diff options
Diffstat (limited to 'content')
32 files changed, 287 insertions, 205 deletions
diff --git a/content/datastructures/datastructures.tex b/content/datastructures/datastructures.tex index 40132a9..a0eea64 100644 --- a/content/datastructures/datastructures.tex +++ b/content/datastructures/datastructures.tex @@ -10,7 +10,7 @@ \subsubsection{Lazy Propagation} Assignment modifications, sum queries \\ - \method{lower\_bound}{erster Index in $[l, r)$ $\geq$ x (erfordert max-combine)}{\log(n)} + \method{lower\_bound}{erster Index in $[l, r)$ $\geq x$ (erfordert max-combine)}{\log(n)} \sourcecode{datastructures/lazyPropagation.cpp} \end{algorithm} @@ -20,6 +20,8 @@ \method{kth}{sort $[l, r)[k]$}{\log(\Sigma)} \method{countSmaller}{Anzahl elemente in $[l, r)$ kleiner als $k$}{\log(\Sigma)} \end{methods} + $\Sigma$ ist die Gr\"o\ss e des Eingabebereichs, d.h. + $\mathit{max} - \mathit{min}$. \sourcecode{datastructures/waveletTree.cpp} \end{algorithm} \columnbreak @@ -27,15 +29,15 @@ \begin{algorithm}{Fenwick Tree} \begin{methods} \method{init}{baut den Baum auf}{n\*\log(n)} - \method{prefix\_sum}{summe von $[0, i]$}{\log(n)} + \method{prefix\_sum}{summe von $[0, i)$}{\log(n)} \method{update}{addiert ein Delta zu einem Element}{\log(n)} \end{methods} \sourcecode{datastructures/fenwickTree.cpp} \begin{methods} \method{init}{baut den Baum auf}{n\*\log(n)} - \method{prefix\_sum}{summe von [$0, i]$}{\log(n)} - \method{update}{addiert ein Delta zu allen Elementen $[l, r)$. $l\leq r$!}{\log(n)} + \method{prefix\_sum}{summe von $[0, i)$}{\log(n)} + \method{update}{addiert ein Delta zu allen Elementen $[l, r)$}{\log(n)} \end{methods} \sourcecode{datastructures/fenwickTree2.cpp} \end{algorithm} @@ -56,7 +58,7 @@ \begin{algorithm}{Range Minimum Query} \begin{methods} \method{init}{baut Struktur auf}{n\*\log(n)} - \method{queryIdempotent}{Index des Minimums in $[l, r)$. $l<r$!}{1} + \method{queryIdempotent}{Index des Minimums in $[l, r)$}{1} \end{methods} \begin{itemize} \item \code{better}-Funktion muss idempotent sein! @@ -64,6 +66,14 @@ \sourcecode{datastructures/sparseTable.cpp} \end{algorithm} +\begin{algorithm}[optional]{Range Aggregate Query} + \begin{methods} + \method{init}{baut Struktur auf}{n\*\log(n)} + \method{query}{Aggregat über $[l,r)$}{1} + \end{methods} + \sourcecode{datastructures/sparseTableDisjoint.cpp} +\end{algorithm} + \begin{algorithm}{STL-Bitset} \sourcecode{datastructures/bitset.cpp} \end{algorithm} @@ -80,30 +90,43 @@ \end{methods} \sourcecode{datastructures/LCT.cpp} \end{algorithm} -\clearpage +\columnbreak -\begin{algorithm}{Lichao} - \sourcecode{datastructures/lichao.cpp} +\begin{algorithm}{Lower Envelope (Convex Hull Optimization)} + Um aus einem Lower Envelope einen Upper Envelope zu machen (oder + umgekehrt), einfach beim Einfügen der Geraden $m$ und $b$ negieren. + \subsubsection{Monotonic} + \begin{methods} + \method{add}{add line $mx + b$, $m$ is decreasing}{1} + \method{query}{minimum value at $x$, $x$ is increasing}{1} + \end{methods} + \sourcecode{datastructures/monotonicConvexHull.cpp} + \subsubsection{Dynamic} + \begin{methods} + \method{add}{add line $mx + b$}{\log(n)} + \method{query}{minimum value at $x$}{\log(n)} + \end{methods} + \sourcecode{datastructures/dynamicConvexHull.cpp} + \subsubsection{Li Chao Tree} + Every pair of functions has at most one intersection. + + \begin{methods} + \method{insert}{add function}{\log(|xs|)} + \method{query}{minimum value at $x$, $x \in xs$}{\log(|xs|)} + \end{methods} + \sourcecode{datastructures/lichao.cpp} \end{algorithm} \begin{algorithm}{Policy Based Data Structures} - \textbf{Wichtig:} Verwende \code{p.swap(p2)} anstatt \code{swap(p, p2)}! - \sourcecode{datastructures/stlPriorityQueue.cpp} - \columnbreak \sourcecode{datastructures/pbds.cpp} \end{algorithm} -\begin{algorithm}{Lower/Upper Envelope (Convex Hull Optimization)} - Um aus einem lower envelope einen upper envelope zu machen (oder umgekehrt), einfach beim Einfügen der Geraden $m$ und $b$ negieren. - \sourcecode{datastructures/monotonicConvexHull.cpp} - \sourcecode{datastructures/dynamicConvexHull.cpp} -\end{algorithm} - \begin{algorithm}{Union-Find} \begin{methods} \method{init}{legt $n$ einzelne Unions an}{n} \method{findSet}{findet den Repräsentanten}{\log(n)} \method{unionSets}{vereint 2 Mengen}{\log(n)} + \method{size}{zählt Elemente in Menge, die $a$ enthält}{\log(n)} \method{m\*findSet + n\*unionSets}{Folge von Befehlen}{n+m\*\alpha(n)} \end{methods} \sourcecode{datastructures/unionFind.cpp} diff --git a/content/datastructures/dynamicConvexHull.cpp b/content/datastructures/dynamicConvexHull.cpp index d669847..2bd67a6 100644 --- a/content/datastructures/dynamicConvexHull.cpp +++ b/content/datastructures/dynamicConvexHull.cpp @@ -19,13 +19,11 @@ struct HullDynamic : multiset<Line, less<>> { auto x = insert({m, b, 0}); while (isect(x, next(x))) erase(next(x)); if (x != begin()) { - x--; - if (isect(x, next(x))) { - erase(next(x)); - isect(x, next(x)); - }} + --x; + while (isect(x, next(x))) erase(next(x)); + } while (x != begin() && prev(x)->p >= x->p) { - x--; + --x; isect(x, erase(next(x))); }} diff --git a/content/datastructures/fenwickTree.cpp b/content/datastructures/fenwickTree.cpp index eb5cd73..7013613 100644 --- a/content/datastructures/fenwickTree.cpp +++ b/content/datastructures/fenwickTree.cpp @@ -1,7 +1,7 @@ vector<ll> tree; void update(int i, ll val) { - for (i++; i < sz(tree); i += i & -i) tree[i] += val; + for (i++; i < ssize(tree); i += i & -i) tree[i] += val; } void init(int n) { @@ -10,6 +10,6 @@ void init(int n) { ll prefix_sum(int i) { ll sum = 0; - for (i++; i > 0; i -= i & -i) sum += tree[i]; + for (; i > 0; i &= i-1) sum += tree[i]; return sum; } diff --git a/content/datastructures/fenwickTree2.cpp b/content/datastructures/fenwickTree2.cpp index 9384e3c..7fcdbb9 100644 --- a/content/datastructures/fenwickTree2.cpp +++ b/content/datastructures/fenwickTree2.cpp @@ -1,21 +1,21 @@ vector<ll> add, mul; void update(int l, int r, ll val) { - for (int tl = l + 1; tl < sz(add); tl += tl & -tl) + for (int tl = l + 1; tl < ssize(add); tl += tl & -tl) add[tl] += val, mul[tl] -= val * l; - for (int tr = r + 1; tr < sz(add); tr += tr & -tr) + for (int tr = r + 1; tr < ssize(add); tr += tr & -tr) add[tr] -= val, mul[tr] += val * r; } -void init(vector<ll>& v) { - mul.assign(sz(v) + 1, 0); - add.assign(sz(v) + 1, 0); - for(int i = 0; i < sz(v); i++) update(i, i + 1, v[i]); +void init(vector<ll> &v) { + mul.assign(size(v) + 1, 0); + add.assign(size(v) + 1, 0); + for(int i = 0; i < ssize(v); i++) update(i, i + 1, v[i]); } ll prefix_sum(int i) { - ll res = 0; i++; - for (int ti = i; ti > 0; ti -= ti & -ti) + ll res = 0; + for (int ti = i; ti > 0; ti &= ti-1) res += add[ti] * i + mul[ti]; return res; } diff --git a/content/datastructures/lazyPropagation.cpp b/content/datastructures/lazyPropagation.cpp index 441590e..82176cb 100644 --- a/content/datastructures/lazyPropagation.cpp +++ b/content/datastructures/lazyPropagation.cpp @@ -1,23 +1,22 @@ struct SegTree { using T = ll; using U = ll; - int n; static constexpr T E = 0; // Neutral element for combine - static constexpr U UF = inf; // Unused value by updates - vector<T> tree; + static constexpr U UF = 1e18; // Unused value by updates + int n; + vector<T> tree; vector<U> lazy; int h; - vector<U> lazy; - vector<int> k; // size of segments (optional) + vector<ll> k; // size of segments (optional) - SegTree(const vector<T>& a) : n(sz(a) + 1), tree(2 * n, E), + SegTree(const vector<T>& a) : n(ssize(a) + 1), tree(2 * n, E), //SegTree(int size, T def = E) : n(size + 1), tree(2 * n, def), - h(__lg(2 * n)), lazy(n, UF), k(2 * n, 1) { - copy(all(a), tree.begin() + n); + lazy(n, UF), h(__lg(2 * n)), k(2 * n, 1) { + ranges::copy(a, tree.begin() + n); for (int i = n - 1; i > 0; i--) { k[i] = 2 * k[2 * i]; tree[i] = comb(tree[2 * i], tree[2 * i + 1]); }} - T comb(T a, T b) {return a + b;} // Modify this + E + T comb(T a, T b) { return a + b; } // Modify this + E void apply(int i, U val) { // And this + UF tree[i] = val * k[i]; @@ -44,17 +43,17 @@ struct SegTree { void update(int l, int r, U val) { l += n, r += n; int l0 = l, r0 = r; - push(l0), push(r0 - 1); + push(l0), push(r0); for (; l < r; l /= 2, r /= 2) { if (l&1) apply(l++, val); if (r&1) apply(--r, val); } - build(l0), build(r0 - 1); + build(l0), build(r0); } T query(int l, int r) { l += n, r += n; - push(l), push(r - 1); + push(l), push(r); T resL = E, resR = E; for (; l < r; l /= 2, r /= 2) { if (l&1) resL = comb(resL, tree[l++]); diff --git a/content/datastructures/monotonicConvexHull.cpp b/content/datastructures/monotonicConvexHull.cpp index 44bff83..e7b9b3e 100644 --- a/content/datastructures/monotonicConvexHull.cpp +++ b/content/datastructures/monotonicConvexHull.cpp @@ -1,27 +1,26 @@ -// Lower Envelope mit MONOTONEN Inserts und Queries. Jede neue -// Gerade hat kleinere Steigung als alle vorherigen. -struct Line { - ll m, b; - ll operator()(ll x) {return m*x+b;} -}; +struct Envelope { + struct Line { + ll m, b; + ll operator()(ll x) { return m*x+b; } + }; -vector<Line> ls; -int ptr = 0; + vector<Line> ls; + int ptr = 0; -bool bad(Line l1, Line l2, Line l3) { - return (l3.b-l1.b)*(l1.m-l2.m) < (l2.b-l1.b)*(l1.m-l3.m); -} + static bool bad(Line l1, Line l2, Line l3) { + return (l3.b-l1.b)*(l1.m-l2.m) < (l2.b-l1.b)*(l1.m-l3.m); + } -void add(ll m, ll b) { // Laufzeit O(1) amortisiert - while (sz(ls) > 1 && bad(ls.end()[-2], ls.end()[-1], {m, b})) { - ls.pop_back(); + void add(ll m, ll b) { + while (sz(ls) > 1 && bad(ls.end()[-2], ls.back(), {m, b})) { + ls.pop_back(); + } + ls.push_back({m, b}); + ptr = min(ptr, sz(ls) - 1); } - ls.push_back({m, b}); - ptr = min(ptr, sz(ls) - 1); -} -ll query(ll x) { // Laufzeit: O(1) amortisiert - ptr = min(ptr, sz(ls) - 1); - while (ptr < sz(ls)-1 && ls[ptr + 1](x) < ls[ptr](x)) ptr++; - return ls[ptr](x); -}
\ No newline at end of file + ll query(ll x) { + while (ptr < sz(ls)-1 && ls[ptr + 1](x) < ls[ptr](x)) ptr++; + return ls[ptr](x); + } +}; diff --git a/content/datastructures/pbds.cpp b/content/datastructures/pbds.cpp index f0889a2..d00b635 100644 --- a/content/datastructures/pbds.cpp +++ b/content/datastructures/pbds.cpp @@ -1,13 +1,21 @@ +#include <ext/pb_ds/priority_queue.hpp> +template<typename T> +using pQueue = __gnu_pbds::priority_queue<T>; //<T, greater<T>> +auto it = pq.push(5); // O(1) +pq.modify(it, 6); // O(log n) +pq.erase(it); // O(log n) +pq.join(pq2); // O(1) +pq.swap(pq2); // O(1) + #include <ext/pb_ds/assoc_container.hpp> using namespace __gnu_pbds; template<typename T> using Tree = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>; -// T.order_of_key(x): number of elements strictly less than x -// *T.find_by_order(k): k-th element +T.order_of_key(x); // number of elements strictly less than x +auto it = T.find_by_order(k); // k-th element -template<typename T> -struct chash { +template<typename T> struct chash { static const uint64_t C = ll(2e18 * acos(-1)) | 199; // random odd size_t operator()(T o) const { return __builtin_bswap64(hash<T>()(o) * C); diff --git a/content/datastructures/segmentTree.cpp b/content/datastructures/segmentTree.cpp index 6b69d0b..1fbf886 100644 --- a/content/datastructures/segmentTree.cpp +++ b/content/datastructures/segmentTree.cpp @@ -4,14 +4,15 @@ struct SegTree { vector<T> tree; static constexpr T E = 0; // Neutral element for combine - SegTree(vector<T>& a) : n(sz(a)), tree(2 * n) { - //SegTree(int size, T val = E) : n(size), tree(2 * n, val) { - copy(all(a), tree.begin() + n); + SegTree(vector<T>& a) : n(ssize(a)), tree(2 * n, E) { + ranges::copy(a, tree.begin() + n); + //SegTree(int size, T val = E) : n(size), tree(2 * n, E) { + // fill(tree.begin() + n, tree.end(), val); for (int i = n - 1; i > 0; i--) { // remove for range update tree[i] = comb(tree[2 * i], tree[2 * i + 1]); }} - T comb(T a, T b) {return a + b;} // modify this + neutral + T comb(T a, T b) { return a + b; } // modify this + neutral void update(int i, T val) { tree[i += n] = val; // apply update code diff --git a/content/datastructures/sparseTable.cpp b/content/datastructures/sparseTable.cpp index b3f946e..5e84236 100644 --- a/content/datastructures/sparseTable.cpp +++ b/content/datastructures/sparseTable.cpp @@ -6,9 +6,9 @@ struct SparseTable { return a[lidx] <= a[ridx] ? lidx : ridx; } - void init(vector<ll>* vec) { - int n = sz(*vec); - a = vec->data(); + void init(vector<ll> &vec) { + int n = sz(vec); + a = vec.data(); st.assign(__lg(n) + 1, vector<int>(n)); iota(all(st[0]), 0); for (int j = 0; (2 << j) <= n; j++) { diff --git a/content/datastructures/sparseTableDisjoint.cpp b/content/datastructures/sparseTableDisjoint.cpp index 55165d4..39c7caa 100644 --- a/content/datastructures/sparseTableDisjoint.cpp +++ b/content/datastructures/sparseTableDisjoint.cpp @@ -7,16 +7,16 @@ struct DisjointST { return x + y; } - void init(vector<ll>* vec) { - int n = sz(*vec); - a = vec->data(); + void init(vector<ll> &vec) { + int n = sz(vec); + a = vec.data(); dst.assign(__lg(n) + 1, vector<ll>(n + 1, neutral)); for (int h = 0, l = 1; l <= n; h++, l *= 2) { for (int c = l; c < n + l; c += 2 * l) { for (int i = c; i < min(n, c + l); i++) - dst[h][i + 1] = combine(dst[h][i], vec->at(i)); + dst[h][i + 1] = combine(dst[h][i], vec[i]); for (int i = min(n, c); i > c - l; i--) - dst[h][i - 1] = combine(vec->at(i - 1), dst[h][i]); + dst[h][i - 1] = combine(vec[i - 1], dst[h][i]); }}} ll query(int l, int r) { diff --git a/content/datastructures/stlHashMap.cpp b/content/datastructures/stlHashMap.cpp deleted file mode 100644 index b107dde..0000000 --- a/content/datastructures/stlHashMap.cpp +++ /dev/null @@ -1,17 +0,0 @@ -#include <ext/pb_ds/assoc_container.hpp> -using namespace __gnu_pbds; - -template<typename T> -struct betterHash { - size_t operator()(T o) const { - size_t h = hash<T>()(o) ^ 42394245; //random value - h = ((h >> 16) ^ h) * 0x45d9f3b; - h = ((h >> 16) ^ h) * 0x45d9f3b; - h = ((h >> 16) ^ h); - return h; -}}; - -template<typename K, typename V, typename H = betterHash<K>> -using hashMap = gp_hash_table<K, V, H>; -template<typename K, typename H = betterHash<K>> -using hashSet = gp_hash_table<K, null_type, H>; diff --git a/content/datastructures/stlPriorityQueue.cpp b/content/datastructures/stlPriorityQueue.cpp deleted file mode 100644 index 32b2455..0000000 --- a/content/datastructures/stlPriorityQueue.cpp +++ /dev/null @@ -1,8 +0,0 @@ -#include <ext/pb_ds/priority_queue.hpp> -template<typename T> -using pQueue = __gnu_pbds::priority_queue<T>; //<T, greater<T>> - -auto it = pq.push(5); -pq.modify(it, 6); -pq.join(pq2); -// push, join are O(1), pop, modify, erase O(log n) amortized diff --git a/content/datastructures/stlTree.cpp b/content/datastructures/stlTree.cpp deleted file mode 100644 index fbb68b9..0000000 --- a/content/datastructures/stlTree.cpp +++ /dev/null @@ -1,13 +0,0 @@ -#include <ext/pb_ds/assoc_container.hpp> -#include <ext/pb_ds/tree_policy.hpp> -using namespace std; using namespace __gnu_pbds; -template<typename T> -using Tree = tree<T, null_type, less<T>, rb_tree_tag, - tree_order_statistics_node_update>; - -int main() { - Tree<int> X; - for (int i : {1, 2, 4, 8, 16}) X.insert(i); - *X.find_by_order(3); // => 8 - X.order_of_key(10); // => 4 = min i, mit X[i] >= 10 -} diff --git a/content/datastructures/treap2.cpp b/content/datastructures/treap2.cpp index c5a60e9..e40da0d 100644 --- a/content/datastructures/treap2.cpp +++ b/content/datastructures/treap2.cpp @@ -3,7 +3,7 @@ struct Treap { struct Node { ll val; int prio, size = 1, l = -1, r = -1; - Node(ll x) : val(x), prio(rng()) {} + Node(ll x): val(x), prio(rng()) {} }; vector<Node> treap; @@ -15,14 +15,14 @@ struct Treap { void upd(int v) { if (v < 0) return; - auto& V = treap[v]; + auto &V = treap[v]; V.size = 1 + getSize(V.l) + getSize(V.r); // Update Node Code } void push(int v) { if (v < 0) return; - //auto& V = treap[v]; + //auto &V = treap[v]; //if (V.lazy) { // Lazy Propagation Code // if (V.l >= 0) treap[V.l].lazy = true; @@ -33,7 +33,7 @@ struct Treap { pair<int, int> split(int v, int k) { if (v < 0) return {-1, -1}; - auto& V = treap[v]; + auto &V = treap[v]; push(v); if (getSize(V.l) >= k) { // "V.val >= k" for lower_bound(k) auto [left, right] = split(V.l, k); @@ -66,7 +66,7 @@ struct Treap { void insert(int i, ll val) { // and i = val auto [left, right] = split(root, i); treap.emplace_back(val); - left = merge(left, sz(treap) - 1); + left = merge(left, ssize(treap) - 1); root = merge(left, right); } diff --git a/content/datastructures/waveletTree.cpp b/content/datastructures/waveletTree.cpp index 090cdb2..55167b6 100644 --- a/content/datastructures/waveletTree.cpp +++ b/content/datastructures/waveletTree.cpp @@ -1,25 +1,20 @@ struct WaveletTree { - using it = vector<ll>::iterator; - WaveletTree *ln = nullptr, *rn = nullptr; + unique_ptr<WaveletTree> ln, rn; vector<int> b = {0}; ll lo, hi; - WaveletTree(vector<ll> in) : WaveletTree(all(in)) {} - - WaveletTree(it from, it to) : // call above one - lo(*min_element(from, to)), hi(*max_element(from, to) + 1) { + WaveletTree(auto in) : lo(*ranges::min_element(in)), + hi(*ranges::max_element(in) + 1) { ll mid = (lo + hi) / 2; - auto f = [&](ll x) {return x < mid;}; - for (it c = from; c != to; c++) { - b.push_back(b.back() + f(*c)); - } + auto f = [&](ll x) { return x < mid; }; + for (ll x: in) b.push_back(b.back() + f(x)); if (lo + 1 >= hi) return; - it pivot = stable_partition(from, to, f); - ln = new WaveletTree(from, pivot); - rn = new WaveletTree(pivot, to); + auto right = ranges::stable_partition(in, f); + ln = make_unique<WaveletTree>( + ranges::subrange(begin(in), begin(right))); + rn = make_unique<WaveletTree>(right); } - // kth element in sort[l, r) all 0-indexed ll kth(int l, int r, int k) { if (k < 0 || l + k >= r) return -1; if (lo + 1 >= hi) return lo; @@ -28,13 +23,10 @@ struct WaveletTree { else return rn->kth(l-b[l], r-b[r], k-inLeft); } - // count elements in[l, r) smaller than k int countSmaller(int l, int r, ll k) { if (l >= r || k <= lo) return 0; if (hi <= k) return r - l; return ln->countSmaller(b[l], b[r], k) + rn->countSmaller(l-b[l], r-b[r], k); } - - ~WaveletTree() {delete ln; delete rn;} }; diff --git a/content/geometry/circle.cpp b/content/geometry/circle.cpp index 6789c52..155b55c 100644 --- a/content/geometry/circle.cpp +++ b/content/geometry/circle.cpp @@ -22,7 +22,7 @@ vector<pt> circleRayIntersection(pt center, double r, double c = norm(orig - center) - r * r; double discr = b * b - 4 * a * c; if (discr >= 0) { - //t in [0, 1] => schnitt mit Segment [orig, orig + dir] + //t in [0, 1] => Schnitt mit Segment [orig, orig + dir] double t1 = -(b + sqrt(discr)) / (2 * a); double t2 = -(b - sqrt(discr)) / (2 * a); if (t1 >= 0) result.push_back(t1 * dir + orig); diff --git a/content/geometry/geometry.tex b/content/geometry/geometry.tex index 92285c4..019a264 100644 --- a/content/geometry/geometry.tex +++ b/content/geometry/geometry.tex @@ -7,7 +7,7 @@ \sourcecode{geometry/closestPair.cpp} \end{algorithm} -\begin{algorithm}{Konvexehülle} +\begin{algorithm}{Konvexe Hülle} \begin{methods} \method{convexHull}{berechnet konvexe Hülle}{n\*\log(n)} \end{methods} @@ -40,7 +40,7 @@ \sourcecode{geometry/formulas3d.cpp} \optional{ - \subsection{3D-Kugeln} + \subsection{3D-Kugeln \opthint} \sourcecode{geometry/spheres.cpp} } @@ -48,15 +48,22 @@ \sourcecode{geometry/hpi.cpp} \end{algorithm} +\begin{algorithm}[optional]{Intersecting Segments} + \begin{methods} + \method{intersect}{finds ids of intersecting segments}{n\*\log(n)} + \end{methods} + \sourcecode{geometry/segmentIntersection.cpp} +\end{algorithm} + \begin{algorithm}[optional]{Delaunay Triangulierung} \begin{methods} \method{delaunay}{berechnet Triangulierung}{n\*\log(n)} \end{methods} - \textbf{WICHTIG:} Wenn alle Punkte kollinear sind gibt es keine Traingulierung! Wenn 4 Punkte auf einem Kreis liegen ist die Triangulierung nicht eindeutig. + \textbf{WICHTIG:} Wenn alle Punkte kollinear sind gibt es keine Triangulierung! Wenn 4 Punkte auf einem Kreis liegen ist die Triangulierung nicht eindeutig. \sourcecode{geometry/delaunay.cpp} \end{algorithm} \optional{ -\subsection{Geraden} +\subsection{Geraden \opthint} \sourcecode{geometry/lines.cpp} } diff --git a/content/graph/LCA_sparse.cpp b/content/graph/LCA_sparse.cpp index 221b5ed..3e87cde 100644 --- a/content/graph/LCA_sparse.cpp +++ b/content/graph/LCA_sparse.cpp @@ -10,7 +10,7 @@ struct LCA { first.assign(sz(adj), 2 * sz(adj)); idx = 0; dfs(adj, root); - st.init(&depth); + st.init(depth); } void dfs(vector<vector<int>>& adj, int v, ll d=0) { diff --git a/content/graph/binary_lifting.cpp b/content/graph/binary_lifting.cpp new file mode 100644 index 0000000..0b8c218 --- /dev/null +++ b/content/graph/binary_lifting.cpp @@ -0,0 +1,28 @@ +struct Lift { + vector<int> dep, par, jmp; + + Lift(vector<vector<int>> &adj, int root): + dep(adj.size()), par(adj.size()), jmp(adj.size(), root) { + function<void(int,int,int)> dfs = [&](int u, int p, int d) { + dep[u] = d, par[u] = p; + jmp[u] = dep[p] + dep[jmp[jmp[p]]] == 2*dep[jmp[p]] + ? jmp[jmp[p]] : p; + for (int v: adj[u]) if (v != p) dfs(v, u, d+1); + }; + dfs(root, root, 0); + } + + int depth(int v) { return dep[v]; } + int lift(int v, int d) { + while (dep[v] > d) v = dep[jmp[v]] < d ? par[v] : jmp[v]; + return v; + } + int lca(int u, int v) { + v = lift(v, dep[u]), u = lift(u, dep[v]); + while (u != v) { + auto &a = jmp[u] == jmp[v] ? par : jmp; + u = a[u], v = a[v]; + } + return u; + } +}; diff --git a/content/graph/graph.tex b/content/graph/graph.tex index 831f4e5..b38e96e 100644 --- a/content/graph/graph.tex +++ b/content/graph/graph.tex @@ -1,12 +1,5 @@ \section{Graphen} -\begin{algorithm}{Kruskal} - \begin{methods}[ll] - berechnet den Minimalen Spannbaum & \runtime{\abs{E}\cdot\log(\abs{E})} \\ - \end{methods} - \sourcecode{graph/kruskal.cpp} -\end{algorithm} - \begin{algorithm}{Minimale Spannbäume} \paragraph{Schnitteigenschaft} Für jeden Schnitt $C$ im Graphen gilt: @@ -16,6 +9,12 @@ \paragraph{Kreiseigenschaft} Für jeden Kreis $K$ im Graphen gilt: Die schwerste Kante auf dem Kreis ist nicht Teil des minimalen Spannbaums. + + \subsection{\textsc{Kruskal}} + \begin{methods}[ll] + berechnet den Minimalen Spannbaum & \runtime{\abs{E}\cdot\log(\abs{E})} \\ + \end{methods} + \sourcecode{graph/kruskal.cpp} \end{algorithm} \begin{algorithm}{Heavy-Light Decomposition} @@ -28,7 +27,7 @@ \sourcecode{graph/hld.cpp} \end{algorithm} -\begin{algorithm}{Lowest Common Ancestor} +\begin{algorithm}[optional]{Lowest Common Ancestor} \begin{methods} \method{init}{baut DFS-Baum über $g$ auf}{\abs{V}\*\log(\abs{V})} \method{getLCA}{findet LCA}{1} @@ -37,6 +36,17 @@ \sourcecode{graph/LCA_sparse.cpp} \end{algorithm} +\begin{algorithm}{Binary Lifting} + % https://codeforces.com/blog/entry/74847 + \begin{methods} + \method{Lift}{constructor}{\abs{V}} + \method{depth}{distance to root of vertex $v$}{1} + \method{lift}{vertex above $v$ at depth $d$}{\log(\abs{V})} + \method{lca}{lowest common ancestor of $u$ and $v$}{\log(\abs{V})} + \end{methods} + \sourcecode{graph/binary_lifting.cpp} +\end{algorithm} + \begin{algorithm}{Centroids} \begin{methods} \method{find\_centroid}{findet alle Centroids des Baums (maximal 2)}{\abs{V}} @@ -99,7 +109,7 @@ Sei $a_{ij}$ die Adjazenzmatrix von $G$ \textcolor{gray}{(mit $a_{ii} = 1$)}, da \sourcecode{graph/connect.cpp} \end{algorithm} -\begin{algorithm}{Erd\H{o}s-Gallai} +\begin{algorithm}{\textsc{Erd\H{o}s-Gallai}} Sei $d_1 \geq \cdots \geq d_{n}$. Es existiert genau dann ein Graph $G$ mit Degreesequence $d$ falls $\sum\limits_{i=1}^{n} d_i$ gerade ist und für $1\leq k \leq n$: $\sum\limits_{i=1}^{k} d_i \leq k\cdot(k-1)+\sum\limits_{i=k+1}^{n} \min(d_i, k)$ \begin{methods} \method{havelHakimi}{findet Graph}{(\abs{V}+\abs{E})\cdot\log(\abs{V})} @@ -170,7 +180,7 @@ Sei $a_{ij}$ die Adjazenzmatrix von $G$ \textcolor{gray}{(mit $a_{ii} = 1$)}, da \sourcecode{graph/virtualTree.cpp} \end{algorithm} -\begin{algorithm}{Maximum Cardinatlity Bipartite Matching} +\begin{algorithm}{Maximum Cardinality Bipartite Matching} \label{kuhn} \begin{methods} \method{kuhn}{berechnet Matching}{\abs{V}\*\min(ans^2, \abs{E})} @@ -197,7 +207,7 @@ Sei $a_{ij}$ die Adjazenzmatrix von $G$ \textcolor{gray}{(mit $a_{ii} = 1$)}, da \subsection{Max-Flow} \optional{ -\subsubsection{Push Relabel} +\subsubsection{Push Relabel \opthint} \begin{methods} \method{maxFlow}{gut bei sehr dicht besetzten Graphen.}{\abs{V}^2\*\sqrt{\abs{E}}} \method{addEdge}{fügt eine \textbf{gerichtete} Kante ein}{1} @@ -205,24 +215,23 @@ Sei $a_{ij}$ die Adjazenzmatrix von $G$ \textcolor{gray}{(mit $a_{ii} = 1$)}, da \sourcecode{graph/pushRelabel.cpp} } +\subsubsection{\textsc{Dinic}'s Algorithm mit Capacity Scaling} +\begin{methods} + \method{maxFlow}{doppelt so schnell wie \textsc{Ford-Fulkerson}}{\abs{V}^2\cdot\abs{E}} + \method{addEdge}{fügt eine \textbf{gerichtete} Kante ein}{1} +\end{methods} +\sourcecode{graph/dinicScaling.cpp} + \begin{algorithm}{Min-Cost-Max-Flow} \begin{methods} \method{mincostflow}{berechnet Fluss}{\abs{V}^2\cdot\abs{E}^2} \end{methods} \sourcecode{graph/minCostMaxFlow.cpp} \end{algorithm} - -\subsubsection{Dinic's Algorithm mit Capacity Scaling} -\begin{methods} - \method{maxFlow}{doppelt so schnell wie Ford Fulkerson}{\abs{V}^2\cdot\abs{E}} - \method{addEdge}{fügt eine \textbf{gerichtete} Kante ein}{1} -\end{methods} -\sourcecode{graph/dinicScaling.cpp} -\vfill\null \columnbreak \optional{ -\subsubsection{Anwendungen} +\subsubsection{Anwendungen \opthint} \begin{itemize} \item \textbf{Maximum Edge Disjoint Paths}\newline Finde die maximale Anzahl Pfade von $s$ nach $t$, die keine Kante teilen. diff --git a/content/latexHeaders/code.sty b/content/latexHeaders/code.sty index 3ebdda3..8a600c5 100644 --- a/content/latexHeaders/code.sty +++ b/content/latexHeaders/code.sty @@ -1,3 +1,6 @@ +\usepackage{ocgx2} +\usepackage{fontawesome} + % Colors, used for syntax highlighting. % To print this document, set all colors to black! \usepackage{xcolor} @@ -101,6 +104,32 @@ % \addtocounter{lstnumber}{-1}% %} +\ifthenelse{\isundefined{\srclink}}{}{ + \lst@AddToHook{Init}{% + \ifthenelse{\equal{\lst@name}{}}{}{% + \begin{minipage}[t][0pt]{\linewidth}% + \vspace{0pt}% + \hfill% + \begin{ocg}[printocg=never]{Source links}{srclinks}{1}% + \hfill\href{\srclink{\lst@name}}{\faExternalLink}% + \end{ocg}% + \end{minipage}% + }% + } +} + +\lst@AddToHook{DeInit}{% + \ifthenelse{\equal{\lst@name}{}}{}{% + \begin{minipage}[b][0pt]{\linewidth}% + \vspace{0pt}% + \hfill% + \begin{ocg}[printocg=never]{Source file names}{srcfiles}{0}% + \hfill\textcolor{gray}{\lst@name}% + \end{ocg}% + \end{minipage}% + }% +} + \newenvironment{btHighlight}[1][] {\begingroup\tikzset{bt@Highlight@par/.style={#1}}\begin{lrbox}{\@tempboxa}} {\end{lrbox}\bt@HL@box[bt@Highlight@par]{\@tempboxa}\endgroup} diff --git a/content/latexHeaders/commands.sty b/content/latexHeaders/commands.sty index edbba1b..73a7dca 100644 --- a/content/latexHeaders/commands.sty +++ b/content/latexHeaders/commands.sty @@ -7,6 +7,11 @@ \newcommand{\code}[1]{\lstinline[breaklines=true]{#1}} \let\codeSafe\lstinline +\ifoptional + \renewcommand{\columnbreak}{} + \newcommand\opthint{\textcolor{gray}{(optional)}} +\fi + \usepackage{tikz} \usetikzlibrary{angles,quotes} @@ -17,7 +22,7 @@ \ifthenelse{\equal{#1}{optional}}{% \optional{ \needspace{4\baselineskip}% - \subsection{#2\textcolor{gray}{(optional)}}% + \subsection{#2 \opthint}% #3% } }{% diff --git a/content/latexmk.opt b/content/latexmk.opt new file mode 100644 index 0000000..88d3463 --- /dev/null +++ b/content/latexmk.opt @@ -0,0 +1,2 @@ +$jobname = 'tcr-opt'; +$pre_tex_code .= '\def\OPTIONAL{}' diff --git a/content/latexmkrc b/content/latexmkrc new file mode 100644 index 0000000..c1a56ab --- /dev/null +++ b/content/latexmkrc @@ -0,0 +1,13 @@ +@default_files = qw(tcr); +$pdf_mode = 1; +$aux_dir = "."; +$out_dir = ".."; +{ + my $commit = `git rev-parse HEAD`; + chomp $commit; + $pre_tex_code .= + '\newcommand{\srclink}[1]' + .'{https://git.gloria-mundi.eu/tcr/plain/content/#1?'.$commit.'}'; +} +&alt_tex_cmds; +$jobname = 'tcr'; diff --git a/content/math/binomial0.cpp b/content/math/binomial0.cpp index 5f2ccaa..f37aea5 100644 --- a/content/math/binomial0.cpp +++ b/content/math/binomial0.cpp @@ -10,5 +10,5 @@ void precalc() { ll calc_binom(ll n, ll k) { if (n < 0 || n < k || k < 0) return 0; - return (inv[k] * inv[n-k] % mod) * fac[n] % mod; + return (fac[n] * inv[n-k] % mod) * inv[k] % mod; } diff --git a/content/math/linearRecurence.cpp b/content/math/linearRecurrence.cpp index 2501e64..2501e64 100644 --- a/content/math/linearRecurence.cpp +++ b/content/math/linearRecurrence.cpp diff --git a/content/math/math.tex b/content/math/math.tex index f99d0d4..f670a70 100644 --- a/content/math/math.tex +++ b/content/math/math.tex @@ -26,7 +26,7 @@ \end{methods}
\sourcecode{math/permIndex.cpp}
\end{algorithm}
-\clearpage
+\columnbreak
\subsection{Mod-Exponent und Multiplikation über $\boldsymbol{\mathbb{F}_p}$}
%\vspace{-1.25em}
@@ -100,8 +100,8 @@ sich alle Lösungen von $x^2-ny^2=c$ berechnen durch: wenn $a\equiv~b \bmod \ggT(m, n)$.
In diesem Fall sind keine Faktoren
auf der linken Seite erlaubt.
- \end{itemize}
- \sourcecode{math/chineseRemainder.cpp}
+ \end{itemize}
+ \sourcecode{math/chineseRemainder.cpp}
\end{algorithm}
\begin{algorithm}{Primzahltest \& Faktorisierung}
@@ -140,7 +140,7 @@ sich alle Lösungen von $x^2-ny^2=c$ berechnen durch: \begin{methods}
\method{kthTerm}{Berechnet $k$-ten Term einer Rekurrenz $n$-ter Ordnung}{\log(k)\cdot n^2}
\end{methods}
- \sourcecode{math/linearRecurence.cpp}
+ \sourcecode{math/linearRecurrence.cpp}
Alternativ kann der \mbox{$k$-te} Term in \runtime{n^3\log(k)} berechnet werden:
$$\renewcommand\arraystretch{1.5}
\setlength\arraycolsep{3pt}
@@ -237,7 +237,7 @@ sich alle Lösungen von $x^2-ny^2=c$ berechnen durch: \sourcecode{math/legendre.cpp}
\end{algorithm}
-\begin{algorithm}{Lineares Sieb und Multiplikative Funktionen}
+\begin{algorithm}{Lineares Sieb und multiplikative Funktionen}
Eine (zahlentheoretische) Funktion $f$ heißt multiplikativ wenn $f(1)=1$ und $f(a\cdot b)=f(a)\cdot f(b)$, falls $\ggT(a,b)=1$.
$\Rightarrow$ Es ist ausreichend $f(p^k)$ für alle primen $p$ und alle $k$ zu kennen.
@@ -251,7 +251,7 @@ sich alle Lösungen von $x^2-ny^2=c$ berechnen durch: \textbf{Wichtig:} Sieb rechts ist schneller für \code{isPrime} oder \code{primes}!
\sourcecode{math/linearSieve.cpp}
- \textbf{\textsc{Möbius}-Funktion:}
+ \textbf{\textsc{Möbius} Funktion:}
\begin{itemize}
\item $\mu(n)=+1$, falls $n$ quadratfrei ist und gerade viele Primteiler hat
\item $\mu(n)=-1$, falls $n$ quadratfrei ist und ungerade viele Primteiler hat
@@ -264,7 +264,7 @@ sich alle Lösungen von $x^2-ny^2=c$ berechnen durch: \item $p$ prim, $k \in \mathbb{N}$:
$~\varphi(p^k) = p^k - p^{k - 1}$
- \item \textbf{Euler's Theorem:}
+ \item \textbf{\textsc{Euler}'s Theorem:}
Für $b \geq \varphi(c)$ gilt: $a^b \equiv a^{b \bmod \varphi(c) + \varphi(c)} \pmod{c}$. Darüber hinaus gilt: $\gcd(a, c) = 1 \Leftrightarrow a^b \equiv a^{b \bmod \varphi(c)} \pmod{c}$.
Falls $m$ prim ist, liefert das den \textbf{kleinen Satz von \textsc{Fermat}}:
$a^{m} \equiv a \pmod{m}$
@@ -359,7 +359,7 @@ Wenn man $k$ Spiele in den Zuständen $X_1, \ldots, X_k$ hat, dann ist die \text \subsection{Kombinatorik}
-\paragraph{Wilsons Theorem}
+\paragraph{\textsc{Wilson}'s Theorem}
A number $n$ is prime if and only if
$(n-1)!\equiv -1\bmod{n}$.\\
($n$ is prime if and only if $(m-1)!\cdot(n-m)!\equiv(-1)^m\bmod{n}$ for all $m$ in $\{1,\dots,n\}$)
@@ -371,14 +371,14 @@ $(n-1)!\equiv -1\bmod{n}$.\\ \end{cases}
\end{align*}
-\paragraph{\textsc{Zeckendorfs} Theorem}
+\paragraph{\textsc{Zeckendorf}'s Theorem}
Jede positive natürliche Zahl kann eindeutig als Summe einer oder mehrerer
verschiedener \textsc{Fibonacci}-Zahlen geschrieben werden, sodass keine zwei
aufeinanderfolgenden \textsc{Fibonacci}-Zahlen in der Summe vorkommen.\\
\emph{Lösung:} Greedy, nimm immer die größte \textsc{Fibonacci}-Zahl, die noch
hineinpasst.
-\paragraph{\textsc{Lucas}-Theorem}
+\paragraph{\textsc{Lucas}'s Theorem}
Ist $p$ prim, $m=\sum_{i=0}^km_ip^i$, $n=\sum_{i=0}^kn_ip^i$ ($p$-adische Darstellung),
so gilt
\vspace{-0.75\baselineskip}
@@ -389,7 +389,7 @@ so gilt %\begin{algorithm}{Binomialkoeffizienten}
\paragraph{Binomialkoeffizienten}
Die Anzahl der \mbox{$k$-elementigen} Teilmengen einer \mbox{$n$-elementigen} Menge.
-
+
\begin{methods}
\method{precalc}{berechnet $n!$ und $n!^{-1}$ vor}{\mathit{lim}}
\method{calc\_binom}{berechnet Binomialkoeffizient}{1}
diff --git a/content/math/tables.tex b/content/math/tables.tex index 53f3758..c422d73 100644 --- a/content/math/tables.tex +++ b/content/math/tables.tex @@ -1,8 +1,12 @@ \enlargethispage{0.2cm} \begin{multicols*}{2} + \refstepcounter{subsection} + \subsectionmark{Tables} + \addcontentsline{toc}{subsection}{\protect\numberline{\thesubsection}Tables} + \input{math/tables/binom} \vfill - \input{math/tables/composite} + \input{math/tables/prime-composite} \vfill \input{math/tables/platonic} \vfill diff --git a/content/math/tables/composite.tex b/content/math/tables/prime-composite.tex index c261db1..99b3348 100644 --- a/content/math/tables/composite.tex +++ b/content/math/tables/prime-composite.tex @@ -1,5 +1,4 @@ - -\begin{tabularx}{\linewidth}{|r||r||r|r||r|r|r||C|} +\begin{tabularx}{\linewidth}{|r|r|rIr|rIrIr|C|} \hline \multicolumn{8}{|c|}{Important Numbers} \\ \hline diff --git a/content/other/other.tex b/content/other/other.tex index b47893f..7a3c52b 100644 --- a/content/other/other.tex +++ b/content/other/other.tex @@ -17,9 +17,9 @@ \begin{expandtable}
\begin{tabularx}{\linewidth}{|Ll|}
\hline
- Bit an Position j lesen & \code{(x & (1 << j)) != 0} \\
+ Bit an Position j lesen & \code{(x \& (1 << j)) != 0} \\
Bit an Position j setzten & \code{x |= (1 << j)} \\
- Bit an Position j löschen & \code{x &= ~(1 << j)} \\
+ Bit an Position j löschen & \code{x \&= ~(1 << j)} \\
Bit an Position j flippen & \code{x ^= (1 << j)} \\
Anzahl an führenden nullen ($x \neq 0$) & \code{__builtin_clzll(x)} \\
Anzahl an schließenden nullen ($x \neq 0$) & \code{__builtin_ctzll(x)} \\
@@ -36,9 +36,9 @@ \begin{expandtable}
\begin{tabularx}{\linewidth}{|lR|}
\hline
- Addition & \code{__builtin_saddll_overflow(a, b, &c)} \\
- Subtraktion & \code{__builtin_ssubll_overflow(a, b, &c)} \\
- Multiplikation & \code{__builtin_smulll_overflow(a, b, &c)} \\
+ Addition & \code{__builtin_saddll_overflow(a, b, \&c)} \\
+ Subtraktion & \code{__builtin_ssubll_overflow(a, b, \&c)} \\
+ Multiplikation & \code{__builtin_smulll_overflow(a, b, \&c)} \\
\hline
\end{tabularx}
\end{expandtable}
diff --git a/content/string/string.tex b/content/string/string.tex index bedabfb..0e482bf 100644 --- a/content/string/string.tex +++ b/content/string/string.tex @@ -63,21 +63,21 @@ \end{algorithm} \clearpage -\begin{algorithm}{Lyndon und De-Bruijn} +\begin{algorithm}{\textsc{Lyndon} und \textsc{De-Bruijn}} \begin{itemize} - \item \textbf{Lyndon-Wort:} Ein Wort das lexikographisch kleiner ist als jede seiner Rotationen. - \item Jedes Wort kann \emph{eindeutig} in eine nicht ansteigende Folge von Lyndon-Worten zerlegt werden. - \item Für Lyndon-Worte $u, v$ mit $u<v$ gilt, dass $uv$ auch ein Lyndon-Wort ist. + \item \textbf{\textsc{Lyndon}-Wort:} Ein Wort das lexikographisch kleiner ist als jede seiner Rotationen. + \item Jedes Wort kann \emph{eindeutig} in eine nicht ansteigende Folge von \textsc{Lyndon}-Worten zerlegt werden. + \item Für \textsc{Lyndon}-Worte $u, v$ mit $u<v$ gilt, dass $uv$ auch ein \textsc{Lyndon}-Wort ist. \end{itemize} \begin{methods} - \method[, Durchschnitt $\Theta(1)$]{next}{lexikographisch nächstes Lyndon-Wort}{n} - \method{duval}{zerlegt $s$ in Lyndon-Worte}{n} + \method[, Durchschnitt $\Theta(1)$]{next}{lexikographisch nächstes \textsc{Lyndon}-Wort}{n} + \method{duval}{zerlegt $s$ in \textsc{Lyndon}-Worte}{n} \method{minrotation}{berechnet kleinste Rotation von $s$}{n} \end{methods} \sourcecode{string/lyndon.cpp} \sourcecode{string/duval.cpp} \begin{itemize} - \item \textbf{De-Bruijn-Sequenze $\boldsymbol{B(\Sigma, n)}$:}~~~ein Wort das jedes Wort der Länge $n$ genau einmal als substring enthält (und minimal ist). Wobei $B(\Sigma, n)$ zyklisch betrachtet wird. + \item \textbf{\textsc{De-Bruijn}-Sequenz $\boldsymbol{B(\Sigma, n)}$:}~~~ein Wort das jedes Wort der Länge $n$ genau einmal als substring enthält (und minimal ist). Wobei $B(\Sigma, n)$ zyklisch betrachtet wird. \item es gibt $\frac{(k!)^{k^{n-1}}}{k^{n}}$ verschiedene $B(\Sigma, n)$ \item $B(\Sigma, n)$ hat Länge $\abs{\Sigma}^n$ \end{itemize} diff --git a/content/tcr.tex b/content/tcr.tex index b327b37..afac53c 100644 --- a/content/tcr.tex +++ b/content/tcr.tex @@ -3,12 +3,20 @@ \documentclass[a4paper,fontsize=7.8pt]{scrartcl} % General information. -\newcommand{\teamname}{Kindergarten Timelimit} +\newcommand{\teamname}{Infinite Loopers} \newcommand{\university}{Karlsruhe Institute of Technology} +% Source code links (optional) +\ifdefined\srclink +\else + \newcommand{\srclink}[1]{https://git.gloria-mundi.eu/tcr/plain/#1} +\fi + % Options \newif\ifoptional -%\optionaltrue +\ifdefined\OPTIONAL + \optionaltrue +\fi % Font encoding. \usepackage[T1]{fontenc} @@ -41,6 +49,7 @@ % Content. \begin{multicols*}{3} + \raggedcolumns \input{datastructures/datastructures} \input{graph/graph} \input{geometry/geometry} @@ -49,17 +58,12 @@ \clearpage \input{math/tables} \begin{multicols*}{3} + \raggedcolumns \input{string/string} \input{python/python} \input{other/other} \input{template/template} \clearpage - \ifodd\value{page} - \else - \null - \thispagestyle{empty} - \clearpage - \fi \input{tests/test} \end{multicols*} \end{document} |