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tsp.cpp
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tsp.cpp
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#include <algorithm>
#include <chrono>
#include <cmath>
#include <cstdint>
#include <iostream>
#include <limits>
#include <random>
#include <stack>
#include "matrix.h"
#ifdef DEBUG
#define LOG(x) std::cerr << x << std::endl
#else
#define LOG(x)
#endif
using namespace std;
// Random number generator.
random_device rd;
default_random_engine rng(rd());
// Size of nearest neighbors matrix.
const static size_t MAX_K = 20;
/*
* Some small helper functions.
*/
// Return the current time.
static inline chrono::time_point<chrono::high_resolution_clock> now() {
return chrono::high_resolution_clock::now();
}
// Output stream operator for durations.
template<typename T, typename E>
inline ostream& operator<<(ostream& out, const chrono::duration<T, E>& d) {
out << chrono::duration_cast<chrono::milliseconds>(d).count() << " ms";
return out;
}
// 3-argument maximum function.
template<typename T>
static inline T maximum(const T& a, const T& b, const T& c) {
return max(a, max(b, c));
}
// 4-argument maximum function.
template<typename T>
static inline T maximum(const T& a, const T& b, const T& c, const T& d) {
return max(a, max(b, max(c, d)));
}
/**
* Returns the shortest distance d[i][j], i != j in the given distance matrix.
*
* @param d Distance matrix.
* @return Minimum distance in d.
*/
uint32_t minDistance(const Matrix<uint32_t>& d) {
size_t N = d.rows();
uint32_t min = numeric_limits<uint32_t>::max();
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
if (i != j)
min = std::min(min, d[i][j]);
}
}
return min;
}
/**
* Returns the total length of a tour.
*
* @param tour The input tour.
* @param d Distance matrix.
* @return The total length of the tour.
*/
inline uint64_t length(const vector<uint16_t>& tour, const Matrix<uint32_t>& d) {
size_t N = tour.size();
uint64_t length = 0;
for (size_t i = 0, j = 1; i < N; ++i, ++j) {
length += d[tour[i]][tour[j % N]];
}
return length;
}
/**
* Reverse a segment of a tour.
*
* This functions reverses the segment [start, end] of the given
* tour and updates the position vector accordingly.
*
* @param tour The input tour.
* @param start Start index of segment to reverse.
* @param end End index of segment to reverse.
* @param position Position of each city in the input tour. Will be updated.
*/
inline void reverse(vector<uint16_t> &tour, size_t start, size_t end,
vector<uint16_t>& position) {
size_t N = tour.size();
size_t numSwaps = (((start <= end ? end - start : (end + N) - start) + 1)/2);
uint16_t i = start;
uint16_t j = end;
for (size_t n = 0; n < numSwaps; ++n) {
swap(tour[i], tour[j]);
position[tour[i]] = i;
position[tour[j]] = j;
i = (i + 1) % N;
j = ((j + N) - 1) % N;
}
}
/**
* Order three edges by tour position.
*
* This function takes as input three disjoint edges GH, IJ and KL, and their
* positions in the tour (G_i, H_i, ..., L_i), and sets AB, CD and EF, and
* associated tour indices A_i, B_i, ..., F_i, to be the input edges in tour
* order.
*
* E.g if GH, IJ and KL have the order ..->GH->..->IJ->..->KL->.., then
* AB = GH, CD = IJ, EF = KL, else AB = IJ, CD = GH, EF = KL.
*
* This is a helper function used in the inner loop of threeOpt(...).
*/
inline void ordered(
uint16_t& A, size_t& A_i, uint16_t& B, size_t& B_i,
uint16_t& C, size_t& C_i, uint16_t& D, size_t& D_i,
uint16_t& E, size_t& E_i, uint16_t& F, size_t& F_i,
uint16_t G, size_t G_i, uint16_t H, size_t H_i,
uint16_t I, size_t I_i, uint16_t J, size_t J_i,
uint16_t K, size_t K_i, uint16_t L, size_t L_i) {
E = K; E_i = K_i;
F = L; F_i = L_i;
if ((I_i < G_i && G_i < K_i) ||
(K_i < I_i && I_i < G_i) ||
(G_i < K_i && K_i < I_i)) {
A = I; A_i = I_i;
B = J; B_i = J_i;
C = G; C_i = G_i;
D = H; D_i = H_i;
} else {
A = G; A_i = G_i;
B = H; B_i = H_i;
C = I; C_i = I_i;
D = J; D_i = J_i;
}
}
/**
* Create a distance matrix from an input stream and return it.
*
* @param in Input stream.
* @return The read distance matrix.
*/
Matrix<uint32_t> createDistanceMatrix(istream& in) {
// Read vertex coordinates.
size_t N;
in >> N;
vector<double> x(N);
vector<double> y(N);
for (size_t i = 0; i < N; ++i) {
in >> x[i] >> y[i];
}
// Calculate distance matrix.
Matrix<uint32_t> d(N, N);
for (size_t i = 0; i < N; ++i) {
for (size_t j = i + 1; j < N; ++j) {
d[i][j] = d[j][i] = round(sqrt(pow(x[i]-x[j], 2) + pow(y[i]-y[j], 2)));
}
}
return d;
}
/**
* Calculate K-nearest neighbors matrix from a distance matrix.
*
* @param d Distance matrix.
* @return d.rows() x K matrix where element i,j is the j:th nearest
* neighbor of city i.
*/
Matrix<uint16_t> createNeighborsMatrix(const Matrix<uint32_t>& d, size_t K) {
size_t N = d.rows();
size_t M = d.cols() - 1;
K = min(M, K);
Matrix<uint16_t> neighbor(N, K);
vector<uint16_t> row(M); // For sorting.
for (size_t i = 0; i < N; ++i) {
// Fill row with 0, 1, ..., i - 1, i + 1, ..., M - 1.
uint16_t k = 0;
for (size_t j = 0; j < M; ++j, ++k) {
row[j] = (i == j) ? ++k : k;
}
// Sort K first elements in row by distance to i.
partial_sort(row.begin(), row.begin() + K, row.end(),
[&](uint16_t j, uint16_t k) {
return d[i][j] < d[i][k];
}
);
// Copy first K elements (now sorted) to neighbor matrix.
copy(row.begin(), row.begin() + K, neighbor[i]);
}
return neighbor;
}
/**
* Calculates a greedy TSP tour.
*
* This is the naive algorithm given in the Kattis problem description.
*
* @param d Distance matrix.
* @return Greedy TSP tour.
*/
inline vector<uint16_t> greedy(const Matrix<uint32_t>& d) {
size_t N = d.rows();
vector<uint16_t> tour(N);
vector<bool> used(N, false);
tour[0] = 0;
used[0] = true;
for (size_t i = 1; i < N; ++i) {
// Find k, the closest city to the (i - 1):th city in tour.
int32_t k = -1;
for (uint16_t j = 0; j < N; ++j) {
if (!used[j] && (k == -1 || d[tour[i-1]][j] < d[tour[i-1]][k])) {
k = j;
}
}
tour[i] = k;
used[k] = true;
}
return tour;
}
/**
* Optimizes the given tour using 2-opt.
*
* This function uses the fast approach described on page 12-13 of "Large-Step
* Markov Chains for the Traveling Salesman Problem" (Martin/Otto/Felten, 1991)
*
* @param tour The tour to optimize.
* @param d Distance matrix.
* @param neighbor Nearest neighbors matrix.
* @param position Position of each city in the input tour. Will be updated.
* @param max Longest inter-city distance in input tour. Will be updated.
* @param min Shortest possible inter-city distance.
*/
inline void twoOpt(vector<uint16_t>& tour, const Matrix<uint32_t>& d,
const Matrix<uint16_t>& neighbor, vector<uint16_t> &position,
uint32_t& max, uint32_t min) {
size_t N = d.rows(); // Number of cities.
// Candidate edges uv, wz and their positions in tour.
uint16_t u, v, w, z;
size_t u_i, v_i, w_i, z_i;
bool locallyOptimal = false;
while (!locallyOptimal) {
locallyOptimal = true;
// For each edge uv.
for (u_i = 0, v_i = 1; u_i < N; ++u_i, ++v_i) {
u = tour[u_i];
v = tour[v_i % N];
// For each edge wz (w k:th closest neighbor of u).
for (size_t k = 0; k < neighbor.cols(); ++k) {
w_i = position[neighbor[u][k]];
z_i = w_i + 1;
w = tour[w_i];
z = tour[z_i % N];
if (v == w || z == u) {
continue; // Skip adjacent edges.
}
// d[u][w] + min is a lower bound on new length.
// d[u][v] + max is an upper bound on old length.
if (d[u][w] + min > d[u][v] + max) {
break; // Go to next edge uv.
}
if (d[u][w] + d[v][z] < d[u][v] + d[w][z]) {
// --u w-- --u-w->
// X ===>
// <-z v-> <-z-v--
reverse(tour, v_i % N, w_i, position);
max = maximum(max, d[u][w], d[v][z]);
locallyOptimal = false;
break;
}
}
}
}
}
/**
* Optimizes the given tour using 3-opt.
*
* This function uses the fast approach described on page 12-15 of "Large-Step
* Markov Chains for the Traveling Salesman Problem" (Martin/Otto/Felten, 1991)
*
* The algorithm will only consider "real" 3-exchanges involving all three
* edges. So for best results, the tour should be preprocessed with the 2-opt
* algorithm first.
*
* @param tour The tour to optimize.
* @param d Distance matrix.
* @param neighbor Nearest neighbors matrix.
* @param position Position of each city in the input tour. Will be updated.
* @param max Longest inter-city distance in input tour. Will be updated.
* @param min Shortest possible inter-city distance.
* @param deadline Deadline at which function will try to return early.
*/
inline void threeOpt(vector<uint16_t>& tour, const Matrix<uint32_t>& d,
const Matrix<uint16_t>& neighbor, vector<uint16_t>& position,
uint32_t& max, uint32_t min,
const chrono::time_point<chrono::high_resolution_clock>& deadline) {
const size_t N = d.rows(); // Number of cities.
// Candidate edges PQ, RS, TU and their positions in tour.
uint16_t P, Q, R, S, T, U;
size_t P_i, Q_i, R_i, S_i, T_i, U_i;
// AB, CD, EF is PQ, RS, TU, but in tour order.
uint16_t A, B, C, D, E, F;
size_t A_i, B_i, C_i, D_i, E_i, F_i;
bool locallyOptimal = false;
while (!locallyOptimal) {
locallyOptimal = true;
// For each edge PQ.
for (size_t i = 0; i < N; ++i) {
P_i = i;
Q_i = (P_i + 1) % N;
P = tour[P_i];
Q = tour[Q_i];
if (chrono::high_resolution_clock::now() > deadline)
return; // Deadline has passed, return early.
// For each edge RS (S j:th nearest neighbor of P).
for (size_t j = 0; j < neighbor.cols(); ++j) {
S_i = position[neighbor[P][j]];
R_i = (S_i + N - 1) % N;
R = tour[R_i];
S = tour[S_i];
if (P == R || R == Q) // RS same as, or follows, PQ.
continue; // Go to next edge RS.
if (d[P][S] + 2 * min > d[P][Q] + 2 * max)
break; // Go to next edge PQ.
if (d[P][S] + 2 * min > d[P][Q] + d[R][S] + max)
continue; // Go to next edge RS.
// For each edge TU (U k:th nearest neighbor of P).
for (size_t k = 0; k < neighbor.cols(); ++k) {
U_i = position[neighbor[P][k]];
T_i = (U_i + N - 1) % N;
T = tour[T_i];
U = tour[U_i];
if (U == S || // TU same as RS.
T == S || // TU follows RS.
U == R || // TU preceeds RS.
T == P || // TU same as PQ.
T == Q) // TU follows PQ.
continue; // Go to next edge TU.
if (d[P][S] + d[Q][U] + min > d[P][Q] + d[R][S] + max)
break; // Go to next edge RS.
// Let AB, CD, EF be the edges PQ, RS, TU in tour order.
ordered(A, A_i, B, B_i, C, C_i, D, D_i, E, E_i, F, F_i,
P, P_i, Q, Q_i, R, R_i, S, S_i, T, T_i, U, U_i);
// Try exchanging AB, CD and EF for another edge triple.
// See 3opt_cases.png for an illustration.
bool changed = false;
uint32_t d_AB_CD_EF = d[A][B] + d[C][D] + d[E][F];
if (d[D][A] + d[F][B] + d[C][E] < d_AB_CD_EF) {
// Change AB, CD, EF to DA, FB, CE.
reverse(tour, F_i, A_i, position);
reverse(tour, D_i, E_i, position);
max = maximum(max, d[D][A], d[F][B], d[C][E]);
changed = true;
} else if (d[B][D] + d[E][A] + d[F][C] < d_AB_CD_EF) {
// Change AB, CD, EF to BD, EA, FC.
reverse(tour, F_i, A_i, position);
reverse(tour, B_i, C_i, position);
max = maximum(max, d[B][D], d[E][A], d[F][C]);
changed = true;
} else if (d[A][C] + d[B][E] + d[D][F] < d_AB_CD_EF) {
// Change AB, CD, EF to AC, BE, DF.
reverse(tour, B_i, C_i, position);
reverse(tour, D_i, E_i, position);
max = maximum(max, d[A][C], d[B][E], d[D][F]);
changed = true;
} else if (d[B][E] + d[D][A] + d[F][C] < d_AB_CD_EF) {
// Change AB, CD, EF to BE, DA, FC.
reverse(tour, A_i, F_i, position);
reverse(tour, B_i, C_i, position);
reverse(tour, D_i, E_i, position);
max = maximum(max, d[B][E], d[D][A], d[F][C]);
changed = true;
}
if (changed) {
locallyOptimal = false;
goto next_PQ; // Go to next edge PQ.
}
}
}
next_PQ: continue;
}
}
}
/**
* Perform a random 4-opt ("double bridge") move on a tour.
*
* E.g.
*
* A--B A B
* / \ /| |\
* H C H------C
* | | --> | |
* G D G------D
* \ / \| |/
* F--E F E
*
* Where edges AB, CD, EF and GH are chosen randomly.
*
* @param tour Input tour (must have at least 8 cities).
* @return The new tour.
*/
inline vector<uint16_t> doubleBridge(const vector<uint16_t>& tour) {
const size_t N = tour.size();
vector<uint16_t> newTour;
newTour.reserve(N);
uniform_int_distribution<size_t> randomOffset(1, N / 4);
size_t A = randomOffset(rng);
size_t B = A + randomOffset(rng);
size_t C = B + randomOffset(rng);
copy(tour.begin(), tour.begin() + A, back_inserter(newTour));
copy(tour.begin() + C, tour.end(), back_inserter(newTour));
copy(tour.begin() + B, tour.begin() + C, back_inserter(newTour));
copy(tour.begin() + A, tour.begin() + B, back_inserter(newTour));
return newTour;
}
/**
* Approximates optimal TSP tour through graph read from the given input stream.
*
* The tour is approximated using iterated local search (ILS), with a greedy initial
* tour and 2-opt + 3-opt as local search methods, and a random 4-exchange ("double
* bridge") as perturbation.
*
* The function will try to return before the given deadline, but expect some
* variations.
*
* @param in Input stream.
* @param deadline Deadline before which function should try to return.
* @return An approximation of the optimal TSP tour.
*/
template<typename T>
vector<uint16_t> approximate(istream &in, const chrono::time_point<T>& deadline) {
/*
* Initialization.
*/
// Deadline for 3-opt inside main loop is 50 ms before hard deadline.
chrono::milliseconds fifty_ms(50);
auto threeOptDeadline = deadline - fifty_ms;
// Calculate distance / K-nearest neighbors matrix.
const Matrix<uint32_t> d = createDistanceMatrix(in);
const Matrix<uint16_t> neighbor = createNeighborsMatrix(d, MAX_K);
const uint32_t min = minDistance(d); // Shortest distance.
const size_t N = d.rows(); // Number of cities.
// Generate initial greedy tour.
vector<uint16_t> tour = greedy(d);
// Create max / position for initial 2-opt + 3-opt.
vector<uint16_t> position(N);
uint32_t max = 0;
for (uint16_t i = 0; i < N; ++i) {
max = std::max(max, d[i][(i + 1) % N]); // Maximum distance in tour.
position[tour[i]] = i; // tour[i] is i:th city in tour.
}
/*
* Main loop.
*
* We repeatedly
*
* 1) Optimize the tour with 2-opt + 3-opt.
* 2) "Kick" the tour with a random 4-exchange.
*
* until only max(50, 2 * average iteration time) milliseconds remains
* before deadline, and then pick the shortest tour we found.
*/
// Some main loop statistics.
size_t i = 0; // Number of iterations of main loop.
chrono::milliseconds totalTime(0); // Total time spent in main loop.
chrono::milliseconds averageTime(0); // Average main loop iteration time.
vector<uint16_t> shortestTour = tour; // Best tour found.
uint64_t shortestTourLength = length(tour, d); // Length of best tour found.
for (i = 0; (now() + std::max(fifty_ms, 2 * averageTime)) < deadline; ++i) {
auto start = now();
// Optimize tour with 2-opt + 3-opt.
twoOpt(tour, d, neighbor, position, max, min);
threeOpt(tour, d, neighbor, position, max, min, threeOptDeadline);
uint64_t tourLength = length(tour, d);
if (tourLength < shortestTourLength) {
// Shorter tour found.
shortestTour = tour;
shortestTourLength = tourLength;
}
if (N >= 8) {
// Perform random 4-opt "double bridge" move.
tour = doubleBridge(tour);
} else {
// Tiny tour, so just shuffle it instead.
shuffle(tour.begin(), tour.end(), rng);
}
// Update max / position needed by fast 2/3-opt.
max = 0;
for (uint16_t j = 0; j < N; ++j) {
max = std::max(max, d[tour[j]][tour[(j + 1) % N]]);
position[tour[j]] = j;
}
// Collect statistics.
totalTime += chrono::duration_cast<chrono::milliseconds>(now() - start);
averageTime = totalTime / (i + 1);
}
LOG("Main Loop Statistics");
LOG(" iterations: " << i);
LOG(" totalTime: " << totalTime);
LOG(" averageTime: " << averageTime);
return shortestTour;
}
int main(int argc, char *argv[]) {
// Approximate/print a TSP tour in ~1950 milliseconds.
for (auto city : approximate(cin, now() + chrono::milliseconds(1950))) {
cout << city << endl;
}
return 0;
}