usaco-guide/content/6_Plat/Bitsets.md

id	title	author	description
bitsets	Bitsets	Benjamin Qi	Three examples of how bitsets give some unintended solutions on recent USACO problems.

Tutorial

tl;dr some operations are 32x-64x faster compared to a boolean array.

• C++ Reference
• Errichto Bitwise Operations Pt 2

Applications

Cowpatibility (Gold)

Label the cows from 0\ldots N-1. For two cows x and y set adj[x][y]=1 if they share a common flavor. Then the number of pairs of cows that are compatible (counting each pair where x and y are distinct twice) is equal to the sum of adj[x].count() over all x. It remains to compute adj[x] for all x.

Unfortunately, storing N bitsets each with N bits takes up 50000^2/32\cdot 4=312.5\cdot 10^6 bytes of memory, which is too much. We can reduce the memory usage by half in exchange for a slight increase in time by first computing the adjacency bitsets for all x satisfying 0<=x<N/2, and then for all x satisfying N/2<=x<N afterwards.

First, we read in all of the flavors.

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef bitset<50000> B;
const int HALF = 25000;

int N; 
B adj[HALF];
vector<int> flav[1000001];
ll ans;

void input() {
	ios_base::sync_with_stdio(0); cin.tie(0);
	freopen("cowpatibility.in","r",stdin);
	freopen("cowpatibility.out","w",stdout);
	cin >> N;
	for (int i = 0; i < N; ++i) 
		for (int j = 0; j < 5; ++j) {
			int x; cin >> x;
			flav[x].push_back(i);
		}
}

Then for each flavor, we can look at all pairs of cows that share that flavor and update the adjacency lists (for those x satisfying x<HALF).

int main() {
	input();
	for (int i = 1; i <= 1000000; ++i) 
		for (int x: flav[i]) if (x < HALF) for (int y: flav[i]) adj[x][y] = 1;
	for (int i = 0; i < HALF; ++i) ans += adj[i].count();
}

adj[i].count() runs quickly enough since its runtime is divided by the bitset constant. However, looping over all cows in flav[i] is too slow if say, flav[i] contains all cows. Then the nested loop could take \Theta(N^2) time! Of course, we can instead write the nested loop in a way that takes advantage of fast bitset operations once again.

for (int i = 1; i <= 1000000; ++i) if (flav[i].size() > 0) {
	B b; for (int x: flav[i]) b[x] = 1;
	for (int x: flav[i]) if (x < HALF) adj[x] |= b;
}

The full main function is as follows:

int main() {
	input();
	for (int i = 1; i <= 1000000; ++i) if (flav[i].size() > 0) {
		B b; for (int x: flav[i]) b[x] = 1;
		for (int x: flav[i]) if (x < HALF) adj[x] |= b;
	}
	for (int i = 0; i < HALF; ++i) ans += adj[i].count();
	for (int i = 0; i < HALF; ++i) adj[i].reset();
	for (int i = 1; i <= 1000000; ++i) if (flav[i].size() > 0) {
		B b; for (int x: flav[i]) b[x] = 1;
		for (int x: flav[i]) if (x >= HALF) adj[x-HALF] |= b;
	}
	for (int i = 0; i < HALF; ++i) ans += adj[i].count();
	cout << ((ll)N*N-ans)/2 << "\n";
}

Lots of Triangles

First, we read in the input data. cross(a,b,c) is positive iff c lies to the left of the line from a to b.

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef pair<ll,ll> P;

#define f first
#define s second

ll cross(P a, P b, P c) {
	b.f -= a.f, b.s -= a.s;
	c.f -= a.f, c.s -= a.s;
	return b.f*c.s-b.s*c.f;
}

vector<P> v;
int N;

void input() {
	ios_base::sync_with_stdio(0); cin.tie(0);
	freopen("triangles.in","r",stdin);
	freopen("triangles.out","w",stdout);
	cin >> N; v.resize(N); 
	for (P& p: v) cin >> p.f >> p.s;
}

There are O(N^3) possible lots. Trying all possible lots and counting the number of trees that lie within each in O(N) for a total time complexity of O(N^4) should solve somewhere between 2 and 5 test cases. Given a triangle t[0], t[1], t[2] with positive area, tree x lies within it iff x is to the left of each of sides (t[0],t[1]), (t[1],t[2]), and (t[2],t[0]).

int main() {
	input();
	vector<int> res(N-2);
	for (int i = 0; i < N; ++i) 
		for (int j = i+1; j < N; ++j) 
			for (int k = j+1; k < N; ++k) {
				vector<int> t = {i,j,k};
				if (cross(v[t[0]],v[t[1]],v[t[2]]) < 0) swap(t[1],t[2]);
				int cnt = 0;
				for (int x = 0; x < N; ++x) {
					if (cross(v[t[0]],v[t[1]],v[x]) <= 0) continue;
					if (cross(v[t[1]],v[t[2]],v[x]) <= 0) continue;
					if (cross(v[t[2]],v[t[0]],v[x]) <= 0) continue;
					cnt ++;
				}
				res[cnt] ++;
			}
	for (int i = 0; i < N-2; ++i) cout << res[i] << "\n";
}

The analysis describes how to count the number of trees within a lot in O(1), which is sufficient to solve the problem. However, O(N) is actually sufficient as long as we divide by the bitset constant. Let b[i][j][k]=1 if k lies to the left of side (i,j). Then x lies within triangle (t[0],t[1],t[2]) as long as b[t[0]][t[1]][x]=b[t[1]][t[2]][x]=b[t[2]][t[0]][x]=1. We can count the number of x such that this holds true by taking the bitwise AND of the bitsets for all three sides and then counting the number of bits in the result.

bitset<300> b[300][300];

int main() {
	input();
	for (int i = 0; i < N; ++i) 
		for (int j = 0; j < N; ++j) if (j != i) 
			for (int k = 0; k < N; ++k) if (cross(v[i],v[j],v[k]) > 0) 
				b[i][j][k] = 1;
	vector<int> res(N-2);
	for (int i = 0; i < N; ++i) 
		for (int j = i+1; j < N; ++j) 
			for (int k = j+1; k < N; ++k) {
				vector<int> t = {i,j,k};
				if (cross(v[t[0]],v[t[1]],v[t[2]]) < 0) swap(t[1],t[2]);
				auto z = b[t[0]][t[1]]&b[t[1]][t[2]]&b[t[2]][t[0]];
				res[z.count()] ++;
			}
	for (int i = 0; i < N-2; ++i) cout << res[i] << "\n";
}

Equilateral Triangles

Again, the intended solution runs in O(N^3). Of course, it is still possible to pass O(N^4) solutions with bitset! See the analysis here.

Other Applications

Using operations such as _Find_first() and _Find_next() mentioned in Errichto's blog above (are these documented?), you can speed up the following:

• BFSing through a dense graph with N vertices in O(N^2)
• bipartite matching in O(N^3)