v1.0.4:增加了多项式、圆方树和修改其他内容

Author: lr580

README.md

@@ -1,4 +1,4 @@
-**当前最新普通版发布版本为 `v1.0.3`** ，  **最新打印版发布版本为 `v1.0.0` (总词数约7.7w(含代码))**
+**当前最新普通版发布版本为 `v1.0.4`** ，  **最新打印版发布版本为 `v1.0.0` (总词数约7.7w(含代码))**
 
 成品为 `template.pdf` (移步 [releases](https://github.com/lr580/algorithm_template/releases) 查看/下载)
 
@@ -46,13 +46,24 @@
 
 ## 更新日志
 
+- 22/06/10 - 22/08/27 (`v1.0.4`)
+
+  - 微改了 STL 的  nth\_element 和 inplace\_merge
+  
+  - 微改了复杂度一节的排版错误
+  - 删除了 STL 的 string 重复内容
+  - 添加了多项式一节，增加 NTT 和分治 FFT 内容
+  - 修正了线段树区间最值例题表述错误
+  - 更换了欧拉筛一道例题
+  - 添加了 tarjan 算法求点双连通分量和圆方树内容
+  
 - 22/05/17 - 22/05/25 (`v1.0.3`)
 
   - 增加了stringstream
   - 增加了普通莫队、带修莫队、树上莫队和回滚莫队算法
   - 增加了set部分内容
   - 增加了2-SAT算法
-  
+
 - 22/04/19 - 22/04/22 (`v1.0.2`)
 
   - 增加了KMP算法周期与border

code/euler_sample.cpp

@@ -0,0 +1,63 @@
+//https://oj.socoding.cn/p/1745
+#include <bits/stdc++.h>
+using namespace std;
+typedef long long ll;
+#define sc(x) scanf("%lld", &x)
+#define mn 1000002
+ll n, k, p[mn], pri[mn], e[mn], pe[mn], g[mn], cnt, ans = 1, mod = 1e9 + 7;
+void euler(ll n)
+{ // e[i]是i质因数分解得到的最大的幂a_i,pe[i]是对应最大的(p^e[i])
+    for (ll i = 2; i <= n; ++i)
+    {
+        if (!p[i])
+        {
+            p[i] = i, pri[++cnt] = i, pe[i] = i, e[i] = 1;
+        }
+        for (ll j = 1; i * pri[j] <= n; ++j)
+        {
+            p[i * pri[j]] = pri[j];
+            if (pri[j] == p[i])
+            {
+                e[i * pri[j]] = e[i] + 1;
+                pe[i * pri[j]] = pe[i] * pri[j];
+                break;
+            }
+            e[i * pri[j]] = 1;
+            pe[i * pri[j]] = pri[j];
+        }
+    }
+}
+ll qpow(ll a, ll b)
+{
+    ll res = 1;
+    for (; b > 0; b >>= 1)
+    {
+        if (b & 1)
+        {
+            res = res * a % mod;
+        }
+        a = a * a % mod;
+    }
+    return res;
+}
+signed main()
+{
+    sc(n), sc(k);
+    g[1] = 1;
+    euler(n);
+    for (ll i = 2; i <= n; ++i)
+    {
+        if (pe[i] == i)
+        {
+            g[i] = (qpow(p[i], e[i] * k + 1) - 1 + mod) % mod * qpow(p[i] - 1, mod - 2) % mod;
+            g[i] = g[i] * (qpow(p[i], e[i] * k) - qpow(p[i], e[i] * k - 1) + mod) % mod;
+        }
+        else
+        {
+            g[i] = g[i / pe[i]] * g[pe[i]] % mod;
+        }
+        ans = (ans + g[i]) % mod;
+    }
+    printf("%lld", ans);
+    return 0;
+}

code/p3803_ntt.cpp

@@ -0,0 +1,75 @@
+#include <bits/stdc++.h>
+using namespace std;
+#define sc(x) scanf("%lld", &x)
+typedef long long ll;
+const ll mn = 3e6 + 10, p = 998244353, g = 3, gi = 332748118;
+ll n, m, lim = 1, l, r[mn], a[mn], b[mn]; // 2^l=lim
+ll qpow(ll a, ll b)
+{
+    ll r = 1;
+    for (; b; b >>= 1)
+    {
+        if (b & 1)
+        {
+            r = r * a % p;
+        }
+        a = a * a % p;
+    }
+    return r;
+}
+void ntt(ll *a, ll type)
+{
+    for (ll i = 0; i < lim; ++i)
+    {
+        if (i < r[i])
+        {
+            swap(a[i], a[r[i]]);
+        }
+    }
+    for (ll mid = 1; mid < lim; mid <<= 1)
+    {
+        ll wn = qpow(type == 1 ? g : gi, (p - 1) / (mid << 1));
+        for (ll j = 0; j < lim; j += (mid << 1))
+        {
+            ll w = 1;
+            for (ll k = 0; k < mid; ++k, w = w * wn % p)
+            {
+                ll x = a[j + k], y = w * a[j + k + mid] % p;
+                a[j + k] = (x + y) % p;
+                a[j + k + mid] = (x - y + p) % p;
+            }
+        }
+    }
+}
+signed main()
+{
+    sc(n), sc(m);
+    for (ll i = 0; i <= n; ++i)
+    {
+        sc(a[i]);
+    }
+    for (ll i = 0; i <= m; ++i)
+    {
+        sc(b[i]);
+    }
+    while (lim <= n + m)
+    {
+        ++l, lim <<= 1;
+    }
+    for (ll i = 0; i < lim; ++i)
+    {
+        r[i] = (r[i >> 1] >> 1 | ((i & 1) << (l - 1)));
+    }
+    ntt(a, 1), ntt(b, 1);
+    for (ll i = 0; i < lim; ++i)
+    {
+        a[i] = a[i] * b[i] % p;
+    }
+    ntt(a, -1);
+    ll inv = qpow(lim, p - 2);
+    for (ll i = 0; i <= n + m; ++i)
+    {
+        printf("%lld ", a[i] * inv % p);
+    }
+    return 0;
+}

code/partition_fft.cpp

@@ -0,0 +1,103 @@
+//https://oj.socoding.cn/p/1778
+#include <bits/stdc++.h>
+using namespace std;
+using ll = long long;
+using pii = pair<int, int>;
+using pll = pair<ll, ll>;
+const ll mod = 998244353;
+
+ll qui(ll a, ll x)
+{
+    ll ret = 1;
+    while (x)
+    {
+        if (x & 1)
+            ret = ret * a % mod;
+        a = a * a % mod;
+        x >>= 1;
+    }
+    return ret;
+}
+
+using Poly = vector<ll>;
+const int BIT = 20;
+int p[1 << BIT];
+const ll maxn = 1e5 + 10;
+ll fac[maxn], inv[maxn];
+
+Poly operator*(const Poly &a, const Poly &b)
+{
+    int n = a.size() - 1, m = b.size() - 1;
+    int L, l = 0;
+    for (L = 1; L <= n + m; l++, L = L << 1)
+        ;
+
+    vector<int> p(L);
+
+    for (int i = 1; i < L; i++)
+        p[i] = ((p[i >> 1] >> 1) | ((i & 1) << (l - 1)));
+    auto u = a, v = b;
+    u.resize(L, 0), v.resize(L, 0);
+    auto ntt = [&L, &l, &p](Poly &g, int type)
+    {
+        for (int i = 0; i < L; i++)
+            if (i < p[i])
+                swap(g[i], g[p[i]]);
+        for (int i = 1; i < L; (i <<= 1))
+        {
+            ll wn = qui(3, (mod - 1) / (i << 1));
+            for (int j = 0; j < L; j += (i << 1))
+            {
+                ll w = 1;
+                for (int k = j; k < j + i; w = w * wn % mod, k++)
+                {
+                    assert(k + i < L);
+                    assert(k < L);
+                    ll t = g[k + i] * w % mod;
+                    g[k + i] = (g[k] - t + mod) % mod;
+                    g[k] = (g[k] + t) % mod;
+                }
+            }
+        }
+        if (type == 1)
+            return;
+        reverse(g.begin() + 1, g.begin() + L);
+        ll ni = qui(L, mod - 2);
+        for (int i = 0; i < L; i++)
+            g[i] = g[i] * ni % mod;
+    };
+    ntt(u, 1), ntt(v, 1);
+
+    Poly g(L, 0);
+    for (int i = 0; i < L; i++)
+        g[i] = u[i] * v[i] % mod;
+    ntt(g, -1);
+
+    return g;
+}
+
+signed main()
+{
+    cin.tie(0)->sync_with_stdio(false);
+    int n, k;
+    cin >> n >> k;
+
+    vector<int> b(n + 1);
+    for (int i = 1; i <= n; i++)
+        cin >> b[i];
+
+    function<Poly(int, int)> calc = [&](int l, int r)
+    {
+        if (l >= r)
+        {
+            assert(l == r);
+            return Poly{1, b[l]};
+        }
+        int mid = (l + r) / 2;
+        return calc(l, mid) * calc(mid + 1, r);
+    };
+
+    Poly ep = calc(0, n);
+    cout << ep[k];
+    return 0;
+}

code/tarjan_bi_conn.cpp

@@ -0,0 +1,29 @@
+ll dfn[mn], low[mn], st, stk[mn], stop, cn;
+vector<ll> c[mn]; //点双
+void tarjan(ll u)
+{
+    dfn[u] = low[u] = ++st;
+    stk[++stop] = u;
+    for (ll i = hd[u]; i; i = e[i].nx)
+    {
+        ll v = e[i].to;
+        if (!dfn[v])
+        {
+            tarjan(v);
+            low[u] = min(low[u], low[v]);
+            if (low[v] >= dfn[u])
+            { //找出新的点双上的所有点
+                c[++cn].push_back(u);
+                for (ll w = 0; w != v;)
+                {
+                    w = stk[stop--];
+                    c[cn].push_back(w);
+                }
+            }
+        }
+        else
+        {
+            low[u] = min(low[u], dfn[v]);
+        }
+    }
+}