Edutorial for 1057 en version

1057/en.md

@@ -0,0 +1,132 @@
+### **Tutorial: LightOJ 1057**
+
+**Prerequisites:** Bitmask DP, Traveling Salesperson Problem (TSP), Chebyshev Distance
+
+#### **Observation 1: Calculating Distance**
+We are given a grid where we can move in all 8 directions (horizontally, vertically, and diagonally). This implies that the shortest distance between any two cells $(r_1, c_1)$ and $(r_2, c_2)$ is calculated using the **Chebyshev distance**: 
+$$\text{dist}((r_1, c_1), (r_2, c_2)) = \max(|r_1 - r_2|, |c_1 - c_2|)$$
+
+*Proof:* To minimize moves, we should move diagonally as much as possible because one diagonal move changes both our X and Y coordinates simultaneously. We do this until we align with our target on either the X or Y axis, after which we move in a straight line. The total number of moves will always be bounded by the maximum of the absolute differences of their coordinates.
+
+#### **Observation 2: Reducing the Problem (TSP)**
+Let $K$ be the total number of gold pieces ('g') on the grid. We can assign each gold piece a unique ID from 0 to $K-1$. The starting position 'x' is our base. 
+The problem now reduces to a variation of the **Traveling Salesperson Problem (TSP)**: find the shortest path that starts at the base, visits all $K$ nodes, and returns to the base. 
+
+Because $K \le 15$, we can solve this efficiently using Dynamic Programming with bitmasking.
+
+---
+
+### **DP Formulation**
+
+**State Definition:**
+Let `dp[mask][i]` be the minimum number of moves required to visit the subset of gold pieces represented by `mask`, ending at gold piece $i$. 
+* `mask` is an integer where the $j$-th bit is 1 if the $j$-th gold piece has been visited, and 0 otherwise.
+* $i$ is the index of the last visited gold piece.
+
+**Base Cases:**
+Initially, we only have one gold piece in our path. For every gold piece $i$ from 0 to $K-1$:
+$dp[1 \ll i][i] = \text{dist}(\text{start}, \text{gold}_i)$
+
+**Transitions:**
+To compute `dp[mask][i]`, we must transition from a previous state where we had visited all nodes in the `mask` *except* node $i$, ending at some node $j$.
+For every node $i$ present in the `mask`, we iterate over every other node $j$ present in the `mask` ($i \neq j$):
+$$dp[\text{mask}][i] = \min_{j}(dp[\text{mask} - (1 \ll i)][j] + \text{dist}(\text{gold}_j, \text{gold}_i))$$
+
+**Final Answer:**
+Once the DP table is fully populated, the final answer requires us to return to the starting position 'x'. We take the minimum over all possible last-visited gold pieces:
+$$\text{Ans} = \min_{i} (dp[(1 \ll K) - 1][i] + \text{dist}(\text{gold}_i, \text{start}))$$
+*(Note: We must handle the edge case where K = 0 seperately. If there is no gold, the answer is just 0!)*
+
+---
+
+### **Complexity Analysis & Optimization**
+
+A loose upper bound for the time complexity is $\mathcal{O}(K^2 2^K)$ per testcase. 
+
+At first glance, this might look too slow. The maximum number of gold pieces is $K = 15$. If we blindly iterate through all $2^{15}$ masks, and for each mask run a $15 \times 15$ nested loop to find $i$ and $j$, the operations per test case would be $15^2 \times 2^{15} \approx 7.3 \times 10^6$. Across 100 test cases, this is roughly $7.4 \times 10^8$ operations, which is beyond the restricted time.
+
+**The Optimization:**
+We can drastically reduce the constant factor by **only iterating through the active bits** (nodes that are actually present in the current mask). 
+
+The exact number of operations with this optimization: For any mask that has exactly $c$ bits turned on, we test $c$ options for the current node $i$, and $c-1$ options for the previous node $j$. The number of possible masks with exactly $c$ bits turned on is $\binom{K}{c}$. 
+
+Summing this up for all possible subset sizes gives us our exact worst-case transitions:
+$$\sum_{c=1}^{K} \binom{K}{c} c(c-1) = K(K-1)2^{K-2}$$
+
+If we plug in $K = 15$:
+$$15 \times 14 \times 2^{13} = 1,720,320 \text{ operations}$$
+
+Across the maximum of 100 test cases, this tight bound results in roughly 172 million operations in total $\approx 1.72 \times 10^8$.
+
+<details>
+<summary><b>Solution code (C++)</b></summary>
+
+```cpp
+#include <bits/stdc++.h>
+using namespace std;
+const int INF = 1e9;
+
+int get_dist(pair<int, int> a, pair<int, int> b) {
+    return max(abs(a.first - b.first), abs(a.second - b.second));
+}
+
+void solve(int tc) {
+    int r, c;
+    cin >> r >> c;
+    pair<int, int> start;
+    vector<pair<int, int>> gold;
+    for (int i = 0; i < r; ++i) {
+        string row;
+        cin >> row;
+        for (int j = 0; j < c; ++j) {
+            if (row[j] == 'x') {
+                start = {i, j};
+            } else if (row[j] == 'g') {
+                gold.push_back({i, j});
+            }
+        }
+    }
+    int K = gold.size();
+    if (K == 0) {
+        cout << "Case " << tc << ": 0\n";
+        return;
+    }
+    vector<vector<int>> dp(1 << K, vector<int>(K, INF));
+    for (int i = 0; i < K; ++i) {
+        dp[1 << i][i] = get_dist(start, gold[i]);
+    }
+    for (int mask = 1; mask < (1 << K); ++mask) {
+        vector<int> active;
+        for (int i = 0; i < K; ++i) {
+            if ((mask >> i) & 1) {
+                active.push_back(i);
+            }
+        }
+        if (active.size() <= 1) continue;
+        for (int i : active) {
+            int prev_mask = mask ^ (1 << i);
+            for (int j : active) {
+                if (i == j) continue;
+                int dist = get_dist(gold[j], gold[i]);
+                dp[mask][i] = min(dp[mask][i], dp[prev_mask][j] + dist);
+            }
+        }
+    }
+    int ans = INF;
+    int full_mask = (1 << K) - 1;
+    for (int i = 0; i < K; ++i) {
+        int return_dist = get_dist(gold[i], start);
+        ans = min(ans, dp[full_mask][i] + return_dist);
+    }
+    cout << "Case " << tc << ": " << ans << "\n";
+}
+
+int main() {
+    int t;
+    if (cin >> t) {
+        for (int i = 1; i <= t; ++i) {
+            solve(i);
+        }
+    }
+    return 0;
+}