Balanced Forest


Problem Statement :


Greg has a tree of nodes containing integer data. He wants to insert a node with some non-zero integer value somewhere into the tree. His goal is to be able to cut two edges and have the values of each of the three new trees sum to the same amount. This is called a balanced forest. Being frugal, the data value he inserts should be minimal. Determine the minimal amount that a new node can have to allow creation of a balanced forest. If it's not possible to create a balanced forest, return -1.

For example, you are given node values c = [15,12,8,14,13]  and edges = [ [1,2], [1, 3], [1, 4]. [4, 5] ]. It is the following tree:

The blue node is root, the first number in a node is node number and the second is its value. Cuts can be made between nodes 1 and 3  and nodes 1 and 4 to have three trees with sums 27, 27  and 8. Adding a new node w of c[w] = 19 to the third tree completes the solution.

Function Description

Complete the balancedForest function in the editor below. It must return an integer representing the minimum value of  that can be added to allow creation of a balanced forest, or -1 if it is not possible.

balancedForest has the following parameter(s):

c: an array of integers, the data values for each node
edges: an array of 2 element arrays, the node pairs per edge



Solution :


                            Solution in C :

In C ++ :



#include <cstdio>
#include <iostream>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <vector>
#include <map>
#include <set>
#include <string>
#include <cstdlib>
#include <ctime>
#include <deque>
#include <unordered_set>
using namespace std;

int q;
map <long long, int> Map1, Map2;
long long ctot;
int c[110000];
vector <int> ve[110000];
long long ans;
int n;
long long sum[1100000];

void dfs1(int x, int f) {
	sum[x] = c[x];
	for (int i = 0; i < (int) ve[x].size(); i++)
		if (ve[x][i] != f) {
			dfs1(ve[x][i], x);
			sum[x] += sum[ve[x][i]];
		}
	Map1[sum[x]] += 1;
}

void test(long long x) {
	long long y = ctot - 2 * x;
	if (y > 0 && y <= x)
		ans = min(ans, x - y);
}

void dfs2(int x, int f) {
	

	
	if (Map2[2 * sum[x]])
		test(sum[x]);
	if (Map2[ctot - sum[x]])
		test(sum[x]);
	if ((ctot - sum[x]) % 2 == 0 && Map2[ctot - (ctot - sum[x]) / 2])
		test((ctot - sum[x]) / 2);

	Map2[sum[x]] += 1;

	if (Map1[sum[x]] > Map2[sum[x]])
		test(sum[x]);

	if (ctot - 2 * sum[x] >= sum[x] && Map1[ctot - 2 * sum[x]] > Map2[ctot - 2 * sum[x]])
		test(sum[x]);

	if ((ctot - sum[x]) % 2 == 0 && (ctot - sum[x]) / 2 >= sum[x] && Map1[(ctot - sum[x]) / 2] > Map2[(ctot - sum[x]) / 2])
		test((ctot - sum[x]) / 2);

	if (sum[x] * 2 == ctot)
		ans = min(ans, sum[x]);
	
	for (int i = 0; i < (int) ve[x].size(); i++)
		if (ve[x][i] != f) {
			dfs2(ve[x][i], x);
		}

	Map2[sum[x]] -= 1;
}

int main() {
	scanf("%d", &q);
	while (q--) {
		Map1.clear();
		Map2.clear();
		ans = 1e18;
		scanf("%d", &n);
		ctot = 0;
		for (int i = 1; i <= n; i++) {
			scanf("%d", &c[i]);
			ctot += c[i];
		}
		for (int i = 1; i <= n; i++)
			ve[i].clear();
		for (int i = 1; i < n; i++) {
			int x, y;
			scanf("%d%d", &x, &y);
			ve[x].push_back(y);
			ve[y].push_back(x);
		}
		dfs1(1, 0);
		dfs2(1, 0);
		if (ans == 1e18)
			printf("-1\n");
		else
			printf("%lld\n", ans);
	}
}







In Java :






import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.BitSet;
import java.util.Comparator;
import java.util.InputMismatchException;

public class E {
	InputStream is;
	PrintWriter out;
	String INPUT = "";
	
	void solve()
	{
		for(int T = ni();T > 0;T--){
			int n = ni();
			int[] a = na(n);
			int[] from = new int[n - 1];
			int[] to = new int[n - 1];
			for (int i = 0; i < n - 1; i++) {
				from[i] = ni() - 1;
				to[i] = ni() - 1;
			}
			int[][] g = packU(n, from, to);
			int[][] pars = parents3(g, 0);
			int[] par = pars[0], ord = pars[1], dep = pars[2];
			int[][] rs = makeRights(g, par, 0);
			int[] iord = rs[1], right = rs[2];
			int[][] spar = logstepParents(par);
			
			long[] des = new long[n];
			for(int i = n-1;i >= 0;i--){
				int cur = ord[i];
				des[cur] += a[cur];
				if(i > 0)des[par[cur]] += des[cur];
			}
			
			long W = des[0];
			
			long ret = Long.MAX_VALUE;
			// t=u
			{
				long[] dd = Arrays.copyOf(des, n);
				Arrays.sort(dd);
				for(int i = 0;i < n-1;i++){
					if(dd[i] == dd[i+1] && W-2*dd[i] <= dd[i] && W-2*dd[i] >= 0){
						ret = Math.min(ret, dd[i]-(W-2*dd[i]));
					}
				}
			}
			
			// s=t
			{
				long[][] poss = new long[n][];
				for(int i = 0;i < n;i++){
					poss[i] = new long[]{des[i], iord[i]};
				}
				Arrays.sort(poss, new Comparator<long[]>() {
					public int compare(long[] a, long[] b) {
						if(a[0] != b[0])return Long.compare(a[0], b[0]);
						return Long.compare(a[1], b[1]);
					}
				});
				long[] posv = new long[n];
				for(int i = 0;i < n;i++)posv[i] = poss[i][0];
				for(int i = 0;i < n;i++){
					long t = des[i];
					long u = W-2L*t;
					if(u >= 0 && t >= u){
						int lb = lowerBound(posv, u);
						int ub = lowerBound(posv, u+1);
						if(lb < ub){
							if((int)poss[lb][1] < iord[i]){
								ret = Math.min(ret, t-u);
							}
							if((int)poss[ub-1][1] > right[iord[i]]){
								ret = Math.min(ret, t-u);
							}
						}
					}
				}
			}
			
			// ireko
			// t=u
			{
				for(int i = 0;i < n;i++){
					long u = des[i];
					long s = W-u*2;
					if(s >= 0 && s <= u){
						int cur = i;
						for(int h = spar.length-1;h >= 0;h--){
							int anc = spar[h][cur];
							if(anc == -1)continue;
							if(des[anc] <= 2*u){
								cur = anc;
							}
						}
						if(des[cur] == 2*u){
							ret = Math.min(ret, u-s);
						}
					}
				}
			}
			
			// s=u
			{
				for(int i = 0;i < n;i++){
					long u = des[i];
					long t = W-u*2;
					if(t >= 0 && t <= u){
						int cur = i;
						for(int h = spar.length-1;h >= 0;h--){
							int anc = spar[h][cur];
							if(anc == -1)continue;
							if(des[anc] <= t+u){
								cur = anc;
							}
						}
						if(des[cur] == t+u){
							ret = Math.min(ret, u-t);
						}
					}
				}
			}
			// s=t
			{
				for(int i = 0;i < n;i++){
					long u = des[i];
					if((W-u)%2 != 0)continue;
					long t = (W-u)/2;
					if(t >= 0 && t >= u){
						int cur = i;
						for(int h = spar.length-1;h >= 0;h--){
							int anc = spar[h][cur];
							if(anc == -1)continue;
							if(des[anc] <= t+u){
								cur = anc;
							}
						}
						if(des[cur] == t+u){
							ret = Math.min(ret, t-u);
						}
					}
				}
			}
			if(ret == Long.MAX_VALUE){
				out.println(-1);
			}else{
				out.println(ret);
			}
		}
	}
	
	public static int lowerBound(long[] a, long v)
	{
		int low = -1, high = a.length;
		while(high-low > 1){
			int h = high+low>>>1;
			if(a[h] >= v){
				high = h;
			}else{
				low = h;
			}
		}
		return high;
	}

	
	public static int[] sortByPreorder(int[][] g, int root){
		int n = g.length;
		int[] stack = new int[n];
		int[] ord = new int[n];
		BitSet ved = new BitSet();
		stack[0] = root;
		int p = 1;
		int r = 0;
		ved.set(root);
		while(p > 0){
			int cur = stack[p-1];
			ord[r++] = cur;
			p--;
			for(int e : g[cur]){
				if(!ved.get(e)){
					stack[p++] = e;
					ved.set(e);
				}
			}
		}
		return ord;
	}
	
	public static int[][] makeRights(int[][] g, int[] par, int root)
	{
		int n = g.length;
		int[] ord = sortByPreorder(g, root);
		int[] iord = new int[n];
		for(int i = 0;i < n;i++)iord[ord[i]] = i;
		
		int[] right = new int[n];
		for(int i = n-1;i >= 0;i--){
			int v = i;
			for(int e : g[ord[i]]){
				if(e != par[ord[i]]){
					v = Math.max(v, right[iord[e]]);
				}
			}
			right[i] = v;
		}
		return new int[][]{ord, iord, right};
	}

	
	public static int lca2(int a, int b, int[][] spar, int[] depth) {
		if (depth[a] < depth[b]) {
			b = ancestor(b, depth[b] - depth[a], spar);
		} else if (depth[a] > depth[b]) {
			a = ancestor(a, depth[a] - depth[b], spar);
		}

		if (a == b)
			return a;
		int sa = a, sb = b;
		for (int low = 0, high = depth[a], t = Integer.highestOneBit(high), k = Integer
				.numberOfTrailingZeros(t); t > 0; t >>>= 1, k--) {
			if ((low ^ high) >= t) {
				if (spar[k][sa] != spar[k][sb]) {
					low |= t;
					sa = spar[k][sa];
					sb = spar[k][sb];
				} else {
					high = low | t - 1;
				}
			}
		}
		return spar[0][sa];
	}

	protected static int ancestor(int a, int m, int[][] spar) {
		for (int i = 0; m > 0 && a != -1; m >>>= 1, i++) {
			if ((m & 1) == 1)
				a = spar[i][a];
		}
		return a;
	}

	public static int[][] logstepParents(int[] par) {
		int n = par.length;
		int m = Integer.numberOfTrailingZeros(Integer.highestOneBit(n - 1)) + 1;
		int[][] pars = new int[m][n];
		pars[0] = par;
		for (int j = 1; j < m; j++) {
			for (int i = 0; i < n; i++) {
				pars[j][i] = pars[j - 1][i] == -1 ? -1 : pars[j - 1][pars[j - 1][i]];
			}
		}
		return pars;
	}


	public static int[][] parents3(int[][] g, int root) {
		int n = g.length;
		int[] par = new int[n];
		Arrays.fill(par, -1);

		int[] depth = new int[n];
		depth[0] = 0;

		int[] q = new int[n];
		q[0] = root;
		for (int p = 0, r = 1; p < r; p++) {
			int cur = q[p];
			for (int nex : g[cur]) {
				if (par[cur] != nex) {
					q[r++] = nex;
					par[nex] = cur;
					depth[nex] = depth[cur] + 1;
				}
			}
		}
		return new int[][] { par, q, depth };
	}

	static int[][] packU(int n, int[] from, int[] to) {
		int[][] g = new int[n][];
		int[] p = new int[n];
		for (int f : from)
			p[f]++;
		for (int t : to)
			p[t]++;
		for (int i = 0; i < n; i++)
			g[i] = new int[p[i]];
		for (int i = 0; i < from.length; i++) {
			g[from[i]][--p[from[i]]] = to[i];
			g[to[i]][--p[to[i]]] = from[i];
		}
		return g;
	}

	
	void run() throws Exception
	{
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		long s = System.currentTimeMillis();
		solve();
		out.flush();
		if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
	}
	
	public static void main(String[] args) throws Exception { new E().run(); }
	
	private byte[] inbuf = new byte[1024];
	private int lenbuf = 0, ptrbuf = 0;
	
	private int readByte()
	{
		if(lenbuf == -1)throw new InputMismatchException();
		if(ptrbuf >= lenbuf){
			ptrbuf = 0;
			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
			if(lenbuf <= 0)return -1;
		}
		return inbuf[ptrbuf++];
	}
	
	private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
	private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private double nd() { return Double.parseDouble(ns()); }
	private char nc() { return (char)skip(); }
	
	private String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){ // when nextLine, (isSpaceChar(b) && b != ' ')
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private char[] ns(int n)
	{
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while(p < n && !(isSpaceChar(b))){
			buf[p++] = (char)b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}
	
	private char[][] nm(int n, int m)
	{
		char[][] map = new char[n][];
		for(int i = 0;i < n;i++)map[i] = ns(m);
		return map;
	}
	
	private int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private int ni()
	{
		int num = 0, b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private long nl()
	{
		long num = 0;
		int b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static void tr(Object... o) 
{ System.out.println(Arrays.deepToString(o)); }
}









In C :





#include <stdio.h>
#include <stdlib.h>
#define HASH_SIZE 123455
typedef struct _lnode{
  int x;
  int w;
  struct _lnode *next;
} lnode;
typedef struct _node{
  long long x;
  int c;
  long long ans;
  struct _node *next;
} node;
void clean_table();
void insert_edge(int x,int y,int w);
void dfs0(int x,int y);
void dfs1(int x,int y);
void insert(long long x,long long ans);
void removee(long long x,long long ans);
void search(long long x);
void freehash();
int a[50000];
long long sub[50000],min,sum;
lnode *table[50000]={0};
node *hash[HASH_SIZE]={0};

int main(){
  int T,n,x,y,i;
  scanf("%d",&T);
  while(T--){
    scanf("%d",&n);
    for(i=sum=0;i<n;i++){
      scanf("%d",a+i);
      sum+=a[i];
    }
    for(i=0;i<n-1;i++){
      scanf("%d%d",&x,&y);
      insert_edge(x-1,y-1,1);
    }
    dfs0(0,-1);
    min=-1;
    dfs1(0,-1);
    printf("%lld\n",min);
    clean_table();
    freehash();
  }
  return 0;
}
void clean_table(){
  int i;
  lnode *p,*pp;
  for(i=0;i<50000;i++)
    if(table[i]){
      p=table[i];
      while(p){
        pp=p->next;
        free(p);
        p=pp;
      }
      table[i]=NULL;
    }
  return;
}
void insert_edge(int x,int y,int w){
  lnode *t=malloc(sizeof(lnode));
  t->x=y;
  t->w=w;
  t->next=table[x];
  table[x]=t;
  t=malloc(sizeof(lnode));
  t->x=x;
  t->w=w;
  t->next=table[y];
  table[y]=t;
  return;
}
void dfs0(int x,int y){
  lnode *p;
  sub[x]=a[x];
  for(p=table[x];p;p=p->next)
    if(p->x!=y){
      dfs0(p->x,x);
      sub[x]+=sub[p->x];
    }
  return;
}
void dfs1(int x,int y){
  lnode *p;
  long long down,up;
  search(sub[x]);
  down=sub[x];
  up=sum-sub[x];
  if(down==up && min==-1)
    min=down;
  if(down%2==0 && down/2*3>=sum)
    insert(down/2,down/2*3-sum);
  if(up<down && up*3>=sum){
    insert(up,up*3-sum);
    insert(down-up,up*3-sum);
  }
  for(p=table[x];p;p=p->next)
    if(p->x!=y)
      dfs1(p->x,x);
  if(down%2==0 && down/2*3>=sum)
    removee(down/2,down/2*3-sum);
  if(up<down && up*3>=sum){
    removee(up,up*3-sum);
    removee(down-up,up*3-sum);
  }
  if(up%2==0 && up/2*3>=sum)
    insert(up/2,up/2*3-sum);
  if(down<up && down*3>=sum){
    insert(down,down*3-sum);
    insert(up-down,down*3-sum);
  }
  return;
}
void insert(long long x,long long ans){
  int bucket=x%HASH_SIZE;
  node *t=hash[bucket];
  while(t){
    if(t->x==x && t->ans==ans){
      t->c++;
      return;
    }
    t=t->next;
  }
  t=(node*)malloc(sizeof(node));
  t->x=x;
  t->ans=ans;
  t->c=1;
  t->next=hash[bucket];
  hash[bucket]=t;
  return;
}
void removee(long long x,long long ans){
  int bucket=x%HASH_SIZE;
  node *t=hash[bucket],*p=NULL;
  while(t){
    if(t->x==x && t->ans==ans){
      t->c--;
      if(!t->c){
        if(!p){
          hash[bucket]=t->next;
          free(t);
        }
        else{
          p->next=t->next;
          free(t);
        }
      }
      return;
    }
    p=t;
    t=t->next;
  }
  return;
}
void search(long long x){
  int bucket=x%HASH_SIZE;
  node *t=hash[bucket];
  while(t){
    if(t->x==x)
      if(min==-1 || t->ans<min)
        min=t->ans;
    t=t->next;
  }
  return;
}
void freehash(){
  int i;
  node *t,*p;
  for(i=0;i<HASH_SIZE;i++){
    t=hash[i];
    while(t){
      p=t->next;
      free(t);
      t=p;
    }
    hash[i]=NULL;
  }
  return;
}









In python3 :







from random import randrange
import sys
sys.setrecursionlimit(10**5)

def r():
    return list(map(int, input().split()))

def bin_search(arr, pred, lo=0):
    hi = len(arr)
    while lo < hi:
        mid = (lo+hi)//2
        if pred(arr[mid]):
            hi = mid
        else: 
            lo = mid + 1
    return lo if lo < len(arr) else None

class UndirectedGraph(object):
    def __init__(self, size):
        self.size = size
        self.M = 0
        self.edges = [set([]) for _ in range(self.size)]

    def add_edge(self, u, v):
        """Adds edge to graph if it doesn't already exist."""
        if v not in self.edges[u]:
            self.edges[u].add(v)
            self.edges[v].add(u)
            self.M += 1

    def neighbors(self, u):
        for v in self.edges[u]:
            yield v

class WeightedTree(UndirectedGraph):
    def __init__(self, size, weights):
        super().__init__(size)
        self.weights = weights
        self.labels = [None]*self.size
        self.cum_weights = [None]*self.size
        self.inverted = [False]*self.size
        self.nextnode = 0
        self.num_children = [None]*self.size
        self.root = 0

    def initialize(self):
        self.set_labels(self.root)
        self.sorted_nodes = sorted(list(range(self.size)), key = lambda x : self.cum_weights[x])

    def set_labels(self, root):
        self.dfs(root, set([]))
        self.total_weight = self.cum_weights[self.root]
        for i, cw in enumerate(self.cum_weights):
            if cw > self.total_weight - cw:
                self.cum_weights[i] = self.total_weight - cw
                self.inverted[i] = True

    def dfs(self, u, visited):
        visited.add(u)
        self.labels[u] = self.nextnode
        self.nextnode += 1
        self.cum_weights[u] = self.weights[u]
        self.num_children[u] = 0
        for v in self.neighbors(u):
            if v not in visited:
                cw, nchildren = self.dfs(v, visited)
                self.cum_weights[u] += cw
                self.num_children[u] += nchildren + 1
        return self.cum_weights[u], self.num_children[u]

    def solve(self):
        tw = self.total_weight
        first_idx = bin_search(self.sorted_nodes, lambda u : self.cum_weights[u] >= (tw+2)//3)
        if first_idx is None:
            return None
        while first_idx < self.size:
            first = self.sorted_nodes[first_idx]
            first_cw = self.cum_weights[first]
            label1 = self.labels[first]
            last_child_label = label1 + self.num_children[first]

            for target_w in (tw - first_cw * 2, first_cw):
                second_idx = bin_search(self.sorted_nodes, lambda u: self.cum_weights[u] >= target_w)
                if second_idx is not None:
                    while (second_idx < self.size
                            and self.cum_weights[self.sorted_nodes[second_idx]] == target_w):
                        second = self.sorted_nodes[second_idx]
                        is_child = label1 < self.labels[second] <= last_child_label
                        is_child = is_child if not self.inverted[first] else not is_child
                        if not is_child and self.labels[second] != label1:
                            return first_cw * 3 - tw
                        second_idx += 1

            first_idx += 1
        if 2*first_cw == tw:
            return first_cw
        return None


def read_input():
    n = int(input())
    weights = r()
    G = WeightedTree(n, weights)
    for _ in range(n-1):
        u, v = r()
        u -= 1
        v -= 1
        G.add_edge(u, v)
    G.initialize()
    return G

def main():
    q = int(input())
    for _ in range(q):
        G = read_input()
        ans = G.solve()
        print(ans if ans is not None else -1)
main()
                        




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