We'll use linearity of expectation. The complexity of a subset is equal to the number of integers $i$ such that the interval $(i,i+1)$ is contained within one of the segments in the subset but $(i-1,i)$ isn't (informally, the number of "start" points). In other words, the segment with left endpoint $i$ contributes one to the complexity as long as it is part of the subset and no other segment in the subset contains $(i-1,i)$.

This is true for exactly $2^{N-1-(\#\text{ of intervals that contain}(i,i+1))}$ subsets. The sum of this quantity over all intervals can be computed in $O(N)$ time with prefix sums and precalculation of powers of 2.

#include "bits/stdc++.h" using namespace std; void setIO(string s) { ios_base::sync_with_stdio(0); cin.tie(0); freopen((s+".in").c_str(),"r",stdin); freopen((s+".out").c_str(),"w",stdout); } #define f first #define s second const int MOD = 1e9+7; int N; int main() { setIO("help"); cin >> N; vector<pair<int,int>> v(N); for (auto& a: v) cin >> a.f >> a.s; vector<int> over(2*N+1), po2(N); po2[0] = 1; for (int i = 1; i < N; ++i) po2[i] = 2*po2[i-1]%MOD; for (auto& t: v) over[t.f] ++, over[t.s] --; for (int i = 1; i <= 2*N; ++i) over[i] += over[i-1]; int ans = 0; for (auto& t: v) ans = (ans+po2[N-1-over[t.f-1]])%MOD; cout << ans << "\n"; }