# Competitive Coding Algorithms

## Dijkstra

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int dijkstraM(int start, int end, vector<vector<pi>> adj, vector<int> &dist) {
vector<bool> visited(N);
priority_queue<pi, vector<pi>, greater<pi>> pq;
pq.push({0, start});
dist[start] = 0;
while (pq.size()) {
pi value = pq.top(); pq.pop();
ll node = value.s;
if (visited[node]) {
continue;
}
visited[node] = true;
for (auto to : adj[node]) {
int newWeight = to.f + dist[node];
if (newWeight < dist[to.s]) {
// nodeInfo[to.s] = nodeInfo[node] + cows[to.s];
prevV[to.s] = node;
dist[to.s] = newWeight;
pq.push({dist[to.s], to.s});
}
}
//dbg(node, dist[node]);
}
return dist[end];
}

## BIT

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struct BIT {
vector<int> tree;
void init(int _n) { tree.resize(_n); }
void upd(int i, int x) {
while (i < tree.size()) {
tree[i] = (tree[i] + x);
//if (tree[i] < 0) tree[i] += M;
if (tree[i] > M) tree[i] -= M;
tree[i] = (tree[i] + M) % M;
i += (i & (-i));
}
}
long long query(int i) {
long long ans = 0;
while (i > 0) {
ans += tree[i];
//if (ans < 0) ans += M;
if (ans > M) ans -= M;
ans = (ans + M) % M;
i -= (i & (-i));
}
return ans;
}
};
BIT allColors;
allColors.query(j);
allColors.upd(j, x);

## Normal Segment Tree

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vector<int> arr;
vector<int> tree;
arr.resize(N + 1);
tree.resize(4*N);
void build(int p, int l, int r) {
if (l == r) {tree[p] = arr[l]; return;}
int mid = (l + r)/2;
build(2*p, l, mid);
build(2*p + 1, mid + 1, r);
tree[p] = min(tree[2*p], tree[2*p + 1]);
}
void update(int p, int l, int r, int x, int y) {
// turn arr[x] into y
if (l == r) { tree[p] = y; return; }
int mid = (l + r)/2;
if (x <= mid) {
update(2*p, l, mid, x, y);
}
else {
update(2*p + 1, mid + 1, r, x, y);
}
tree[p] = min(tree[2*p], tree[2*p + 1]);
}
int query(int p, int l, int r, int x, int y) {
if (l == r) { return tree[p]; }
int mid = (l + r)/2;
int ans = INT_MAX;
if (x <= mid) {
ans = min(ans, query(2 * p, l, mid, x, y));
}
if (y >= mid + 1) {
ans = min(ans, query(2 * p + 1, mid + 1, r, x, y));
}
return ans;
}

## Lazy Segment Tree

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template<class T, int MX> struct LazySeg {
T treesum[MX], treemin[MX], lazy[MX];
void build(int n, int tl, int tr) {
if (tl == tr) {
treesum[n] = arr[tl];
treemin[n] = arr[tl];
return;
}
int mid = (tl + tr)/2;
build(n << 1, tl, mid);
build((n << 1) + 1, mid + 1, tr);
// change me
treesum[n] = (treesum[n << 1] + treesum[(n << 1) + 1]);
treemin[n] = min(treemin[2*n], treemin[2*n + 1]);
}
void pushdown(int n, int tl, int tr) {
if (lazy[n] > 0) {
// change me
treesum[n] += (tr - tl + 1) * lazy[n];
treemin[n] += lazy[n];
if (tl != tr) {
lazy[2*n] += lazy[n];
lazy[2*n+1] += lazy[n];
}
lazy[n] = 0;
}
}
void upd(int n, int tl, int tr, int st, int ed, T v) {
pushdown(n, tl, tr);
if (tl > tr || tl > ed || tr < st) return;
if (tl >= st && tr <= ed) {
// change me: only for min/max
treesum[n] += (tr - tl + 1) * v;
treemin[n] += v;
// only for min
if (tl != tr) {
lazy[2*n] += v;
lazy[2*n + 1] += v;
}
return;
}
else {
int mid = (tl + tr)/2;
upd(2*n, tl, mid, st, ed, v);
upd(2*n+1, mid+1, tr, st, ed, v);
// change me
treesum[n] = (treesum[2*n] + treesum[2*n+1]);
treemin[n] = min(treemin[2*n+1], treemin[2*n]);
}
}
T querySum(int n, int tl, int tr, int st, int ed) {
// change me
if (tl > tr || ed < tl || st > tr) return 0;
pushdown(n, tl, tr);
// segment completely inside query
if (st <= tl && tr <= ed) {
return treesum[n];
}
else {
int mid = (tl + tr)/2;
T a = querySum(2*n, tl, mid, st, ed);
T b = querySum(2*n + 1, mid + 1, tr, st, ed);
// change me
return a+b;
}
}
T queryMin(int n, int tl, int tr, int st, int ed) {
// change me
if (tl > tr || ed < tl || st > tr) return MAX;
pushdown(n, tl, tr);
// segment completely inside query
if (st <= tl && tr <= ed) {
return treemin[n];
}
int mid = (tl + tr)/2;
T a = queryMin(2*n, tl, mid, st, ed);
T b = queryMin(2*n + 1, mid + 1, tr, st, ed);
// change me
return min(a, b);
}
};
const int SEGSZ = 4*MXV;
LazySeg<long long, SEGSZ> seg;

## Fast Segment Tree

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template<class T> struct Seg { // comb(ID,b) = b
const T ID = 1e18; T comb(T a, T b) { return min(a,b); }
int n; vector<T> seg;
void init(int _n) { n = _n; seg.assign(2*n,ID); }
void pull(int p) { seg[p] = comb(seg[2*p],seg[2*p+1]); }
void upd(int p, T val) { // set val at position p
seg[p += n] = val; for (p /= 2; p; p /= 2) pull(p); }
T query(int l, int r) { // min on interval [l, r]
T ra = ID, rb = ID;
for (l += n, r += n+1; l < r; l /= 2, r /= 2) {
if (l&1) ra = comb(ra,seg[l++]);
if (r&1) rb = comb(seg[--r],rb);
}
return comb(ra,rb);
}
};
Seg<int> st;
st.upd(i, x);
int a = st.query(x, y);

## LCA

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int T = 1;
vector<int> st, ed, depth;
vector<vector<int>> up(2e5 + 1, vector<int> (20));
//euler tour to flattern the tree
void dfs(int n, int p) {
st[n] = T;
depth[n] = depth[p] + 1;
//jump table for node n
up[n][0] = p;
for (int i = 1; i < 20; i++) {
up[n][i] = up[up[n][i - 1]][i - 1];
}
T++;
for (auto c : adj[n]) {
if (c != p) {
dfs(c, n);
}
}
ed[n] = T - 1;
}
int lca(int a, int b) {
if (depth[a] < depth[b]) swap(a, b);
int diff = depth[a] - depth[b];
for (int i = 0; i < 20; i++) {
if ((diff >> i) & 1) {
a = up[a][i];
}
}
if (a == b) return a;
for (int i = 19; i >= 0; i--) {
int ap = up[a][i];
int bp = up[b][i];
if (ap != bp) {
a = ap; b = bp;
}
}
return up[a][0];
}
//dfs to get depth and initialize jump table
depth[0] = -1;
dfs(1, 0);
//lca
int l = lca(a, b);

## HLD

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//dfs1 to get depth, parent and sz of each node
void dfs(int n, int p) {
// make trees
depth[n] = depth[p] + 1;
sz[n] = 1;
for (auto i : adj[n]) {
if (i != p) {
parent[i] = n;
dfs(i, n);
sz[n] += sz[i];
}
}
}
int T = 1;
//dfs2 to flatten and get index T of node, head of node
void dfs2(int n, int h, int p) {
st[n] = T++;
//do some calculation.e.g seg.upd(st[n], values[n]);
head[n] = h;
int hv = -1;
for (auto i : adj[n]) {
if (i != p) {
if (hv == -1 || sz[i] > sz[hv]) {
hv = i;
}
}
}
// leaf node, then return
if (hv == -1) return;
// heavy edge = inherit
dfs2(hv, h, n);
// other eddge, start a new segment
for (auto i : adj[n]) {
if (i != p && i != hv) {
// light edge = new segment
dfs2(i, i, n);
}
}
}
//do something for 2 nodes ...
int hldquery(int a, int b) {
int ans = 0;
while (head[a] != head[b]) {
if (depth[head[a]] < depth[head[b]]) swap(a, b);
int q1 = seg.query(st[head[a]], st[a]);
a = parent[head[a]];
ans = max(ans, q1);
}
if (depth[a] < depth[b]) swap(a, b);
int q2 = seg.query(st[b], st[a]);
ans = max(ans, q2);
return ans;
}

### Union Find

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int parent[N]; // initialize parent[i] = i
#ifndef PATH_COMPRESSION
int findRoot (int a) {
if (parent[a] == a) {
return a;
}
return findRoot(parent[a]);
}
#else
//Path Compression - O(log N) per query:
int findRoot (int a) {
if (parent[a] == a) {
return a;
}
return parent[a] = findRoot(parent[a]);
}
#endif
bool isConnected (int a, int b) {
return findRoot(a) == findRoot(b);
}
#ifdef KEEP_SIZE_SMALL
int size[N]; // initialize size[i] = 1
void join (int a, int b) {
a = findRoot(a);
b = findRoot(b);
if (a == b) {
return;
}
if (size[a] < size[b]) {
parent[a] = b;
size[b] += size[a];
}
else {
parent[b] = a;
size[a] += size[b];
}
}
#elseif KEEP_DEPTH_SMALL
int depth[N]; // initialize depth[i] = 1
void join (int a, int b) {
a = findRoot(a);
b = findRoot(b);
if (a == b) {
return;
}
if (depth[a] < depth[b]) {
parent[a] = b;
}
else {
parent[b] = a;
depth[a] = max(depth[a], depth[b] + 1);
}
}
#else
void join (int a, int b) {
parent[ findRoot(a) ] = findRoot(b);
}
#endif

## KMP

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typedef vector<int> VI;
void buildTable(string& w, VI& t)
{
t = VI(w.length());
int i = 2, j = 0;
t[0] = -1; t[1] = 0;
while(i < w.length())
{
if(w[i-1] == w[j]) {
t[i] = j+1;
i++;
j++;
}
else if(j > 0) {
j = t[j];
}
else {
t[i] = 0;
i++;
}
}
}
int KMP(string& s, string& w)
{
int m = 0, i = 0;
VI t;
buildTable(w, t);
while(m+i < s.length())
{
if(w[i] == s[m+i])
{
i++;
if(i == w.length()) return m;
}
else
{
m += i-t[i];
if(i > 0) i = t[i];
}
}
return s.length();
}

## Math

### Fast Power

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int mul(int x,int n,int p)
{
int ans=0;
while(n) {
if (n&1)ans=ans+x%p;
x=x+x%p;
n>>=1;
}
return ans%p;
}
int fpow(int x,int n,int p)
{
int ans=1;
while (n) {
if (n&1) ans=mul(ans,x,p);
x=mul(x,x,p);
n>>=1;
}
return ans%p;
}

### GCD

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int gcd(int x,int y){
return y==0?x:gcd(y,x%y);
}
int exgcd(int a,int b,int &x,int &y)
{
if (b==0) {
x=1;y=0;
return a;
}
int g=exgcd(b,a%b,x,y);
int t=x;x=y;y=t-a/b*x;
return g;
}

## Data Structure

### Unordered Map Custom Comparitor

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struct hashPi {
size_t operator()(const pair<int, int>& p) const { return (p.first*100001) + p.second; }
};
unordered_map<pair<int, int>, int, hashPi> findlca;

### Priority queue

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priority_queue<int> q; //biggest on top
priority_queue<int,vector<int>,less<int> > q; //biggest on top
priority_queue<int,vectot<int>,greater<int> > q; //smallest on top
struct cmp{
bool operator () (pair<int,int> &a, pair<int,int> &b){
if (a.first!= b.first) return a.first<b.first;
else return a.second<b.second;
}
};
priority_queue<rec,vector<rec>,cmp>q;

## Sorting

### Quick Sort

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void quick_sort(int q[], int l, int r)
{
if (l >= r) return;
int i = l - 1, j = r + 1, x = q[l + r >> 1];
while (i < j)
{
do i ++ ; while (q[i] < x);
do j -- ; while (q[j] > x);
if (i < j) swap(q[i], q[j]);
}
quick_sort(q, l, j), quick_sort(q, j + 1, r);
}

### Merge Sort

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void merge_sort(int q[], int l, int r)
{
if (l >= r) return;
int mid = l + r >> 1;
merge_sort(q, l, mid);
merge_sort(q, mid + 1, r);
int k = 0, i = l, j = mid + 1;
while (i <= mid && j <= r)
if (q[i] <= q[j]) tmp[k ++ ] = q[i ++ ];
else tmp[k ++ ] = q[j ++ ];
while (i <= mid) tmp[k ++ ] = q[i ++ ];
while (j <= r) tmp[k ++ ] = q[j ++ ];
for (i = l, j = 0; i <= r; i ++, j ++ ) q[i] = tmp[j];
}

## Binary Search

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bool check(int x) {/* ... */}
/*FFFTT*/
int binary_search() {
int l = 0;
int r = 1e9 + 1;
while (l != r) {
int mid = (l + r)/2;
if (check(mid)) {
r = mid;
}
else {
l = mid + 1;
}
}
return l;
}
/*TTFF*/
int binary_search(vector<ll> x) {
int l = 0;
int r = 1e9 + 1;
while (l != r) {
int mid = (l + r + 1)/2;
if (check(x, mid)) {
l = mid;
}
else {
r = mid - 1;
}
}
return l;
}

## Math

### Binary Search for Equation Solving

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bool check(double x) {/* ... */}
double bsearch_3(double l, double r)
{
const double eps = 1e-6;
while (r - l > eps)
{
double mid = (l + r) / 2;
if (check(mid)) r = mid;
else l = mid;
}
return l;
}

### Big Integer

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// C = A + B, A >= 0, B >= 0
vector<int> add(vector<int> &A, vector<int> &B)
{
if (A.size() < B.size()) return add(B, A);
vector<int> C;
int t = 0;
for (int i = 0; i < A.size(); i ++ )
{
t += A[i];
if (i < B.size()) t += B[i];
C.push_back(t % 10);
t /= 10;
}
if (t) C.push_back(t);
return C;
}
// C = A - B, A >= B, A >= 0, B >= 0
vector<int> sub(vector<int> &A, vector<int> &B)
{
vector<int> C;
for (int i = 0, t = 0; i < A.size(); i ++ )
{
t = A[i] - t;
if (i < B.size()) t -= B[i];
C.push_back((t + 10) % 10);
if (t < 0) t = 1;
else t = 0;
}
while (C.size() > 1 && C.back() == 0) C.pop_back();
return C;
}
// C = A * b, A >= 0, b > 0
vector<int> mul1(vector<int> &A, int b)
{
vector<int> C;
int t = 0;
for (int i = 0; i < A.size() || t; i ++ )
{
if (i < A.size()) t += A[i] * b;
C.push_back(t % 10);
t /= 10;
}
while (C.size() > 1 && C.back() == 0) C.pop_back();
return C;
}
// C = A * B
vector<int> mul2(vector<int> &A, vector<int> &B) {
vector<int> C(A.size() + B.size(), 0);
for (int i = 0; i < A.size(); i++)
for (int j = 0; j < B.size(); j++)
C[i + j] += A[i] * B[j];
int t = 0;
for (int i = 0; i < C.size(); i++) {
t += C[i];
C[i] = t % 10;
t /= 10;
}
while (C.size() > 1 && C.back() == 0) C.pop_back();
return C;
}
// A / b = C ... r, A >= 0, b > 0
vector<int> div(vector<int> &A, int b, int &r)
{
vector<int> C;
r = 0;
for (int i = A.size() - 1; i >= 0; i -- )
{
r = r * 10 + A[i];
C.push_back(r / b);
r %= b;
}
reverse(C.begin(), C.end());
while (C.size() > 1 && C.back() == 0) C.pop_back();
return C;
}

## Prefix Sum

### 1D Prefix Sum

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for (int i=1; i<=N; i++) {
pre[i] = val[i] + pre[i-1];
}
//sum from a to b inclusive
pre[b] - pre[a-1]

### Reverse 1D Prefix Sum

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val[a] += x;
val[b+1] -= x;
for (int i=1; i< N; i++) {
val[i] += val[i-1];
}

### 2D Prefix Sum

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for (int i=1; i<=N; i++) {
for (int j=1; j<=N; j++) {
pre[i][j] = val[i][j] + pre[i-1][j] + pre[i][j-1] - pre[i-1][j-1];
}
}
//sum of the range from (x1, y1) to (x2, y2) is:
pre[x2][y2] - pre[x1-1][y2] - pre[x2][y1-1] + pre[x1-1][y1-1]

### Reverse 2D Prefix Sum

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// add v to the range from (x1, y1) to (x2, y2), inclusive
val[x1][y1] += v;
val[x1][y2+1] -= v;
val[x2+1][y1] -= v;
val[x2+1][y2+1] += v;
// get value at (i,j)
for (int i=1; i<=N; i++) {
for (int j=1; j<=N; j++) {
val[i][j] += + pre[i-1][j] + pre[i][j-1] - pre[i-1][j-1];
}
}

## Bit Operation

### The k-th bit of an integer n

1

n >> k & 1

### The first LSB 1 and itâ€™s trailing 0s

1

lowbit(n) = n & -n

## 2 Pointers

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for (int i = 0, j = 0; i < n; i ++ )
{
while (j < i && check(i, j)) j ++ ;
// your code
}

## Coordinate Compression

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vector<int> alls; // store all input values
sort(alls.begin(), alls.end());
alls.erase( unique( alls.begin(), alls.end() ), alls.end()); // remove duplicates
// use binary search to find coordinate of value x
// can also be done using upper_bound(alls.begin(), all.end(), x)
int find(int x)
{
return upper_bound(alls.begin(), alls.end(), x) - alls.begin();
}
//add is array of pairs, first is the uncompressed index, second is the operation to add
//a is the compressed array, each index is compressed from original index
map<int, int> a;
for(auto item : add)
{
int compressed_index = find(item.first);
a[compressed_index] += item.second;
}

## Segments Merger

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// combine the over lapping ranges
void merge(vector<pair<int,int>> &segs)
{
vector<pair<int,int>> res;
sort(segs.begin(), segs.end());
int st = -2e9, ed = -2e9;
//[st, ed]: the last range
for (auto seg : segs)
//case 1: can't merge new range
if (ed < seg.first)
{
if (st != -2e9)
res.push_back({st, ed});
st = seg.first, ed = seg.second;
}
else
//case 2: the new range's end is smaller than last range's end
//case 3: the new range's end is bigger than last range's end
//in both cases, extend the new end for the last range
ed = max(ed, seg.second);
if (st != -2e9)
res.push_back({st, ed});
segs = res;
}

## Monotonous Stack

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//Open a stack and decrease monotonically.
//For the top element on the stack, what you can see is the number of elements behind
stack<int>s;
for (int i = 0; i < n; i++) {
int temp;
cin >> temp;
//When the input is bigger, remove the smaller from the stack
while (!s.empty() && temp >= s.top())s.pop();
s.push(temp);
}

## Sliding Window

Use monotonous queue to find minimal in a sliding window.

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deque<int> dq;
//find the minimal number in the sliding window
for(int i = 0; i < n; i ++)
{
//keep the front of the deque always inside sliding window of size k
if( !dq.empty() && k < i - dq.front() + 1)
dq.pop_front();
//maintain monotonous queue
//if the queue is not empty and the tail element is not smaller than a[i]
//pop all the bigger elements until it is smaller than a[i]
while( !dq.empty() && a[dq.back()] >= a[i])
dq.pop_back();
dq.push_back(i);
if(i + 1 >= k)
cout << a[dq.front()] << " ";
}

## KMP

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//ne: based on pattern string p
//ne[i] = j, j the maximal length of prefix: ne[1:j] == ne[i-j+1:i]
for (int i = 2, j = 0; i <= m; i ++ )
{
while (j && p[i] != p[j + 1])
j = ne[j];
if (p[i] == p[j + 1])
j ++ ;
ne[i] = j;
}
for (int i = 1, j = 0; i <= n; i ++ )
{
while (j && s[i] != p[j + 1])
j = ne[j];
if (s[i] == p[j + 1])
j++;
if (j == m)
{
j = ne[j];
}
}

## Trie

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int son[N][26], cnt[N], idx;
// 0 is the root, and empty node
// son[p][u] stores p's subtree root node
// cnt[p] stores number of words ending with p node
void insert(string str)
{
int p = 0;
for (int i = 0; i < str.size(); i ++ )
{
int u = str[i] - 'a';
if (!son[p][u])
son[p][u] = ++idx;
p = son[p][u];
}
cnt[p]++ ;
}
int query(string str)
{
int p = 0;
for (int i = 0; str.size(); i ++ )
{
int u = str[i] - 'a';
if (!son[p][u])
return 0;
p = son[p][u];
}
return cnt[p];
}

## State machine

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def bessie_substrings(x):
max_bessie_counts = []
for i in range(len(x)):
count = 0
j = i
while j < len(x) and count < 6:
if x[j] == 'bessie'[count]:
count += 1
j += 1
max_bessie_counts.append(count)
return sum(max_bessie_counts)
def total_bessie_counts(x):
total_count = 0
for i in range(len(x)):
total_count += bessie_substrings(x[i:])
return total_count