The problem of finding the minimal string obtained by XORing a binary string with all possible values can be approached using suffix sorting techniques. Each XOR operation with (2^k) swaps adjacent blocks of length (2^k), and any XOR value is a composition of such transformations.

Define arrays (sa_{k,x}) and (rk_{k,x}) representing the sorted order of prefixes of length (2^k) for strings formed by XORing with value (x). The prefix of length (2^{k+1}) is formed by concatenating two such prefixes. This leads to a suffix sorting adaptation where (rk[i+w]) becomes (rk[i \oplus w]).

#include <iostream>
#include <vector>
#include <cstring>
using namespace std;

int main() {
    int n;
    cin >> n;
    string str;
    cin >> str;
    int size = 1 << n;
    vector<int> bucket(27, 0), suffixArray(size), rank(size), tempRank(size), order(size);
    for (int i = 0; i < size; ++i) {
        rank[i] = str[i] - 'a' + 1;
        bucket[rank[i]]++;
    }
    for (int i = 1; i <= 26; ++i) bucket[i] += bucket[i - 1];
    for (int i = size - 1; i >= 0; --i) suffixArray[--bucket[rank[i]]] = i;
    int distinct = 26;
    for (int step = 1; step <= size; step <<= 1) {
        fill(bucket.begin(), bucket.begin() + distinct + 1, 0);
        for (int i = 0; i < size; ++i) bucket[rank[suffixArray[i] ^ step]]++;
        for (int i = 1; i <= distinct; ++i) bucket[i] += bucket[i - 1];
        for (int i = size - 1; i >= 0; --i) order[--bucket[rank[suffixArray[i] ^ step]]] = suffixArray[i];
        fill(bucket.begin(), bucket.begin() + distinct + 1, 0);
        for (int i = 0; i < size; ++i) bucket[rank[order[i]]]++;
        for (int i = 1; i <= distinct; ++i) bucket[i] += bucket[i - 1];
        for (int i = size - 1; i >= 0; --i) suffixArray[--bucket[rank[order[i]]]] = order[i];
        tempRank = rank;
        rank[suffixArray[0]] = 1;
        distinct = 1;
        for (int i = 1; i < size; ++i) {
            int curr = suffixArray[i], prev = suffixArray[i - 1];
            if (tempRank[curr] == tempRank[prev] && tempRank[curr ^ step] == tempRank[prev ^ step]) {
                rank[curr] = distinct;
            } else {
                rank[curr] = ++distinct;
            }
        }
        if (distinct == size) break;
    }
    for (int i = 0; i < size; ++i) cout << str[i ^ suffixArray[0]];
    cout << endl;
    return 0;
}

For the Snowy Mountain problem, consider a path of length (L) ending at a basin. Define height (h_x) via multi-source BFS from basins. The kinetic energy dynamics show that after (h_x) steps increasing energy and (L - h_x) steps decreasing it, the remaining energy is (2h_x - L).

A greedy strategy involves tarversing edges where (h_u = h_v). If kinetic energy remains when crossing such an edge, oscillating along it can extend the path until energy depletes. This leads to an energy remainder (w = h_u = h_v), giving (L = 2h_x - w). The objective is to minimize (w) for each starting point (x), where (w) is the minimal value among reachable nodes with an edge satisfying (h_u = h_v).

Let (S = {u \mid \exists (u,v) \in E, h_u = h_v}). The sum (\sum_{u \in S} h_u) is (O(n)), implying the number of distinct (h_u) values in (S) is (O(\sqrt{n})). This reduces the state space.

Define (dp_{u,i}) as the negative of the minimum kinetic energy (allowing negative values) on any path from (u) to a node with height (i), where (i) ranges over (O(\sqrt{n})) distinct heights. Transitions require handling nodes with equal heights separately, which are disjoint connected components. Perform upward and downward dynamic progarmming within each component to compute values correctly.

#include <iostream>
#include <vector>
#include <queue>
#include <climits>
using namespace std;

const int MAX_N = 200005;
const int INF = INT_MAX / 2;
vector<int> graph[MAX_N];
vector<int> heightGroups[MAX_N];
int height[MAX_N];
int compId[MAX_N];
int dp[MAX_N][500];
int distinctHeights[500];
int heightIndex[MAX_N];
bool visited[MAX_N];
int totalHeights = 0;

void minimize(int &a, int b) { if (b < a) a = b; }
void maximize(int &a, int b) { if (b > a) a = b; }

void processUpward(int node, int parent, int currentHeight) {
    visited[node] = true;
    for (int neighbor : graph[node]) {
        if (neighbor == parent || height[neighbor] != currentHeight) continue;
        processUpward(neighbor, node, currentHeight);
        for (int idx = 1; idx <= totalHeights; ++idx) {
            minimize(dp[node][idx], dp[neighbor][idx] + 1);
        }
        dp[node][heightIndex[currentHeight]] = 1;
    }
}

void processDownward(int node, int parent, int currentHeight) {
    for (int neighbor : graph[node]) {
        if (neighbor == parent || height[neighbor] != currentHeight) continue;
        for (int idx = 1; idx <= totalHeights; ++idx) {
            minimize(dp[neighbor][idx], dp[node][idx] + 1);
        }
        dp[neighbor][heightIndex[currentHeight]] = 1;
        processDownward(neighbor, node, currentHeight);
    }
}

int main() {
    int n;
    cin >> n;
    vector<int> basins;
    for (int i = 1; i <= n; ++i) {
        int isBasin;
        cin >> isBasin;
        if (isBasin) {
            basins.push_back(i);
            height[i] = 0;
        } else {
            height[i] = INF;
        }
    }
    for (int i = 0; i < n - 1; ++i) {
        int u, v;
        cin >> u >> v;
        graph[u].push_back(v);
        graph[v].push_back(u);
    }
    queue<int> bfsQueue;
    for (int basin : basins) {
        bfsQueue.push(basin);
        visited[basin] = true;
    }
    while (!bfsQueue.empty()) {
        int current = bfsQueue.front();
        bfsQueue.pop();
        for (int neighbor : graph[current]) {
            if (!visited[neighbor]) {
                height[neighbor] = height[current] + 1;
                visited[neighbor] = true;
                bfsQueue.push(neighbor);
            }
        }
    }
    fill(visited, visited + n + 1, false);
    for (int node = 1; node <= n; ++node) {
        heightGroups[height[node]].push_back(node);
        if (heightIndex[height[node]]) continue;
        for (int neighbor : graph[node]) {
            if (height[node] == height[neighbor]) {
                distinctHeights[++totalHeights] = height[node];
                heightIndex[height[node]] = totalHeights;
                break;
            }
        }
    }
    for (int h = 0; h <= n; ++h) {
        for (int node : heightGroups[h]) {
            for (int idx = 1; idx <= totalHeights; ++idx) dp[node][idx] = INF;
            for (int neighbor : graph[node]) {
                if (height[neighbor] != h - 1) continue;
                for (int idx = 1; idx <= totalHeights; ++idx) {
                    if (dp[neighbor][idx]) minimize(dp[node][idx], dp[neighbor][idx] - 1);
                    else dp[node][idx] = 0;
                }
            }
        }
        for (int node : heightGroups[h]) {
            if (!visited[node]) {
                processUpward(node, 0, h);
                processDownward(node, 0, h);
            }
        }
    }
    for (int node = 1; node <= n; ++node) {
        int best = height[node];
        for (int idx = 1; idx <= totalHeights; ++idx) {
            if (dp[node][idx] == 0) maximize(best, 2 * height[node] - distinctHeights[idx]);
        }
        cout << best << " ";
    }
    cout << endl;
    return 0;
}