usaco-guide/content/5_Gold/BIT.md

id	title	author	prerequisites
bit	Binary Indexed Trees	Benjamin Qi	Silver - Prefix Sums

A Binary Indexed Tree allows you to perform the following tasks in O(\log N) time each on an array of size N:

• Update the element at a single position (point).
• Query the sum of a prefix of the array.

Binary Indexed Tree

Aka Fenwick Tree.

Sample Problems

• CSES Range Sum Queries II
  • can also do range XOR queries w/ update
• SPOJ Inversion Counting

Tutorials

• CPH 9.2 (very good)
• CSAcademy BIT (also very good)
• cp-algorithms Fenwick Tree
  • extends to range increment and range query, although this is not necessary for gold
• Topcoder BIT

My implementation can be found here, and can compute range sum queries for any number of dimensions.

Indexed Set

In the special case where all elements of the array are either zero or one (which is the case for several gold problems), users of C++ will find indexed set useful. Using this, we can solve "Inversion Counting" in just a few lines (with template). Tree<int> behaves mostly the same way as set<int> with the additional functions

• order_of_key(x): counts the number of elements in the indexed set that are strictly less than x.
• find_by_order(k): similar to find, returns the iterator corresponding to the k-th lowest element in the set (0-indexed).

See the link for more examples of usage.

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

#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
using namespace __gnu_pbds;
template <class T> using Tree = tree<T, null_type, less<T>, 
  rb_tree_tag, tree_order_statistics_node_update>; 

int main() {
  int T; cin >> T;
  for (int i = 0; i < T; ++i) {
    int n; cin >> n;
    Tree<int> TS; long long numInv = 0;
    for (int j = 0; j < n; ++j) { 
      int x; cin >> x;
      numInv += j-TS.order_of_key(x); // gives # elements before it > x
      TS.insert(x);
    }
    cout << numInv << "\n";
  }
}

Note that if it were not the case that all elements of the input array were distinct, then this code would be incorrect since Tree<int> would remove duplicates. Instead, we would use an indexed set of pairs (Tree<pair<int,int>>), where the first element of each pair would denote the value while the second would denote the array position.

Practice Problems

• USACO Gold
  • The first three problems are just variations on inversion counting.
  • Haircut
  • Balanced Photo
  • Circle Cross
  • Sleepy Cow Sort
    • as far as I know, all gold problems have had only one possible output ...
  • Out of Sorts (harder?)
• Other Problems:
  • USACO Plat Mincross
  • Mega Inversions
    • also just inversion counting
  • Out of Sorts (USACO Silver)
    • aka Sorting Steps
    • Of course, this doesn't require anything other than sorting but fast range sum queries may make this easier.
  • Twin Permutations
    • Offline 2D queries can be done with a 1D data structure