v][1] = suc;
pq.offer(new long[]{v, dist[v][0]});
} else if (suw + uvw == svw) {
dist[v][1] = (dist[v][1] + suc) % 1000000007;
}
}
}
return dist;
}
/**
* 稠密图 邻接矩阵
*/
public int[] dijkstra(int n, int[][] edges, int start) {
int INF = Integer.MAX_VALUE / 2;
int[][] graph = new int[n][n];
int[] dist = new int[n];
for (int i = 0; i < n; i++) {
Arrays.fill(graph[i], INF);
dist[i] = INF;
}
for (int[] edge : edges) {
graph[edge[0]][edge[1]] = edge[2];
}
boolean[] vis = new boolean[n];
dist[start] = 0;
for (int i = 0; i < n; i++) {
// 找到当前距离最小的未访问节点
int x = -1;
for (int y = 0; y < n; ++y) {
if (!vis[y] && (x == -1 || dist[y] < dist[x])) {
x = y;
}
}
// 访问标记
vis[x] = true;
for (int y = 0; y < n; ++y) {
// 更新最短路长度
dist[y] = Math.min(dist[y], dist[x] + graph[x][y]);
}
}
return dist;
}
}
//多源最短路径
class Floyd {
public int[][] floyd(int n, int[][] edges) {
int INF = Integer.MAX_VALUE;
int[][] dist = new int[n][n];
for (int i = 0; i < n; ++i) {
Arrays.fill(dist[i], INF);
dist[i][i] = 0;
}
for (int[] e : edges) {
dist[e[0]][e[1]] = e[2];
}
for (int k = 0; k < n; k++) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (dist[i][k] == INF || dist[k][j] == INF) {
continue;
}
dist[i][j] = Math.max(dist[i][j], dist[i][k] + dist[k][j]);
}
}
}
return dist;
}
}
class Prim {
public void primMST(int n, int[][] edges) {
}
}
}
动态规划
- 爬楼梯
- 打家劫舍
- 子数组
- 子序列
- 背白 01背包 完全背包
- 划分
- 区间
class DpTemplates {
/**
* 入门
* 爬楼梯 打家劫舍
*/
class Base {
/**
* 爬楼梯-每次相同方式二选一
*/
public int climbingStairs(int n) {
int[] dp = new int[n + 1];
dp[0] = 1;
dp[1] = 1;
for (int i = 2; i <= n; i++) {
dp[i] = dp[i - 1] + dp[i - 2];
}
return dp[n];
/*int p1 = 0, p2 = 1;
for (int i = 1; i <= n; i++) {
p2 += p1;
p1 = p2 - p1;
}
return p2;*/
}
/**
* 花费代价爬楼梯 每次相同方式二选一,并选择较少消费
*/
public int climbingStairsByMinCost(int[] cost) {
int len = cost.length;
for (int i = 2; i < len; i++) {
cost[i] = Math.min(cost[i - 1], cost[i - 2]) + cost[i];
}
return Math.min(cost[len - 1], cost[len - 2]);
}
/**
* 打家劫舍 根据要求选或不选
*/
public int rob(int[] nums) {
int[] dp = new int[2];
for (int num : nums) {
//选 上一个只能是不选
int doit = dp[1] + num;
//不选 上一个选或不选
int not = Math.max(dp[0], dp[1]);
dp[0] = doit;
dp[1] = not;
}
return Math.max(dp[0], dp[1]);
}
}
/**
* 子数组dp
*/
class Subarray {
/**
* 子数组最大值
*/
public int maxSubArrayDp(int[] nums) {
int[] dp = new int[nums.length];
dp[0] = nums[0];
int res = nums[0];
for (int i = 1; i < nums.length; i++) {
dp[i] = Math.max(dp[i - 1] + nums[i], nums[i]);
res = Math.max(res, dp[i]);
}
return res;
}
/**
* 子数组最大值
*/
public int maxSubArray(int[] nums) {
int max = nums[0];
int pre = max;
for (int i = 1; i < nums.length; i++) {
pre = Math.max(pre + nums[i], nums[i]);
max = Math.max(max, pre);
}
return max;
}
/**
* 单调递增/减 最长子数组
*/
public int longestMonotonicSubarray(int[] nums) {
int res = 1;
int i = 0, n = nums.length;
while (i < n - 1) {
//相等直接跳过
if (nums[i + 1] == nums[i]) {
i++;
continue;
}
// 记录开始位置
int start = i;
//定下基调：递增/递减
boolean inc = nums[i + 1] > nums[i];
// i 和 i+1 已经满足要求，从 i+2 开始判断
i += 2;
while (i < n && nums[i] != nums[i - 1] && (nums[i] > nums[i - 1]) == inc) {
i++;
}
// 从 start 到 i-1 是满足题目要求的（并且无法再延长的）子数组
res = Math.max(res, i - start);
i--;
}
return res;
}
}
/**
* 子序列dp
*/
class Subsequence {
/**
* 最长公共子序列LCS
*/
public int longestCommonSubsequence(int[] arr1, int[] arr2) {
int m = arr1.length, n = arr2.length;
int[][] dp = new int[m + 1][n + 1];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
dp[i + 1][j + 1] = arr1[i] == arr2[j] ? dp[i][j] + 1 : Math.max(dp[i + 1][j], dp[i][j + 1]);
}
}
return dp[m][n];
}
/**
* 子序列数量
*/
public int diffSubsequenceCount(int[] arr1, int[] arr2) {
int MOD = 1_000_000_007;
int m = arr1.length;
int n = arr2.length;
int[][] dp = new int[m + 1][n + 1];
//init
for (int i = 0; i <= m; i++) {
dp[i][0] = 1;
}
for (int i = 1; i <= m; i++) {
for (int j = 1; j <= i && j <= n; j++) {
dp[i][j] = dp[i - 1][j];
if (arr1[i - 1] == arr2[j - 1]) {
dp[i][j] = (dp[i][j] + dp[i - 1][j - 1]) % MOD;
}
}
}
return dp[m][n];
}
/**
* 最长递增子序列 LIS
*/
public int longestIncreasingSubsequence(int[] nums) {
int n = nums.length;
int[] dp = new int[n];
int res = 0;
for (int i = 0; i < n; i++) {
for (int j = 0; j < i; j++) {
if (nums[j] < nums[i]) {
dp[i] = Math.max(dp[i], dp[j] + 1);
}
}
res = Math.max(res, dp[i]);
}
return res + 1;
}
}
/**
* 背包dp
*/
class Knapsack {
/**
* 01背包
*/
public int zeroOneKnapsack(int[] ws, int[] vs, int c) {
int n = ws.length;
int[][] dp = new int[n + 1][c + 1];
dp[0][0] = 1;
// 动态规划求解
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= c; j++) {
if (ws[i - 1] > j) {
dp[i][j] = dp[i - 1][j];
} else {
dp[i][j] = Math.max(dp[i - 1][j], dp[i - 1][j - ws[i - 1]] + vs[i - 1]);
}
}
}
return dp[n][c];
}
/**
* 完全背包
*/
public int completeKnapsack(int[] ws, int[] vs, int c) {
int n = ws.length;
int[][] dp = new int[n + 1][c + 1];
dp[0][0] = 1;
// 动态规划求解
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= c; j++) {
if (ws[i - 1] > j) {
// 物品 i 不被选入背包
dp[i][j] = dp[i - 1][j];
} else {
// 物品 i 被选入背包
// 可以重复选取，所以是 dp[i][j - weights[i - 1]] + values[i - 1]
dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - ws[i - 1]] + vs[i - 1]);
}
}
}
return dp[n][c];
}
}
/**
* 划分dp
*/
class Partition {
/**
* 能否划分
* f[i] 表示 a[:i]能否划分;
* 枚举最右侧划分左端l,判断a[l,i]是否符合条件
* f[i] = min/max(f[i],f[l-1]+1)
*/
boolean canPartition(int[] nums) {
int n = nums.length;
boolean[] dp = new boolean[n + 1];
dp[0] = true;
for (int i = 1; i <= n; i++) {
for (int l = 0; l <= i; l++) {
if (dp[l] && check(nums, l, i)) {//num[l:i] 是否符合条件
dp[i] = true;
break;
}
}
}
return dp[n];
}
/**
* 划分个数 f[i] 表示a[:i]在约束下 能划分的最大/小个数
* 枚举右侧划分左端l,判断a[l,i]是否符合条件
* f[i] = min/max(f[i],f[l-1]+1)
*/
int partitionNum(int[] nums) {
int n = nums.length;
int[] dp = new int[n + 1];
for (int i = 0; i < n; i++) {
dp[i + 1] = dp[i] + 1;
for (int l = 0; l <= i; l++) {
//符合划分条件
if (check(nums, l, i)) {
//min / max
dp[i + 1] = Math.min(dp[i + 1], dp[l] + 1);
}
}
}
return dp[n];
}
/**
* 约束划分数量 划分为k个,计算最优解;
* f[i][j]: 将a[:i]划分为j个部分的最优解;
* 枚举右侧左端l,从f[l][j-1]转移到f[i][j],考虑最最优解的影响
*/
public int partitionCnt(int[] nums, int k) {
int n = nums.length;
int[] pre = new int[n + 1];
for (int i = 0; i < n; i++) {
pre[i + 1] = pre[i] + nums[i];
}
int[][] dp = new int[n + 1][k + 1];
for (int i = 0; i <= n; i++) {
Arrays.fill(dp[i], Integer.MAX_VALUE);
}
dp[0][0] = 0;
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= Math.min(i, k); j++) {
for (int l = j - 1; l < i; l++) {
//最小化 划分子数组的和
dp[i][j] = Math.min(dp[i][j], Math.max(dp[l][j - 1], pre[i] - pre[l]));
}
}
}
return dp[n][k];
}
/**
* 约束划分长度 子数组长度<=k时,最大和
* f[i] 表示nums[:i] 划分后,子数组最大值
* 枚举最后子数组左端点l
* f[i] = f[r]+val
*/
public int partitionLen(int[] nums, int k) {
int n = nums.length;
int[] dp = new int[n + 1];
for (int i = 1; i <= n; i++) {
//逆序维护区间最大值
int max = 0;
for (int l = i - 1; l >= i - k && l >= 0; l--) {
max = Math.max(max, nums[l]);
dp[i] = Math.max(dp[i], dp[l] + max * (i - l));
}
}
return dp[n];
}
private boolean check(int[] nums, int l, int r) {
return true;
}
/**
* 区间不相交划分 限定区间范围(1~n)或 [is[0],is[1]]区间范围较小
*/
public long intervalPartition(int n, int[][] is) {
//按区间结束排序
List<int[]>[] list = new List[n + 1];
for (int[] interval : is) {
if (list[interval[1]] == null) {
list[interval[1]] = new ArrayList<>();
}
list[interval[1]].add(new int[]{interval[0], interval[1] - interval[0] + interval[2]});
}
//dpi 表示 到达第i个点时,能获得的最大价值
long[] dp = new long[n + 1];
for (int i = 1; i <= n; i++) {
dp[i] = dp[i - 1];
if (list[i] == null) {
continue;
}
for (int[] r : list[i]) {
dp[i] = Math.max(dp[i], dp[r[0]] + r[1]);
}
}
return dp[n];
}
/**
* 区间不相交划分 [is[0],is[1]]区间范围较大时
*/
public int intervalPartition(int[][] is) {
int n = is.length;
//按区间排序
Arrays.sort(is, (a, b) -> a[1] == b[1] ? a[0] - b[0] : a[1] - b[1]);
//dpi: 在i项中 能获得的最大价值
int[] dp = new int[n + 1];
for (int i = 0; i < n; i++) {
int s = is[i][0], p = is[i][2];
//二分找到上一个区间
int l = 0, r = i - 1;
while (l <= r) {
int m = l + (r - l) / 2;
//可无缝衔接<=,否则<
if (is[m][1] <= s) {
l = m + 1;
} else {
r = m - 1;
}
}
dp[i + 1] = Math.max(dp[i], dp[r + 1] + p);
}
return dp[n];
}
}
}
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