(Analysis by Mark Gordon)

This problem asks us to come up with a deterministic strategy for Bessie to find the exit of the barn using only her observed segment lengths and angles of the side of the barn that minimizes the worst case additional distance that Bessie will have to travel.

One way to think about this is to abstract away the polygon aspect of the problem and instead treat the angles and side lengths as a string. Now we need to help Bessie explore a string instead. To solve this we'll want to apply dynamic programming, but first we must determine what should be included in our state.

The order that Bessie explores the string is unimportant. Only what has been explored, where she started, and where she is now are relevant. Note that her starting point within the explored string is important. Bessie may choose different strategies for otherwise equal states if she started somewhere different in the string. This is because she's not trying to minimize her worst case distance to the exit; she's trying to minimize the worst case additional distance over lights-on case which is sensitive to the starting location.

Therefore our state can be given by the explored string, start position, and current location. There are only $N^2$ possible strings, $N$ starting locations, and 2 possible current locations if we restrict ourselves to be only at the ends of the string. For each state we want to determine if we should extend the string on the left or right to minimize the worst case distance we'll have to travel from here on out over the lights on case for any starting point.

In the base case where we're at the exit then we just yield the negative distance in the lights-on case for our starting position (which must now be unique because the exit can be distinguished from all other corners).

Here's my solution to this problem

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

#define MAXN 210
#define INF 0x3FFFFFFF

int opt[MAXN];
int psum[MAXN];
int canon[MAXN*2][MAXN*2];
vector<int> lparents[MAXN][MAXN];
vector<int> rparents[MAXN][MAXN];

int dp[MAXN][MAXN][MAXN][2];

int main() {
int N; cin >> N;
vector<pair<long long, long long> > A(N);
for (int i = 0; i < N; i++) {
cin >> A[i].first >> A[i].second;
}

/* Create the underlying string from the polygon.  Represent the exit as
* 0.  Represent clockwise turns as -1, counter clockwise as -2.  Represent
* lengths by their length.
*/
vector<int> S(1, 0);
for (int i = 0; i < N; i++) {
int j = (i + 1) % N;
int k = (i + 2) % N;
S.push_back(abs(A[i].first - A[j].first) +
abs(A[i].second - A[j].second));

/* Use a cross product to determine which way the polygon turned. */
if ((A[i].first - A[j].first) * (A[k].second - A[j].second) -
(A[k].first - A[j].first) * (A[i].second - A[j].second) > 0) {
S.push_back(-1);
} else {
S.push_back(-2);
}
}
S.back() = 0;

/* Compute the lights-on cost for each corner. */
for (int i = 0; i < N; i++) {
psum[i + 1] = opt[i + 1] = opt[i] + S[2 * i + 1];
}
opt[N] = 0;
for (int i = N - 1; i >= 0; i--) {
opt[i] = min(opt[i], opt[i + 1] + S[2 * i + 1]);
}

/* Compute j = canon[i][ln] to be the first j s.t. S[j:j+ln-1] = S[i:i+ln-1].
The canonical starting position of the string S[i:i+ln-1].
*/
for (int ln = 1; ln <= S.size(); ln++) {
for (int i = 0; i + ln <= S.size(); i++) {
for (int& j = canon[i][ln]; j < i; j++) {
/* If ln - 1 matches and the last character matches... */
if (canon[j][ln - 1] == canon[i][ln - 1] &&
S[j + ln - 1] == S[i + ln - 1]) {
break;
}
}
}
}

/* Pre-compute the state transitions; how to extend left and right for each
* possible string. */
for (int i = 0; i < S.size(); i += 2) {
for (int ln = 3; i + ln <= S.size(); ln += 2) {
if (i != canon[i][ln]) {
continue;
}
lparents[canon[i + 2][ln - 2] / 2][ln / 2].push_back(i / 2);
rparents[canon[i][ln - 2] / 2][ln / 2].push_back(i / 2);
}
}

int result = 0;
for (int ln = N; ln >= 1; ln--) {
for (int i = 0; i + ln <= N + 1; i++) {
if (canon[2 * i][2 * ln - 1] != 2 * i) {
/* Non-canonical string, skip. */
continue;
}

int dist_across = psum[i + ln - 1] - psum[i];
for (int strt = 0; strt < ln; strt++) {
for (int side = 0; side < 2; side++) {
if (i == 0 || i + ln == N + 1) {
/* We're at the exit. */
dp[i][ln][strt][side] = -opt[i + strt];
continue;
}

/* Compute the worst case cost for left and right. */
int lft_cst = -INF;
for (int j : lparents[i][ln]) {
lft_cst = max(lft_cst, S[2 * j + 1] + dp[j][ln + 1][strt + 1][0]);
}

int rht_cst = -INF;
for (int j : rparents[i][ln]) {
rht_cst = max(rht_cst,
S[2 * (j + ln) - 1] + dp[j][ln + 1][strt][1]);
}

/* Add in the cost for crossing the string if we choose to extend the
* far side of the string. */
(side ? lft_cst : rht_cst) += dist_across;

/* Result is just the min of these two choices. */
dp[i][ln][strt][side] = min(lft_cst, rht_cst);

/* Update the result if this is a 1 length string. */
if (ln == 1) {
result = max(result, dp[i][ln][strt][side]);
}
}
}
}
}
cout << result << endl;

return 0;
}