https://www.spoj.com/problems/COT3/
https://blog.csdn.net/huzecong/article/details/9142121
https://my.oschina.net/u/4321213/blog/3694105
非常 nice 的题目… 同时考察组合博弈和线段树合并。。。
不妨先考虑简单的问题,如何输出第一步的方案。
我们枚举第一步落在哪里,那么形成的黑链会把原树划分为若干子树,每组子树相互独立,于是变成游戏的和,
根据 SG 定理,我们只要能求出每个节点为根的 sg 值即可。
void gao(int u = 1, int p = -1, int SG = 0) { // 递归进来的 SG 表示子树外的游戏 for (auto v: adj[u]) if (v != p) SG ^= sg[v]; // 再 xor 上子树内的游戏 if (!col[u] && !SG) Z.PB(u); for (auto v: adj[u]) if (v != p) gao(v, u, SG^sg[v]); }
再考虑如何求以 u 为根的子树的 sg 值,我们考虑第一步怎么走,并且维护所有能转移到的 sg 值的集合。
讨论节点 u 的颜色。
如果是黑色,我们还是枚举第一步落在子树中哪一个节点里,那么和上面的情况一样,也是拆分成了若干游戏的和。
如果是白色,那么多一种操作,把这个 sg 也丢到集合里。(对应下面 !col[u] 的情况)
最后把所有 sg 值的集合 mex 一下即可。
这样暴力 O(n2) 的算法就有了。
void dfs(int u = 1, int p = -1) { int s = 0; for (auto v: adj[u]) if (v != p) { dfs(v, u); s ^= sg[v]; } if (!col[u]) Init(rt[u], LV, s); for (auto v: adj[u]) if (v != p) { put_xor(rt[v], LV, s ^ sg[v]); rt[u] = Merge(rt[u], rt[v]); } sg[u] = Mex(rt[u], LV); }
考虑优化,我们可以用 01-Trie 去维护 sg 值的集合,那么上面的枚举过程就不需要进入子树枚举。
而只要分成 v 节点中的 sg 集合,和 v 节点外的组成的 sg 两个状态即可,而这个相当于给整个集合打 xor tag 然后再合并。
而 01-Trie 和线段树结构几乎一样,稍微改改即可。
复杂度 O(nlogn)。
/* This code has been written by MinakoKojima, feel free to ask me question. Blog: https://www.shuizilong.com/house Template Date: 2015.10.12 Note: ... */ #pragma comment(linker, "/STACK:36777216") //#pragma GCC optimize ("O2") #define LOCAL #include <functional> #include <algorithm> #include <iostream> #include <fstream> #include <sstream> #include <iomanip> #include <numeric> #include <cstring> #include <climits> #include <cassert> #include <complex> #include <cstdio> #include <string> #include <vector> #include <bitset> #include <queue> #include <stack> #include <cmath> #include <ctime> #include <list> #include <set> #include <map> //#include <tr1/unordered_set> //#include <tr1/unordered_map> //#include <array> using namespace std; #define REP(i, n) for (int i=0;i<n;++i) #define FOR(i, a, b) for (int i=a;i<b;++i) #define DWN(i, b, a) for (int i=b-1;i>=a;--i) #define REP_1(i, n) for (int i=1;i<=n;++i) #define FOR_1(i, a, b) for (int i=a;i<=b;++i) #define DWN_1(i, b, a) for (int i=b;i>=a;--i) #define REP_C(i, n) for (int n____=n,i=0;i<n____;++i) #define FOR_C(i, a, b) for (int b____=b,i=a;i<b____;++i) #define DWN_C(i, b, a) for (int a____=a,i=b-1;i>=a____;--i) #define REP_N(i, n) for (i=0;i<n;++i) #define FOR_N(i, a, b) for (i=a;i<b;++i) #define DWN_N(i, b, a) for (i=b-1;i>=a;--i) #define REP_1_C(i, n) for (int n____=n,i=1;i<=n____;++i) #define FOR_1_C(i, a, b) for (int b____=b,i=a;i<=b____;++i) #define DWN_1_C(i, b, a) for (int a____=a,i=b;i>=a____;--i) #define REP_1_N(i, n) for (i=1;i<=n;++i) #define FOR_1_N(i, a, b) for (i=a;i<=b;++i) #define DWN_1_N(i, b, a) for (i=b;i>=a;--i) #define REP_C_N(i, n) for (int n____=(i=0,n);i<n____;++i) #define FOR_C_N(i, a, b) for (int b____=(i=0,b);i<b____;++i) #define DWN_C_N(i, b, a) for (int a____=(i=b-1,a);i>=a____;--i) #define REP_1_C_N(i, n) for (int n____=(i=1,n);i<=n____;++i) #define FOR_1_C_N(i, a, b) for (int b____=(i=a,b);i<=b____;++i) #define DWN_1_C_N(i, b, a) for (int a____=(i=b,a);i>=a____;--i) #define ECH(it, A) for (__typeof((A).begin()) it=(A).begin(); it != (A).end(); ++it) #define rECH(it, A) for (__typeof((A).rbegin()) it=(A).rbegin(); it != (A).rend(); ++it) #define REP_S(i, str) for (char*i=str;*i;++i) #define REP_L(i, hd, suc) for (int i=hd;i;i=suc[i]) #define REP_G(i, u) REP_L(i,hd[u],suc) #define REP_SS(x, s) for (int x=s;x;x=(x-1)&s) #define DO(n) for ( int ____n = n; ____n-->0; ) #define REP_2(i, j, n, m) REP(i, n) REP(j, m) #define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m) #define REP_3(i, j, k, n, m, l) REP(i, n) REP(j, m) REP(k, l) #define REP_3_1(i, j, k, n, m, l) REP_1(i, n) REP_1(j, m) REP_1(k, l) #define REP_4(i, j, k, ii, n, m, l, nn) REP(i, n) REP(j, m) REP(k, l) REP(ii, nn) #define REP_4_1(i, j, k, ii, n, m, l, nn) REP_1(i, n) REP_1(j, m) REP_1(k, l) REP_1(ii, nn) #define ALL(A) A.begin(), A.end() #define LLA(A) A.rbegin(), A.rend() #define CPY(A, B) memcpy(A, B, sizeof(A)) #define INS(A, P, B) A.insert(A.begin() + P, B) #define ERS(A, P) A.erase(A.begin() + P) #define LBD(A, x) (lower_bound(ALL(A), x) - A.begin()) #define UBD(A, x) (upper_bound(ALL(A), x) - A.begin()) #define CTN(T, x) (T.find(x) != T.end()) #define SZ(A) int((A).size()) #define PB push_back #define MP(A, B) make_pair(A, B) #define PTT pair<T, T> #define Ts *this #define rTs return Ts #define fi first #define se second #define re real() #define im imag() #define Rush for(int ____T=RD(); ____T--;) #define Display(A, n, m) { \ REP(i, n){ \ REP(j, m-1) cout << A[i][j] << " "; \ cout << A[i][m-1] << endl; \ } \ } #define Display_1(A, n, m) { \ REP_1(i, n){ \ REP_1(j, m-1) cout << A[i][j] << " "; \ cout << A[i][m] << endl; \ } \ } typedef long long LL; //typedef long double DB; typedef double DB; typedef unsigned uint; typedef unsigned long long uLL; typedef vector<int> VI; typedef vector<char> VC; typedef vector<string> VS; typedef vector<LL> VL; typedef vector<DB> VF; typedef set<int> SI; typedef set<string> SS; typedef map<int, int> MII; typedef map<string, int> MSI; typedef pair<int, int> PII; typedef pair<LL, LL> PLL; typedef vector<PII> VII; typedef vector<VI> VVI; typedef vector<VII> VVII; template<class T> inline T& RD(T &); template<class T> inline void OT(const T &); //inline int RD(){int x; return RD(x);} inline LL RD(){LL x; return RD(x);} inline DB& RF(DB &); inline DB RF(){DB x; return RF(x);} inline char* RS(char *s); inline char& RC(char &c); inline char RC(); inline char& RC(char &c){scanf(" %c", &c); return c;} inline char RC(){char c; return RC(c);} //inline char& RC(char &c){c = getchar(); return c;} //inline char RC(){return getchar();} template<class T> inline T& RDD(T &); inline LL RDD(){LL x; return RDD(x);} template<class T0, class T1> inline T0& RD(T0 &x0, T1 &x1){RD(x0), RD(x1); return x0;} template<class T0, class T1, class T2> inline T0& RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2); return x0;} template<class T0, class T1, class T2, class T3> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3); return x0;} template<class T0, class T1, class T2, class T3, class T4> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4); return x0;} template<class T0, class T1, class T2, class T3, class T4, class T5> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5); return x0;} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6); return x0;} template<class T0, class T1> inline void OT(const T0 &x0, const T1 &x1){OT(x0), OT(x1);} template<class T0, class T1, class T2> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2){OT(x0), OT(x1), OT(x2);} template<class T0, class T1, class T2, class T3> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);} template<class T0, class T1, class T2, class T3, class T4> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4, const T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4, const T5 &x5, const T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);} inline char& RC(char &a, char &b){RC(a), RC(b); return a;} inline char& RC(char &a, char &b, char &c){RC(a), RC(b), RC(c); return a;} inline char& RC(char &a, char &b, char &c, char &d){RC(a), RC(b), RC(c), RC(d); return a;} inline char& RC(char &a, char &b, char &c, char &d, char &e){RC(a), RC(b), RC(c), RC(d), RC(e); return a;} inline char& RC(char &a, char &b, char &c, char &d, char &e, char &f){RC(a), RC(b), RC(c), RC(d), RC(e), RC(f); return a;} inline char& RC(char &a, char &b, char &c, char &d, char &e, char &f, char &g){RC(a), RC(b), RC(c), RC(d), RC(e), RC(f), RC(g); return a;} inline DB& RF(DB &a, DB &b){RF(a), RF(b); return a;} inline DB& RF(DB &a, DB &b, DB &c){RF(a), RF(b), RF(c); return a;} inline DB& RF(DB &a, DB &b, DB &c, DB &d){RF(a), RF(b), RF(c), RF(d); return a;} inline DB& RF(DB &a, DB &b, DB &c, DB &d, DB &e){RF(a), RF(b), RF(c), RF(d), RF(e); return a;} inline DB& RF(DB &a, DB &b, DB &c, DB &d, DB &e, DB &f){RF(a), RF(b), RF(c), RF(d), RF(e), RF(f); return a;} inline DB& RF(DB &a, DB &b, DB &c, DB &d, DB &e, DB &f, DB &g){RF(a), RF(b), RF(c), RF(d), RF(e), RF(f), RF(g); return a;} inline void RS(char *s1, char *s2){RS(s1), RS(s2);} inline void RS(char *s1, char *s2, char *s3){RS(s1), RS(s2), RS(s3);} template<class T0,class T1>inline T0& RDD(T0&a, T1&b){RDD(a),RDD(b); return a;} template<class T0,class T1,class T2>inline T1& RDD(T0&a, T1&b, T2&c){RDD(a),RDD(b),RDD(c); return a;} template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));} template<class T> inline void FLC(T &A, int x){memset(A, x, sizeof(A));} template<class T> inline void CLR(T &A){A.clear();} template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);} template<class T0, class T1, class T2> inline void RST(T0 &A0, T1 &A1, T2 &A2){RST(A0), RST(A1), RST(A2);} template<class T0, class T1, class T2, class T3> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3){RST(A0), RST(A1), RST(A2), RST(A3);} template<class T0, class T1, class T2, class T3, class T4> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5), RST(A6);} template<class T0, class T1> inline void FLC(T0 &A0, T1 &A1, int x){FLC(A0, x), FLC(A1, x);} template<class T0, class T1, class T2> inline void FLC(T0 &A0, T1 &A1, T2 &A2, int x){FLC(A0, x), FLC(A1, x), FLC(A2, x);} template<class T0, class T1, class T2, class T3> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, int x){FLC(A0, x), FLC(A1, x), FLC(A2, x), FLC(A3, x);} template<class T0, class T1, class T2, class T3, class T4> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, int x){FLC(A0, x), FLC(A1, x), FLC(A2, x), FLC(A3, x), FLC(A4, x);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, int x){FLC(A0, x), FLC(A1, x), FLC(A2, x), FLC(A3, x), FLC(A4, x), FLC(A5, x);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6, int x){FLC(A0, x), FLC(A1, x), FLC(A2, x), FLC(A3, x), FLC(A4, x), FLC(A5, x), FLC(A6, x);} template<class T> inline void CLR(priority_queue<T> &Q){while (!Q.empty()) Q.pop();} template<class T> inline void CLR(stack<T> &S){while (!S.empty()) S.pop();} template<class T> inline void CLR(queue<T> &Q){while (!Q.empty()) Q.pop();} template<class T0, class T1> inline void CLR(T0 &A0, T1 &A1){CLR(A0), CLR(A1);} template<class T0, class T1, class T2> inline void CLR(T0 &A0, T1 &A1, T2 &A2){CLR(A0), CLR(A1), CLR(A2);} template<class T0, class T1, class T2, class T3> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3){CLR(A0), CLR(A1), CLR(A2), CLR(A3);} template<class T0, class T1, class T2, class T3, class T4> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5), CLR(A6);} template<class T> inline void CLR(T &A, int n){REP(i, n) CLR(A[i]);} template<class T> inline bool EPT(T &a){return a.empty();} template<class T> inline T& SRT(T &A){sort(ALL(A)); return A;} template<class T, class C> inline T& SRT(T &A, C cmp){sort(ALL(A), cmp); return A;} template<class T> inline T& RVS(T &A){reverse(ALL(A)); return A;} template<class T> inline T& UNQQ(T &A){A.resize(unique(ALL(A))-A.begin());return A;} template<class T> inline T& UNQ(T &A){SRT(A);return UNQQ(A);} template<class T, class C> inline T& UNQ(T &A, C cmp){SRT(A, cmp);return UNQQ(A);} //} /** Constant List .. **/ //{ const int MOD = int(1e9) + 7; const int INF = 0x3f3f3f3f; const LL INFF = 0x3f3f3f3f3f3f3f3fLL; const DB EPS = 1e-9; const DB OO = 1e20; const DB PI = acos(-1.0); //M_PI; const int dx[] = {-1, 1, 0, 0}; const int dy[] = {0, 0, 1, -1}; //} /** Add On .. **/ //{ // <<= '0. Nichi Joo ., //{ template<class T> inline bool checkMin(T &a,const T b){return b < a ? a = b, 1 : 0;} template<class T> inline bool checkMax(T &a,const T b){return a < b ? a = b, 1 : 0;} template <class T, class C> inline bool checkUpd(T& a, const T b, C c){return c(b,a) ? a = b, 1 : 0;} template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);} template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);} template<class T> inline T min(T a, T b, T c, T d){return min(min(a, b), min(c, d));} template<class T> inline T max(T a, T b, T c, T d){return max(max(a, b), max(c, d));} template<class T> inline T min(T a, T b, T c, T d, T e){return min(min(min(a,b),min(c,d)),e);} template<class T> inline T max(T a, T b, T c, T d, T e){return max(max(max(a,b),max(c,d)),e);} template<class T> inline T sqr(T a){return a*a;} template<class T> inline T cub(T a){return a*a*a;} template<class T> inline T ceil(T x, T y){return (x - 1) / y + 1;} template<class T> T abs(T x){return x>0?x:-x;} inline int sgn(DB x){return x < -EPS ? -1 : x > EPS;} inline int sgn(DB x, DB y){return sgn(x - y);} inline DB cos(DB a, DB b, DB c){return (sqr(a)+sqr(b)-sqr(c))/(2*a*b);} inline DB cot(DB x){return 1./tan(x);}; inline DB sec(DB x){return 1./cos(x);}; inline DB csc(DB x){return 1./sin(x);}; //} // <<= '1. Bitwise Operation ., //{ namespace BO{ inline bool _1(int x, int i){return bool(x&1<<i);} inline bool _1(LL x, int i){return bool(x&1LL<<i);} inline LL _1(int i){return 1LL<<i;} inline LL _U(int i){return _1(i) - 1;}; inline int reverse_bits(int x){ x = ((x >> 1) & 0x55555555) | ((x << 1) & 0xaaaaaaaa); x = ((x >> 2) & 0x33333333) | ((x << 2) & 0xcccccccc); x = ((x >> 4) & 0x0f0f0f0f) | ((x << 4) & 0xf0f0f0f0); x = ((x >> 8) & 0x00ff00ff) | ((x << 8) & 0xff00ff00); x = ((x >>16) & 0x0000ffff) | ((x <<16) & 0xffff0000); return x; } inline LL reverse_bits(LL x){ x = ((x >> 1) & 0x5555555555555555LL) | ((x << 1) & 0xaaaaaaaaaaaaaaaaLL); x = ((x >> 2) & 0x3333333333333333LL) | ((x << 2) & 0xccccccccccccccccLL); x = ((x >> 4) & 0x0f0f0f0f0f0f0f0fLL) | ((x << 4) & 0xf0f0f0f0f0f0f0f0LL); x = ((x >> 8) & 0x00ff00ff00ff00ffLL) | ((x << 8) & 0xff00ff00ff00ff00LL); x = ((x >>16) & 0x0000ffff0000ffffLL) | ((x <<16) & 0xffff0000ffff0000LL); x = ((x >>32) & 0x00000000ffffffffLL) | ((x <<32) & 0xffffffff00000000LL); return x; } template<class T> inline bool odd(T x){return x&1;} template<class T> inline bool even(T x){return !odd(x);} template<class T> inline T low_bit(T x) {return x & -x;} template<class T> inline T high_bit(T x) {T p = low_bit(x);while (p != x) x -= p, p = low_bit(x);return p;} template<class T> inline T cover_bit(T x){T p = 1; while (p < x) p <<= 1;return p;} template<class T> inline int cover_idx(T x){int p = 0; while (_1(p) < x ) ++p; return p;} inline int clz(int x){return __builtin_clz(x);} inline int clz(LL x){return __builtin_clzll(x);} inline int ctz(int x){return __builtin_ctz(x);} inline int ctz(LL x){return __builtin_ctzll(x);} inline int lg2(int x){return !x ? -1 : 31 - clz(x);} inline int lg2(LL x){return !x ? -1 : 63 - clz(x);} inline int low_idx(int x){return !x ? -1 : ctz(x);} inline int low_idx(LL x){return !x ? -1 : ctz(x);} inline int high_idx(int x){return lg2(x);} inline int high_idx(LL x){return lg2(x);} inline int parity(int x){return __builtin_parity(x);} inline int parity(LL x){return __builtin_parityll(x);} inline int count_bits(int x){return __builtin_popcount(x);} inline int count_bits(LL x){return __builtin_popcountll(x);} } using namespace BO;//} // <<= '2. Number Theory .,//{ namespace NT{ //#define gcd __gcd inline LL gcd(LL a, LL b){return b?gcd(b,a%b):a;} inline LL lcm(LL a, LL b){return a*b/gcd(a,b);} inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;} inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;} /* 模数两倍刚好超 int 时。 inline int sum(uint a, int b){a += b; a %= MOD;if (a < 0) a += MOD; return a;} inline void INC(int &a, int b){a = sum(a, b);} */ inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;} inline int dff(int a, int b){a -= b; if (a < 0) a += MOD; return a;} inline void MUL(int &a, int b){a = (LL)a * b % MOD;} //inline int pdt(int a, int b){return (LL)a * b % MOD;} inline int pdt(int x,int y) { int ret; __asm__ __volatile__ ("\tmull %%ebx\n\tdivl %%ecx\n":"=d"(ret):"a"(x),"b"(y),"c"(MOD)); return ret; } inline int gcd(int m, int n, int &x, int &y){ x = 1, y = 0; int xx = 0, yy = 1, q; while (1){ q = m / n, m %= n; if (!m){x = xx, y = yy; return n;} DEC(x, pdt(q, xx)), DEC(y, pdt(q, yy)); q = n / m, n %= m; if (!n) return m; DEC(xx, pdt(q, x)), DEC(yy, pdt(q, y)); } } inline int sum(int a, int b, int c){return sum(a, sum(b, c));} inline int sum(int a, int b, int c, int d){return sum(sum(a, b), sum(c, d));} inline int pdt(int a, int b, int c){return pdt(a, pdt(b, c));} inline int pdt(int a, int b, int c, int d){return pdt(pdt(a, b), pdt(c, d));} inline int pow(int a, LL b){ int c(1); while (b){ if (b&1) MUL(c, a); MUL(a, a), b >>= 1; } return c; } template<class T> inline T pow(T a, LL b){ T c(1); while (b){ if (b&1) c *= a; a *= a, b >>= 1; } return c; } template<class T> inline T pow(T a, int b){ return pow(a, (LL)b); } inline int _I(int b){ int a = MOD, x1 = 0, x2 = 1, q; while (1){ q = a / b, a %= b; if (!a) return x2; DEC(x1, pdt(q, x2)); q = b / a, b %= a; if (!b) return x1; DEC(x2, pdt(q, x1)); } } inline void DIV(int &a, int b){MUL(a, _I(b));} inline int qtt(int a, int b){return pdt(a, _I(b));} struct Int{ int val; operator int() const{return val;} Int(int _val = 0):val(_val){ val %= MOD; if (val < 0) val += MOD; } Int(LL _val):val(_val){ _val %= MOD; if (_val < 0) _val += MOD; val = _val; } Int& operator +=(const int& rhs){INC(val, rhs);rTs;} Int operator +(const int& rhs) const{return sum(val, rhs);} Int& operator -=(const int& rhs){DEC(val, rhs);rTs;} Int operator -(const int& rhs) const{return dff(val, rhs);} Int& operator *=(const int& rhs){MUL(val, rhs);rTs;} Int operator *(const int& rhs) const{return pdt(val, rhs);} Int& operator /=(const int& rhs){DIV(val, rhs);rTs;} Int operator /(const int& rhs) const{return qtt(val, rhs);} Int operator-()const{return MOD-*this;} }; } using namespace NT;//} //} /** I/O Accelerator Interface .. **/ //{ #define g (c=getchar()) #define d isdigit(g) #define p x=x*10+c-'0' #define n x=x*10+'0'-c #define pp l/=10,p #define nn l/=10,n template<class T> inline T& RD(T &x){ char c;while(!d);x=c-'0';while(d)p; return x; } template<class T> inline T& RDD(T &x){ char c;while(g,c!='-'&&!isdigit(c)); if (c=='-'){x='0'-g;while(d)n;} else{x=c-'0';while(d)p;} return x; } inline DB& RF(DB &x){ //scanf("%lf", &x); char c;while(g,c!='-'&&c!='.'&&!isdigit(c)); if(c=='-')if(g=='.'){x=0;DB l=1;while(d)nn;x*=l;} else{x='0'-c;while(d)n;if(c=='.'){DB l=1;while(d)nn;x*=l;}} else if(c=='.'){x=0;DB l=1;while(d)pp;x*=l;} else{x=c-'0';while(d)p;if(c=='.'){DB l=1;while(d)pp;x*=l;}} return x; } #undef nn #undef pp #undef n #undef p #undef d #undef g inline char* RS(char *s){ //gets(s); scanf("%s", s); return s; } LL last_ans; int Case; template<class T> inline void OT(const T &x){ //printf("Case #%d: ", ++Case); //printf("%lld\n", x); //printf("%I64d\n", x); //printf("%.9f\n", x); printf("%d\n", x); //cout << x << endl; //last_ans = x; } //}/* .................................................................................................................................. */ const int N = int(1e5) + 9; #define ri register int #define rep(io, st, ed) for(ri io = st; io <= ed; io ++) #define drep(io, ed, st) for(ri io = ed; io >= st; io --) namespace Chairman_Tree { #define lx c[0][x] #define rx c[1][x] #define ly c[0][y] #define ry c[1][y] #define ml ((l+r)>>1) #define mr (ml+1) #define lc lx, lv #define rc rx, lv const int NN = 20*N; const int LV = 17; int c[2][NN], xr[NN]; bool cv[NN]; int tot; int new_node() { ++tot; return tot; } void put_xor(int x, int lv, int v) { if (_1(v, lv)) swap(lx, rx); xr[x] ^= v; } void upd(int x) { cv[x] = cv[lx] && cv[rx]; } void rls(int x, int lv) { if (xr[x]) { --lv; put_xor(rc, xr[x]); put_xor(lc, xr[x]); xr[x] = 0; } } void Init(int &x, int lv, int v) { x = new_node(); if (!~lv) cv[x] = 1; else { if (_1(v, lv--)) Init(rc, v); else Init(lc, v); } } int Merge(int x, int y, int lv = LV) { if (!x || !y) return x | y; if (!~lv) { cv[x] |= cv[y]; return x; } rls(x, lv); rls(y, lv--); lx = Merge(lx, ly, lv); rx = Merge(rx, ry, lv); upd(x); return x; } int Mex(int x, int lv = LV) { if (!x || !~lv) return 0; rls(x, lv--); return cv[lx] ? _1(lv+1) + Mex(rc) : Mex(lc); } } using namespace Chairman_Tree; VI adj[N]; int rt[N], col[N], sg[N]; int n; void dfs(int u = 1, int p = -1) { int s = 0; for (auto v: adj[u]) if (v != p) { dfs(v, u); s ^= sg[v]; } if (!col[u]) Init(rt[u], LV, s); for (auto v: adj[u]) if (v != p) { put_xor(rt[v], LV, s ^ sg[v]); rt[u] = Merge(rt[u], rt[v]); } sg[u] = Mex(rt[u], LV); } VI Z; void gao(int u = 1, int p = -1, int SG = 0) { for (auto v: adj[u]) if (v != p) SG ^= sg[v]; if (!col[u] && !SG) Z.PB(u); for (auto v: adj[u]) if (v != p) gao(v, u, SG^sg[v]); } int main() { #ifndef ONLINE_JUDGE freopen("in.txt", "r", stdin); #endif RD(n); REP_1(i, n) RD(col[i]); DO(n-1) { int x, y; RD(x, y); adj[x].PB(y); adj[y].PB(x); } dfs(); if (sg[1]) { gao(); SRT(Z); for (auto z: Z) OT(z); } else { puts("-1"); } }