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September 17, 2020 Single Round Match 790 Editorials

Div2-Easy: Alice’s Birthday

For a given \(K\), we have to partition the first \(K\) fibonacci numbers in two sets with equal sum. The fact that \(F_k = F_{k-1} + F_{k-2}\) means that if \(K\) is a multiple of 3, then I can give \(F_k\) to Charlie and \(F_{k-1}, F_{k-2}\) to Eric, \(F_{k-3}\) to Charlie and \(F_{k-4}, F_{k-5}\) to Eric, so on and so forth eventually giving \(F_3\) to Charlie and \(F_2, F_1\) to Eric. The other two cases are \(K \equiv 2 \pmod{3}\) and \(K \equiv 1 \pmod{3}\).

For the first case, let us try to re-use our previous strategy for the last \(K – 2\) boxes starting with \(F_k\) going to Charlie and \(F_{k-1}, F_{k-2}\) to Eric. In the end, we will be left with the first two boxes containing \(F_1\) and \(F_2\), respectively. Since, \(F_1 = F_2 = 1\) so we can simply give one box to Charlie and other one to Eric.

For the second case, we can try our previous strategy but will always be left with one extra box. The fact that the sum of \(F_1, F_2, \ldots, F_k\) is odd means that we cannot divide them evenly among Charlie and Eric. Had it been possible to divide it equally, it would mean that the total sum is even, which is not possible and can be proven using induction.

Since we only have to run a loop till \(K\), so time complexity is \(O(K)\).

Reference sol in Java

import java.io.*; 
import java.util.*; 
import javafx.util.Pair;

public class AlicesBirthday {
    public int[] partition(int k) {
	int[] arr = {-1} ;
	if(k % 3 == 1) return arr ; 
	
        ArrayList<Integer> charlie = new ArrayList<Integer>(), eric = new ArrayList<Integer>(); 
	while(k > 2) {
	    charlie.add(k--);
	    eric.add(k--); eric.add(k--);
	}
	if(k == 2) { 
	    eric.add(k--);
	    charlie.add(k--);
	}
	int[] ret = new int[charlie.size()]; int index = 0;
	for (Integer value: charlie) 
   		ret[index++] = value;
	return ret;
    }
}

Reference sol in Python

class AlicesBirthday:
	def partition(self, K):
		F = [1, 1]
		while len(F) < K: F.append( F[-1] + F[-2] )
		F = F[:K]
		charlie, eric, answer = 0, 0, []
		for k in reversed(range(K)):
			if charlie > eric:
				eric += F[k]
			else:
				charlie += F[k]
				answer.append(k+1)
		if charlie == eric:
			return answer
		return [-1]

Div2-Medium: Bob the Builder

Suppose we are at the given height \(B\). At any step, we can either move to one of its factors or add \(K\) to it. While we can move to its factors without any cost, we have to spend \(1\)$ to add \(K\). So, basically, we have a given state, few possible options from there, and we want to find out end whether a specific end state (i.e. height \(H\)) is reachable or not. If reachable, we have to do so in minimum cost. This naturally appeals to be modeled by graphs with states representing the nodes and edges representing the transitions. Thus, we have to find the shortest path in a graph, so Dijkstra definitely comes as the first solution in mind.

One may use Dijkstra to solve the problem, but it is not immediately clear when to stop exploring if there does not exist any path. We would never actually need to go beyond \(10^6\) (proof is shown below). But going so high, Dijkstra may get TLE. The fact that edge weights are only \(0\) and \(1\) means that we can use \(0 – 1\ BFS\) instead of Dijkstra and the time limit was high enough to allow that in case exploration goes so high. However, in practice, the max step explored before reaching a goal seems to be quite low (the worst case we could find was not more than \(5 \cdot 10^4\)); hence normal Dijkstra passes easily.

Reference sol in Java using 0-1 BFS

import java.util.*;
import java.lang.*;
import java.io.*;

public class BobtheBuilder {
    public int N = (int) 5e5 + 2 ; 
    public int maxStep = -1 ; 
    public List <Integer>[] factors = new ArrayList[N];
    public int dist[] = new int[N] ; 
    public boolean seen[] = new boolean[N] ; 
    
    void prep() {
	    for(int i = 1; i < N; i++) {
	        factors[i] = new ArrayList<Integer> (); 
	        dist[i] = (int) 2e9 ; seen[i] = false ; 
	        int sq = (int) Math.sqrt(i);
	        for(int j = 1; j <= sq; j++) {
	            if(i % j == 0) {
	                factors[i].add(j); factors[i].add(i/j);
	            }
	        }
	    }
    }
    public int minimumPrice(int b, int k, int h) {
	prep();
        Deque<Integer> deq = new ArrayDeque<Integer>(); // seen[b] = true ; 
        dist[b] = 0 ;
        deq.addFirst(b);
        while(deq.size() > 0) {
            int top = deq.removeFirst(); if(top > maxStep) maxStep = top ; 
            if(top == h) return dist[top] ;
            for(int f : factors[top]) {
                if(dist[top] < dist[f]) {
                    deq.addFirst(f);
                    dist[f] = dist[top] ; 
                }
            }
            if( (top + k) < N) {
                if( (dist[top] + 1) < dist[top + k]) {
                    dist[top + k] = dist[top] + 1 ;
                    deq.addLast(top + k); 
                }
            }
        }
        return -1 ; 
    }
}

Reference sol in C++ using Dijkstra

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

struct record {
    int id;
    int dist;
};

bool operator<(const record &amp;A, const record &amp;B) {
    if (A.dist != B.dist) return A.dist < B.dist;
    return A.id < B.id;
}

int solve(int start, int add, int goal) {
    int div = __gcd(add,goal);
    if (start % div) return -1;

    vector<int> D(1002000,2000000);
    D[start] = 0;
    set<record> Q;
    Q.insert( {start, 0} );
    while (!Q.empty()) {
        record kde = *Q.begin();
        Q.erase( Q.begin() );
        if (kde.id >= 1002000) continue;
        if (kde.id == goal) break;
        for (int div=1; div*div<=kde.id; ++div) if (kde.id % div == 0) {
            vector<int> opts = {div, kde.id/div};
            for (int kam : opts) {
                int ndist = kde.dist;
                if (ndist >= D[kam]) continue;
                Q.erase( {kam, D[kam]} );
                D[kam] = ndist;
                Q.insert( {kam, D[kam]} );
            }
        }
        vector<int> opts = { kde.id + add };
        for (int kam : opts) {
            int ndist = kde.dist + 1;
            if (ndist >= D[kam]) continue;
            Q.erase( {kam, D[kam]} );
            D[kam] = ndist;
            Q.insert( {kam, D[kam]} );
        }
    }
    if (D[goal] == 2000000) return -1;
    return D[goal];
}

struct BobtheBuilder {
    int minimumPrice(int B, int K, int H) { return solve(B,K,H); }
};

Proof
If we are at some state \(C\), then at anytime, we can go to one of its factors (possibly \(C\) itself) or add \(K\). We do this starting from \(B\) until we reach \(H\). Suppose it is possible to reach \(H\), then thinking about the process in reverse is basically from \(H\) we can go to one of its multiple (possibly \(H\) itself) or subtract \(K\) and continue this loop until we reach \(B\). Thus, we can model this in the following form: $$ ( .. (((Hx_1 – K)x_2 – K)x_3 – K) ..) = B $$ Simplifying this, we reduce it to a well-known form: $$ Hx – Ky = B \quad (\text{for some } x, y \in \mathbb{Z}) $$ So, we end up with a linear diophantine equation. Let \(g\ =\ gcd(H, K)\). This is only solvable when \(g\ |\ B\) so we can instantly return -1 if \(B\) is not a multiple of \(g\). The fact that \(H, K\) have opposite signs means that this admits infinite positive solutions. From Bezuot’s Identity, we can always find a pair \(s, t\) such that \(Hs – Kt = g\) and \(|s| \le K/2\ \&\ |t| \le H/2\). So we can always find \(s, t\) lie in the range \([-500, 500]\) and since infinite positive solutions are possible and we can shift \(s, t\), so it’s always possible to find a pair \((s, t)\) with both lying in \([0, 1000]\). Since \(H, K \le 10^3\), so the max state we need to explore to ensure we always find a solution is bounded by \(10^6\). Once we reach \(g\) from \(H\), we can repeat the same process to reach \(B\) from \(g\). While we did the derivation in the reverse form, it obviously holds in the forward manner starting from \(B\). Some useful links:

  1. Details on Bezout’s Identity
  2. Positive Solution of Linear Diophantine Eq

Div2-Hard / Div1-Easy: The Social Network

We are given an undirected connected graph \(G\) with exponential weights of the form \(2^x\), and we have to find its minimum cut. The problem may look intimidating but what is in favor for us is the fact that all edge weights are distinct.

Let the given weights sorted in decreasing order be \(w_1, w_2, \ldots w_m\) (i.e. edge weight is \(2^{w_i}\)). Then, for any valid \(j\) between \(1\) and \(m – 1\), we have

So, we would like to keep the edge with the highest weight out of the cut edges set at the cost of ending up with all other edges in the cut since their total sum is strictly less than the highest weight. Thus, we can keep adding edges one by one in the graph (in decreasing order of their weight) as long as one whole connected component does not form.

The edges which do not get added in the graph form our cut edges, and their sum is the answer. This can be done by DSU but was not required since the constraints were significantly small. If we do use DSU, then sorting dominates the runtime with total complexity as \(O(M \text{log}(M))\).

Reference Sol in Java

import java.util.*;
import java.lang.*;
import java.io.*;

public class TheSocialNetwork {
    
    class Edge implements Comparable<Edge> {
        public int u;
        public int v;
        public int w;
        
        public Edge(int uu, int vv, int cc) {
            this.u = uu;
            this.v = vv;
            this.w = cc;
        }
        
        public int compareTo(Edge c) {
            if (w < c.w) return 1;
            if (w > c.w) return -1;
            return 0;
        }
    }
        
    int M = 1002 ; int C = 100002 ; int N = 302 ; 
    List <Edge> edges = new ArrayList<Edge>() ; 
    int[] par = new int[N] ; int[] sz = new int[N] ; int[] pwr = new int[C] ; 
    int MOD = (int) (1e9 + 7); int comp ; 
    
    boolean[] seen = new boolean[C] ;
    List < Integer >[] adj = new ArrayList[N] ; 
    boolean[] visited = new boolean[N] ;
    int connected ;
    
    void prepare() {
        pwr[0] = 1 ; seen[0] = false ; 
        for(int i = 1; i < C; i++) {
            pwr[i] = (pwr[i-1] * 2) % MOD ;
            seen[i] = false ;
        }
        for(int i = 1; i < N; i++) {
            par[i] = i ; sz[i] = 1 ;
            adj[i] = new ArrayList<Integer>(); 
            visited[i] = false ;
        }
    }
    
    int getPar(int v) {
        if(par[v] == v) return v ;
        else return par[v] = getPar(par[v]); 
    }
    
    int dsu(int u, int v) {
        int parU = getPar(u) ; int parV = getPar(v);
        if(parU == parV) return 0;
        
        if(sz[u] < sz[v]) { // swap
            u ^= v ; v ^= u ; u ^= v ; 
        }
        sz[u] += sz[v] ;
        par[parV] = parU ; 
        return 1 ;
    }
    
    public int minimumCut(int n, int m, int[] u, int[] v, int[] l) {
        prepare();
        for(int i = 0; i < m; i++) {
            Edge e = new Edge(u[i], v[i], l[i]);
            edges.add(e); 
        }
        Collections.sort(edges);
        int ans = 0; comp = n ; 
        for(Edge e : edges) {
            System.out.println(e.w);
            int parU = getPar(e.u) ; int parV = getPar(e.v);
            if(parU == parV) {
            }
            else {
                if(comp > 2) { 
                    comp-- ;
                    dsu(e.u, e.v);
                }
                else {
                    ans += pwr[e.w] ; 
                    ans %= MOD ; 
                }
            }
        }
        return ans ; 
    }    
}

Reference Sol in Python

def pow2(n):
	if n == 0: return 1
	t = pow2(n//2)
	t *= t
	if n % 2: t *= 2
	return t % 1000000007

class TheSocialNetwork:
	def minimumCut(self, n, m, u, v, l):
		edges = [ (-z,x-1,y-1) for x,y,z in zip(u,v,l) ]
                edges.sort()
                component_count = n
                component = list(range(n))
		answer = 0
		for w,x,y in edges:
			if component[x] == component[y]: continue
			if component_count == 2:
				answer = (answer + pow2(-w)) % 1000000007
			else:
				cx, cy = component[x], component[y]
				component = [ cx if c==cy else c for c in component ]
				component_count -= 1
		return answer

Side Thought: We were wondering whether it is possible to solve this problem using max-flow?

Div1-Medium: Proposal Optimization

Such kind of ratio problems can be solved by using a niche trick. I think it is hard to develop this line of thinking unless you have encountered it before.

Let us say the optimal ratio is \(Q\). Thus, \(\frac{\sum R_i}{\sum T_i} = Q\). What we can do is a binary search on \(Q\), and check if a ratio is feasible or not. So, for a given \(Q\), we have to check whether \(\frac{\sum R_i}{\sum T_i} \ge Q\) is possible, which is the same thing as \(\sum R_i – Q*\sum T_i \ge 0\) (let’s call the L.H.S. \(P\)). The constraints on \(R_i, T_i\) seem to be quite high to do any form of *DP* while constraints on the grid’s dimension, \(N, M\) are quite low although obviously not permitting brute force solutions. However, meet-in-the-middle looks promising since we can store \(P\) while doing brute force from one corner of the grid and checking from another (meeting in cells lying on some sort of diagonal of the grid).

In fact, it is the intended solution, and we also have to take care of the costs along with \(P\) to actually know if any of the options is actually feasible. So, when doing brute from one side, we store \((P_i, Cost_i)\) pairs on those diagonal cells and later sort them according to \(Cost_i\) to ease our process in the next brute. While doing brute from the other end, we have \((P_j, Cost_j)\), and we want the highest \(P_i\) whose cost satisfies \(Cost_i \le K – Cost_j\). By storing \(P_i\) in prefix maximum manner, we can binary search on the highest index option with cost \(\le K – Cost_j\) and take its \(P_i\) (since we already sorted) to get our answer.

The best diagonal cells for a grid of size \((N, M)\) would be the set \({(i, j) : i + j == (N+M)/2 }\). Since \(N\)x\(M \le 300\), so the number of such diagonal cells (let us denote that by \(C\)) can’t exceed \(17 \quad (\sqrt{300} = 17 )\). We can focus on the analysis of square matrix since it will dominate the rectangular grids. Let the number of binary search iterations required to find the optimal ratio be \(L\) (80 iterations suffice for \(10^{-9}\) error). To reach a diagonal cell \((i, j)\) there are \(i + j \choose i\) ways and then we have to binary search on those many paths which we got from first brute force. Thus, doing meet-in-the-middle takes $$ \sum^C_{i = 1} {C \choose i} \cdot log( {C \choose i} )\ \le \ C \cdot 2^C $$ Hence, the total time required is bounded by \(O(L \cdot C \cdot 2^C)\).

Reference Sol in C++

#include <bits/stdc++.h>
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/tree_policy.hpp>

using namespace std ;
using namespace __gnu_pbds;

template <typename T> // *s.find_by_order(0), s.order_of_key(2) ;
using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;

#define reMin(a, b) a = min(a, b)
#define reMax(a, b) a = max(a, b)

#define lint long long
#define pb push_back
#define F first 
#define S second 
#define sz(x) (int)x.size()
#define all(x) begin(x), end(x)
#define SET(x, val) memset(x, val, sizeof(x))
#define fastio ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0)

typedef vector < int > vi ;
typedef pair < int, int > pii ;

const int N = 300 + 2 ;
const int MOD = 1e9 + 7 ;
const lint INF = 1e18 ;

double best = 0 ;
int n, m, k ;
int roses[N][N], tulips[N][N], cost[N][N] ;

vector < pair < int, double > > meet[N][N] ;
vector < double > ratios[N][N] ; 
int diagonal ;
bool possible = 0 ;
bool ratioSatisfied = 0 ; 

void down(int i, int j, int rose, int tulip, int totalCost, double&amp; R) {
	if(i >= n or j >= m) return ;
	rose += roses[i][j] ; tulip += tulips[i][j] ; totalCost += cost[i][j] ; 
	if(totalCost > k) return ; 
	if(i + j == diagonal) {
		double cur = rose ; cur -= R * tulip ; 
		meet[i][j].pb({totalCost, cur});
		return ; 
	}
	down(i+1, j, rose, tulip, totalCost, R); down(i, j+1, rose, tulip, totalCost, R); 
}

void up(int i, int j, int rose, int tulip, int totalCost, double&amp; R) {
	if(i < 0 or j < 0) return ;
	rose += roses[i][j] ; tulip += tulips[i][j] ; totalCost += cost[i][j] ; 
	if(totalCost > k) return ; 
	if(i + j == diagonal) {
		rose -= roses[i][j] ; tulip -= tulips[i][j] ; totalCost -= cost[i][j] ; 
		double cur = rose ; cur -= R * tulip ; 

		pair < int, double > item = {k - totalCost, 2e9} ;
		int idx = upper_bound(all(meet[i][j]), item) - meet[i][j].begin();
		idx-- ;
		if(idx >= 0) {
			possible = 1 ; assert(idx < sz(ratios[i][j]) and sz(ratios[i][j]) == sz(meet[i][j]));
			double mx = ratios[i][j][idx] ; 
			
			if((mx + cur) >= 0) // should be EPS!! 
				ratioSatisfied = 1 ; 
		}
		return ; 
	}
	up(i-1, j, rose, tulip, totalCost, R); up(i, j-1, rose, tulip, totalCost, R); 
}

void check(double ratio) {
	down(0, 0, 0, 0, 0, ratio);

	for(int i = 0; i < n; i++) {
		int j = diagonal - i ;
		if(j >= m or j < 0) continue ; 
		
        if(sz(meet[i][j]) <= 0) continue ; 
		sort(all(meet[i][j]));

		ratios[i][j].pb(meet[i][j][0].S);
		for(int p = 1; p < sz(meet[i][j]); p++)
			ratios[i][j].pb(max(meet[i][j][p].S, ratios[i][j].back()));
	}
	up(n-1, m-1, 0, 0, 0, ratio);
}

double solve() {
	double low = 0, high = 1e6 ; ratioSatisfied = 0 ;
	
	for(int iter = 0; iter < 70; iter++) {
		double mid = (low + high) / 2 ;
		check(mid); 
		if(ratioSatisfied) low = mid ;
		else high = mid ;
		
		// clear stuff
		ratioSatisfied = 0 ; 
		for(int i = 0; i < n; i++) {
			int j = diagonal - i ;
			if(j < 0 or j >= m) continue ; 
			ratios[i][j].clear(); meet[i][j].clear();
		}
	}
	return low ; 
}

class ProposalOptimization {
public:
	double bestPath(int rows, int cols, int K, vi roz,  vi tolip, vi expense);
};

double ProposalOptimization::bestPath(int rows, int cols, int K, vi roz,  vi tolip, vi expense ){
	n = rows ; m = cols ; k = K ;
	for(int i = 0; i < n; i++) for(int j = 0; j < m; j++) roses[i][j] = roz[i*cols + j] ; 
	for(int i = 0; i < n; i++) for(int j = 0; j < m; j++) tulips[i][j] = tolip[i* cols + j] ; 
	for(int i = 0; i < n; i++) for(int j = 0; j < m; j++) cost[i][j] = expense[i*cols + j] ; 
	double best = solve();
	if(!possible) return -1 ;
	else return best ; 
}

Reference Sol in Java

import java.util.*;
import java.math.*;
 
public class ProposalOptimization {
    class Option implements Comparable<Option> {
        public int orchids;
        public int tulips;
        public int cost;
 
        public Option(int orchids, int tulips, int cost) {
            this.orchids = orchids;
            this.tulips = tulips;
            this.cost = cost;
        }
 
        public int compareTo(Option c) {
            if (cost < c.cost) return -1;
            if (cost > c.cost) return 1;
            if (orchids < c.orchids) return -1;
            if (orchids > c.orchids) return 1;
            if (tulips < c.tulips) return -1;
            if (tulips > c.tulips) return 1;
            return 0;
        }
    }
 
    Option[][] getOptions(int[][] orchids, int[][] tulips, int[][] costs, int steps, boolean secondPass) {
        int R = orchids.length, C = orchids[0].length;
        int[][] options = new int[][] { {0,0,0,0,0} };
        for (int d=1; d<=steps; ++d) {
            int newOptionCount = 0;
            for (int i=0; i<options.length; ++i) {
                if (options[i][0] + 1 < R) ++newOptionCount;
                if (options[i][1] + 1 < C) ++newOptionCount;
            }
            int[][] newOptions = new int[newOptionCount][5];
            for (int i=0, j=0; i<options.length; ++i) {
                if (options[i][0] + 1 < R) {
                    int r = newOptions[j][0] = options[i][0] + 1;
                    int c = newOptions[j][1] = options[i][1];
                    int mul = (secondPass &amp;&amp; d == steps) ? 0 : 1;
                    newOptions[j][2] = options[i][2] + mul * orchids[r];
                    newOptions[j][3] = options[i][3] + mul * tulips[r];
                    newOptions[j][4] = options[i][4] + mul * costs[r];
                    ++j;
                }
                if (options[i][1] + 1 < C) {
                    int r = newOptions[j][0] = options[i][0];
                    int c = newOptions[j][1] = options[i][1] + 1;
                    int mul = (secondPass &amp;&amp; d == steps) ? 0 : 1;
                    newOptions[j][2] = options[i][2] + mul * orchids[r];
                    newOptions[j][3] = options[i][3] + mul * tulips[r];
                    newOptions[j][4] = options[i][4] + mul * costs[r];
                    ++j;
                }
            }
            options = newOptions;
        }
        int[] answerCounts = new int[R];
        for (int i=0; i<options.length; ++i) ++answerCounts[options[i][0]];
        Option[][] answer = new Option[R][];
        for (int r=0; r<R; ++r) answer[secondPass ? R-1-r : r] = new Option[answerCounts[r]];
        for (int i=0; i<options.length; ++i) {
            int r = options[i][0];
            answer[secondPass ? R-1-r : r][ --answerCounts[r] ] = new Option( options[i][2], options[i][3], options[i][4] );
            // System.out.println("row "+r+" added option "+options[i][2]+" "+options[i][3]+" "+options[i][4] );
        }
        return answer;
    }
 
    int[][] flip(int[][] array) {
        int R = array.length, C = array[0].length;
        int[][] answer = new int[R][C];
        for (int r=0; r<R; ++r) for (int c=0; c<C; ++c) answer[r] = array[R-1-r][C-1-c];
        return answer;
    }
 
    boolean solvable(double x, Option[] options1, Option[] options2, int K) {
        int b = 0;
        double bestb = -1e20;
        for (int a=options1.length-1; a>=0; --a) {
            while (b < options2.length &amp;&amp; options1[a].cost + options2[b].cost <= K) {
                bestb = Math.max( bestb, options2[b].orchids - x * options2[b].tulips );
                ++b;
            }
            if (options1[a].orchids - x * options1[a].tulips + bestb >= 0) return true;
        }
        return false;
    }
 
    public double bestPath(int R, int C, int K, int[] _orchids, int[] _tulips, int[] _costs) {
 
        int[][] orchids = new int[R][C];
        int[][] tulips = new int[R][C];
        int[][] costs = new int[R][C];
        for (int r=0; r<R; ++r) for (int c=0; c<C; ++c) {
            orchids[r] = _orchids[r*C + c];
            tulips[r] = _tulips[r*C + c];
            costs[r] = _costs[r*C + c];
        }
 
        Option[][] options1 = getOptions( orchids, tulips, costs, (R+C-2)/2, false);
        Option[][] options2 = getOptions( flip(orchids), flip(tulips), flip(costs), (R+C-1)/2, true);
 
        for (int i=0; i<options1.length; ++i) Arrays.sort(options1[i]);
        for (int i=0; i<options2.length; ++i) Arrays.sort(options2[i]);
 
        int bestCost = K+1;
        for (int i=0; i<options1.length; ++i) if (options1[i].length > 0 &amp;&amp; options2[i].length > 0) bestCost = Math.min( bestCost, options1[i][0].cost + options2[i][0].cost );
        if (bestCost > K) return -1;
 
        double lo = 0, hi = 1e6 + 1;
        for (int step=0; step<200; ++step) {
            double med = (lo+hi) / 2;
            boolean isSolvable = false;
            for (int i=0; i<options1.length; ++i) if (solvable(med,options1[i],options2[i],K)) {
                isSolvable = true;
                break;
            }
            if (isSolvable) lo = med; else hi = med;
        }
        return lo;
    }
}

Div1-Hard: Tale Of Two Squares:

A number can be expressed as a sum of two sqaures if and only if all its prime factors of the form \(4k+3\) occur even times. So while taking the product of few numbers, we are only interested in its prime decomposition; hence, we can store a bit-vector for each \(A_i\) denoting odd/even occurrence of such prime factors.

From now on, whenever I talk about primes it is assumed to be of the form \(4k + 3\) (for some \(k \in \mathbb{Z}\)) and that the reader is familar with basis of a vector space and gaussian elimination. Please go through this if you are not familiar with the usage of those concepts in CP:
Introductory Blog on using Linear Algebraic techniques to solve XOR related problems

Now, \(A_i \le 10^7\) so #primes is around \(3* 10^5\). Normal Gaussian elimination would easily TLE, even with bitsets. Still, lets imagine doing Gaussian elimination traversing from the highest bit to the lowest and xoring when necessary. Notice that, when you are traversing, if any prime factor > \(\sqrt A_i\) exists (at max one can exist), the moment you xor it, no more bits > \(\sqrt A_i\) will be on anymore (since only one such was on earlier and now you have xored it). Now, you can simply simulate Gaussian elimination on lower-order primes (i.e. \(primes \le \sqrt A_i\)) and handle that one highest bit (if it exists) separately. This way, combined with bitset for lower-order primes, solves the problem without queries.

For queries, let us process them offline. Let \(queries[l]\) store all the queries with their left endpoint as \(l\). We will insert \(A_i\) into our basis from the right side, \(i\) going from \(N-1\) to \(0\). After we have inserted the element at index \(i\), we will answer all the queries which begin their, i.e., the ones in \(queries[i]\).

We will use a simple fact about linear combination of vectors but in a clever manner to answer the queries. Suppose we are about to try and insert \(v\) into our \(Basis\) (may not insert if it turns out to be dependant). If \(v = b_1 + b_2 + \cdots b_k\) (where \(b_i \in Basis\)), then we can remove one of the \(b_i\) and instead insert \(v\) into its place, keeping the basis intact. So, we will use this fact while inserting — we will try to keep such elements in the basis which have leftmost index so as to answer as many queries as possible. We can do this greedily: while inserting \(v\), whenever we are about to xor it with some \(b_i\), we will first check which has the least index and \(swap(b_i, v)\) if \(v\)‘s index is smaller than \(b_i\). If we do end up swapping, then \(v\) becomes a part of the basis (in place of \(b_i\)), and we progress doing Gaussian Elimination with \(b_i\) instead. Otherwise, we move forward with \(v\) like we normally do. Please refer to the attached code for more details.

Now, when we answer queries which have left index \(i\), we will have the basis vectors from \(A[i:N]\) having the least possible index. So for a query \(i\ R\), we simply need to count how many vectors in our basis have index \(\le R\). This can be easily done with some BIT or set.

Let \(K\) denote the number of primes \(\le \sqrt A_i \ ( = 225)\). We store a bitset of size \(K\) for every prime. While doing Gaussian elimination, we handle the large bit manually while following the normal procedure for lower-order primes; hence it takes \(O(K^2/64)\) time to insert a \(A_i\). After that, we answer queries at index \(i\) in \(O(log(N))\) time by querying in our BIT. Hence, the overall complexity is \(O(N\cdot \frac{K^2}{64} + Q\cdot log(N))\), dominated by Gaussian elimination using bitsets:

Reference Sol in Java

import java.util.*;
import java.lang.*;
import java.io.*;

public class TaleOfTwoSquares {
    
    class Query {
        public int L;
        public int R;
        public int idx;
        
        public Query(int l, int r, int j) {
            this.L = l;
            this.R = r;
            this.idx = j;
        }
    }
    
    int K = (int) 1e7 + 2 ;  int N = 100002 ; int Q = 100002 ;
    int BS = 256 ; int SQRT = 3200 ; int MAX_PRIMES = (int) 4e5 ; 
    
    int spf[] = new int[K] ;
    List<Integer> arr = new ArrayList<Integer>();
    
    Map<Integer, Integer> map = new HashMap<Integer, Integer>();
    Map<Integer, Integer> highMap = new HashMap<Integer, Integer>();
    
    boolean filled[] = new boolean[SQRT] ;
    
    int indices[] = new int[K] ;
    List<BitSet> basis = new ArrayList<BitSet>();
    List<BitSet> basisHigh = new ArrayList<BitSet>();
    
    List<Query>[] queries = new ArrayList[Q] ; 
    int basisIndex[] = new int[K] ; 
    int bit[] = new int[N] ;
    int pwr2[] = new int[MAX_PRIMES] ; 
    
    int MOD = 998244353 ;
    Scanner s = new Scanner(System.in);
    
    void spfsieve()
    {
        for(int i = 2; i < K; i++) spf[i]=-1;
        int cnt = 0 ;
        for(int i = 2; i < K; i++)
        {
            if(spf[i] != -1) continue;
            if(i % 4 == 3) indices[i] = cnt++ ; 
            for(int j = i; j < K; j+=i)
                if(spf[j]==-1) 
                    spf[j]=i;
        }
        assert cnt < MAX_PRIMES ;
    }
    
    void update(int i, int x) {
        for(; i < N; i += i &amp; -i)
            bit[i] += x ;
    }
    
    int getPrefixSum(int i) {
        int s = 0;
        for(; i > 0; i -= i &amp; -i)
            s += bit[i] ;
        return s ; 
    }
    
    void prepare() {
        for(int i = 0; i < Q; i++) 
            queries[i] = new ArrayList < Query > (); 
        pwr2[0] = 1 ;
        for(int i = 0; i < MAX_PRIMES; i++) {
            BitSet lst = new BitSet(BS);
            basisHigh.add(lst); 
            if(i > 0) 
                pwr2[i] = (pwr2[i-1] * 2) % MOD ;
        }
        for(int i = 0; i < SQRT; i++) {
            filled[i] = false ; 
            BitSet lst = new BitSet(BS);
            basis.add(lst); 
        }
    }
    
    int insertVector(int highest, int idx, BitSet bs) {
        // handle highest bit manually 
        if(highest >= SQRT) {
            int bitIndex = indices[highest] ; 
            
            if(highMap.containsKey(bitIndex)) {
                
                if(idx < basisIndex[bitIndex]) {
                    update(basisIndex[bitIndex], -1); update(idx, 1); 
                    
                    // swap 
                    bs.xor(basisHigh.get(bitIndex));
                    BitSet cpy = basisHigh.get(bitIndex); 
                    cpy.xor(bs); basisHigh.set(bitIndex, cpy);
                    bs.xor(basisHigh.get(bitIndex)); 
                    
                    int tmp = basisIndex[bitIndex] ;
                    basisIndex[bitIndex] = idx ;
                    idx = tmp; 
                }
                bs.xor(basisHigh.get(bitIndex));
            }
            else {
                highMap.put(bitIndex, idx);
                basisHigh.set(bitIndex, bs); 
                basisIndex[bitIndex] = idx ; update(idx, 1); 
                return 0;
            }
        }
        // gauss 
        for(int i = SQRT - 1; i >= 0; i--) {
            if(!bs.get(i)) continue ;
            if(!filled[i]) {
                filled[i] = true ; 
                basis.set(i, bs) ; update(idx, 1); 
                basisIndex[i] = idx ; 
                return 0 ;
            }
            if(idx < basisIndex[i]) {
                update(basisIndex[i], -1); update(idx, 1); 
                
                // swap 
                bs.xor(basis.get(i));
                BitSet cpy = basis.get(i); cpy.xor(bs); basis.set(i, cpy);
                bs.xor(basis.get(i)); 
                
                int tmp = basisIndex[i] ;
                basisIndex[i] = idx ;
                idx = tmp; 
            }
            bs.xor(basis.get(i)); 
        }
        return 1; 
    }
    
    public int count(int n, int[] aprefix, int query_cnt, int[] ql, int[] qr, int seed) {
        spfsieve(); prepare(); long state = seed ; long modo = 1 ; modo <<= 31 ; 

	for(int x : aprefix) arr.add(x);
        for(int i = aprefix.length; i < n; i++) {
            state = (state * 1103515245 + 12345) % modo ;
            arr.add( (int) (1 + (state % 10000000)) );
        }
        
	int qprefix = ql.length; assert qprefix == qr.length ; 
	for(int i = 0; i < qprefix; i++) {
	    Query q = new Query(ql[i] + 1, qr[i] + 1, i);
	    queries[ql[i] + 1].add(q);
	}

        for(int i = qprefix; i < query_cnt; i++) {
            state = (state * 1103515245 + 12345) % modo ;
            int x = (int) state % n ; x++ ; 
            state = (state * 1103515245 + 12345) % modo ;
            int y = (int) state % n ; y++ ; 
            int l = Math.min(x, y); int r = Math.max(x, y);
            Query q = new Query(l, r, i);
            queries[l].add(q); 
        }
        
	int sum = 0 ; 
        for(int i = n; i > 0; i--) {
            int now = arr.get(i - 1) ;
            BitSet bs = new BitSet(BS); 
            int highest = -1 ; 
            
            while(now > 1) {
                int primeFactor = spf[now] ; 
                int occurs = 0 ;
                while(now % primeFactor == 0) {
                    now /= primeFactor ;
                    occurs++ ; 
                }
                if(primeFactor % 4 == 3 &amp;&amp; occurs % 2 == 1) {
                    // System.out.println(arr.get(i));
                    if(primeFactor >= SQRT) {
                        assert highest == -1 ; 
                        highest = primeFactor ; 
                    }
                    else {
                        int pos = indices[primeFactor] ;
                        bs.set(pos); 
                    }
                }
                
            }
            insertVector(highest, i, bs); 
            for(Query qq : queries[i]) {
                int sz = getPrefixSum(qq.R); 
                int tot = qq.R - qq.L + 1 ; 
                int dependant = tot - sz ; 
                // System.out.println(qq.L + " " + qq.R + " " + sz + " - " + dependant);
                int ans = pwr2[dependant] - 1 ;
                if(ans < 0) ans += MOD ; 
		sum += ans ; sum %= MOD ; 
            }
        }
        return sum ; 
    }
}

Reference sol in C++

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

const int SQRTMAX = 3162;
const int SMALL_PRIME_COUNT = 225;
const int SMALL_PRIMES[] = {3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571, 587, 599, 607, 619, 631, 643, 647, 659, 683, 691, 719, 727, 739, 743, 751, 787, 811, 823, 827, 839, 859, 863, 883, 887, 907, 911, 919, 947, 967, 971, 983, 991, 1019, 1031, 1039, 1051, 1063, 1087, 1091, 1103, 1123, 1151, 1163, 1171, 1187, 1223, 1231, 1259, 1279, 1283, 1291, 1303, 1307, 1319, 1327, 1367, 1399, 1423, 1427, 1439, 1447, 1451, 1459, 1471, 1483, 1487, 1499, 1511, 1523, 1531, 1543, 1559, 1567, 1571, 1579, 1583, 1607, 1619, 1627, 1663, 1667, 1699, 1723, 1747, 1759, 1783, 1787, 1811, 1823, 1831, 1847, 1867, 1871, 1879, 1907, 1931, 1951, 1979, 1987, 1999, 2003, 2011, 2027, 2039, 2063, 2083, 2087, 2099, 2111, 2131, 2143, 2179, 2203, 2207, 2239, 2243, 2251, 2267, 2287, 2311, 2339, 2347, 2351, 2371, 2383, 2399, 2411, 2423, 2447, 2459, 2467, 2503, 2531, 2539, 2543, 2551, 2579, 2591, 2647, 2659, 2663, 2671, 2683, 2687, 2699, 2707, 2711, 2719, 2731, 2767, 2791, 2803, 2819, 2843, 2851, 2879, 2887, 2903, 2927, 2939, 2963, 2971, 2999, 3011, 3019, 3023, 3067, 3079, 3083, 3119 };
vector<int> SMALL_PRIME_ID, pow2;

struct Fenwick1D { // {{{
    int size;
    vector<int> T;

    Fenwick1D(int maxval) {
        size = 1;
        while (size < maxval) size <<= 1;
        T.clear();
        T.resize(size+1,0);
    }

    void update(int x, int delta) { // assumes 1 <= x <= init_maxval
        while (x <= size) { T[x] += delta; x += x &amp; -x; }
    }

    int sum(int x1, int x2) { // sum in the closed interval [x1,x2]
        int res=0;
        --x1;
        while (x2) { res += T[x2]; x2 -= x2 &amp; -x2; }
        while (x1) { res -= T[x1]; x1 -= x1 &amp; -x1; }
        return res;
    }

    int find(int sum) { // largest z such that sum( [1,z] ) <= sum
        int idx = 0, bitMask = size;
        while (bitMask &amp;&amp; (idx < size)) {
            int tIdx = idx + bitMask;
            if (sum >= T[tIdx]) { idx=tIdx; sum -= T[tIdx]; }
            bitMask >>= 1;
        }
        return idx;
    }
}; // }}}

void init() {
    SMALL_PRIME_ID.clear();
    SMALL_PRIME_ID.resize(SQRTMAX+1, -1);
    for (int i=0; i<SMALL_PRIME_COUNT; ++i) SMALL_PRIME_ID[ SMALL_PRIMES[i] ] = i;
    pow2.clear();
    pow2.push_back(1);
    for (int i=1; i<100005; ++i) pow2.push_back( (pow2.back()*2) % 998244353 );
}

struct signature {
    bitset< SMALL_PRIME_COUNT > small_bits;
    int large_bit;
    signature() : large_bit(-1) {}

    int highest_prime() {
        if (large_bit != -1) return large_bit;
        for (int i=SMALL_PRIME_COUNT-1; i>=0; --i) if (small_bits.test(i)) return SMALL_PRIMES[i];
        return 0;
    }

    void do_xor(const signature &amp;other) {
        if (other.large_bit != -1) {
            if (large_bit != -1) {
                assert( large_bit == other.large_bit );
                large_bit = -1;
            } else {
                large_bit = other.large_bit;
            }
        }
        small_bits ^= other.small_bits;
    }
};

ostream&amp; operator<< (ostream&amp; out, const signature &amp;S) {
    /*
    int max = SMALL_PRIME_COUNT - 1;
    while (max > 0 &amp;&amp; !S.small_bits.test(max)) --max;
    for (int i=0; i<=max; ++i) out << int( S.small_bits.test(i) );
    */
    if (S.large_bit != -1) out << S.large_bit << "+";
    for (int i=9; i>=0; --i) out << int(S.small_bits.test(i));
    return out;
}

struct query {
    int l, r;
};

bool operator< (const query &amp;A, const query &amp;B) {
    if (A.l != B.l) return A.l > B.l;
    return A.r < B.r;
}

signature get_signature(int N) {
    signature answer;
    for (int d=2; d*d<=N; ++d) {
        if (N % d) continue;
        int cnt = 0;
        while (N % d == 0) { ++cnt; N /= d; }
        if (d % 4 != 3) continue;
        if (cnt % 2 == 0) continue;
        answer.small_bits.set( SMALL_PRIME_ID[d] );
    }
    if (N > 1 &amp;&amp; N % 4 == 3) {
        if (N > SQRTMAX) answer.large_bit = N; else answer.small_bits.set( SMALL_PRIME_ID[N] );
    }
    return answer;
}

struct TaleOfTwoSquares {
    int count(int N, vector<int> Aprefix, int Q, vector<int> Lprefix, vector<int> Rprefix, int seed) {
        init();

        long long state = seed;

        vector<int> A = Aprefix;
        while (int(A.size()) < N) {
            state = (state * 1103515245 + 12345) % (1LL << 31);
            A.push_back(1 + (state % 10000000));
        }

        if (N <= 20) for (int n=0; n<N; ++n) cout << A[n] << ", "; cout << endl;

        vector<int> L = Lprefix, R = Rprefix;
        while (int(L.size()) < Q) {
            state = (state * 1103515245 + 12345) % (1LL << 31);
            int x = state % N;
            state = (state * 1103515245 + 12345) % (1LL << 31);
            int y = state % N;
            L.push_back( min(x,y) );
            R.push_back( max(x,y) );
        }

        if (Q <= 20) for (int n=0; n<Q; ++n) cout << L[n] << ", "; cout << endl;
        if (Q <= 20) for (int n=0; n<Q; ++n) cout << R[n] << ", "; cout << endl;

        vector<signature> B;
        for (int n=0; n<N; ++n) B.push_back( get_signature(A[n]) );

        vector< vector<int> > query_by_L(N);
        for (int q=0; q<Q; ++q) query_by_L[ L[q] ].push_back( R[q] );

        Fenwick1D InBase(N+2);
        int answer = 0;
        unordered_map<int,int> base_by_highest_prime;

        for (int n=N-1; n>=0; --n) {
            // two dummy updates
            InBase.update(N+1,+1);
            InBase.update(N+1,-1);
            // add number A[n] to the base
            if (B[n].highest_prime() != 0) {
                InBase.update(n+1,+1);
                int where = n;
                while (true) {
                    int p = B[where].highest_prime();
                    if (p == 0) {
                        InBase.update(where+1,-1);
                        break;
                    }
                    if (!base_by_highest_prime.count(p)) {
                        base_by_highest_prime[p] = where;
                        break;
                    }
                    int nxt = base_by_highest_prime[p];
                    if (nxt > where) {
                        base_by_highest_prime[p] = where;
                        B[nxt].do_xor( B[where] );
                        where = nxt;
                    } else {
                        B[where].do_xor( B[nxt] );
                    }
                }
            }
            // answer queries that begin here
            for (int r : query_by_L[n]) {
                int length = r-n+1;
                int base_size = InBase.sum(n+1,r+1);
                answer += pow2[length-base_size] - 1;
                answer %= 998244353;
            }
        }

        return answer;
    }
};


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