The Algorithms/**
* @brief
* [A* search algorithm](https://en.wikipedia.org/wiki/A*_search_algorithm)
* @details
* A* is an informed search algorithm, or a best-first search, meaning that it
* is formulated in terms of weighted graphs: starting from a specific starting
* node of a graph (initial state), it aims to find a path to the given goal
* node having the smallest cost (least distance travelled, shortest time,
* etc.). It evaluates by maintaining a tree of paths originating at the start
* node and extending those paths one edge at a time until it reaches the final
* state.
* The weighted edges (or cost) is evaluated on two factors, G score
* (cost required from starting node or initial state to current state) and H
* score (cost required from current state to final state). The F(state), then
* is evaluated as:
* F(state) = G(state) + H(state).
*
* To solve the given search with shortest cost or path possible is to inspect
* values having minimum F(state).
* @author [Ashish Daulatabad](https://github.com/AshishYUO)
*/
#include <algorithm> /// for `std::reverse` function
#include <array> /// for `std::array`, representing `EightPuzzle` board
#include <cassert> /// for `assert`
#include <functional> /// for `std::function` STL
#include <iostream> /// for IO operations
#include <map> /// for `std::map` STL
#include <set> /// for `std::set` STL
#include <vector> /// for `std::vector` STL
/**
* @namespace machine_learning
* @brief Machine learning algorithms
*/
namespace machine_learning {
/**
* @namespace aystar_search
* @brief Functions for [A*
* Search](https://en.wikipedia.org/wiki/A*_search_algorithm) implementation.
*/
namespace aystar_search {
/**
* @class EightPuzzle
* @brief A class defining [EightPuzzle/15-Puzzle
* game](https://en.wikipedia.org/wiki/15_puzzle).
* @details
* A well known 3 x 3 puzzle of the form
* `
* 1 2 3
* 4 5 6
* 7 8 0
* `
* where `0` represents an empty space in the puzzle
* Given any random state, the goal is to achieve the above configuration
* (or any other configuration if possible)
* @tparam N size of the square Puzzle, default is set to 3 (since it is
* EightPuzzle)
*/
template <size_t N = 3>
class EightPuzzle {
std::array<std::array<uint32_t, N>, N>
board; /// N x N array to store the current state of the Puzzle.
std::vector<std::pair<int, int>> moves = {
{0, 1},
{1, 0},
{0, -1},
{-1,
0}}; /// A helper array to evaluate the next state from current state;
/**
* @brief Finds an empty space in puzzle (in this case; a zero)
* @returns a pair indicating integer distances from top and right
* respectively, else returns -1, -1
*/
std::pair<uint32_t, uint32_t> find_zero() {
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
if (!board[i][j]) {
return {i, j};
}
}
}
return {-1, -1};
}
/**
* @brief check whether the index value is bounded within the puzzle area
* @param value index for the current board
* @returns `true` if index is within the board, else `false`
*/
inline bool in_range(const uint32_t value) const {
return value >= 0 && value < N;
}
public:
/**
* @brief get the value from i units from right and j units from left side
* of the board
* @param i integer denoting ith row
* @param j integer denoting column
* @returns non-negative integer denoting the value at ith row and jth
* column
* @returns -1 if invalid i or j position
*/
uint32_t get(size_t i, size_t j) const {
if (in_range(i) && in_range(j)) {
return board[i][j];
}
return -1;
}
/**
* @brief Returns the current state of the board
*/
std::array<std::array<uint32_t, N>, N> get_state() { return board; }
/**
* @brief returns the size of the EightPuzzle (number of row / column)
* @return N, the size of the puzzle.
*/
inline size_t get_size() const { return N; }
/**
* @brief Default constructor for EightPuzzle
*/
EightPuzzle() {
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
board[i][j] = ((i * 3 + j + 1) % (N * N));
}
}
}
/**
* @brief Parameterized Constructor for EightPuzzle
* @param init a 2-dimensional array denoting a puzzle configuration
*/
explicit EightPuzzle(const std::array<std::array<uint32_t, N>, N> &init)
: board(init) {}
/**
* @brief Copy constructor
* @param A a reference of an EightPuzzle
*/
EightPuzzle(const EightPuzzle<N> &A) : board(A.board) {}
/**
* @brief Move constructor
* @param A a reference of an EightPuzzle
*/
EightPuzzle(const EightPuzzle<N> &&A) noexcept
: board(std::move(A.board)) {}
/**
* @brief Destructor of EightPuzzle
*/
~EightPuzzle() = default;
/**
* @brief Copy assignment operator
* @param A a reference of an EightPuzzle
*/
EightPuzzle &operator=(const EightPuzzle &A) {
board = A.board;
return *this;
}
/**
* @brief Move assignment operator
* @param A a reference of an EightPuzzle
*/
EightPuzzle &operator=(EightPuzzle &&A) noexcept {
board = std::move(A.board);
return *this;
}
/**
* @brief Find all possible states after processing all possible
* moves, given the current state of the puzzle
* @returns list of vector containing all possible next moves
* @note the implementation is compulsory to create A* search
*/
std::vector<EightPuzzle<N>> generate_possible_moves() {
auto zero_pos = find_zero();
// vector which will contain all possible state from current state
std::vector<EightPuzzle<N>> NewStates;
for (auto &move : moves) {
if (in_range(zero_pos.first + move.first) &&
in_range(zero_pos.second + move.second)) {
// swap with the possible moves
std::array<std::array<uint32_t, N>, N> new_config = board;
std::swap(new_config[zero_pos.first][zero_pos.second],
new_config[zero_pos.first + move.first]
[zero_pos.second + move.second]);
EightPuzzle<N> new_state(new_config);
// Store new state and calculate heuristic value, and depth
NewStates.emplace_back(new_state);
}
}
return NewStates;
}
/**
* @brief check whether two boards are equal
* @returns `true` if check.state is equal to `this->state`, else
* `false`
*/
bool operator==(const EightPuzzle<N> &check) const {
if (check.get_size() != N) {
return false;
}
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
if (board[i][j] != check.board[i][j]) {
return false;
}
}
}
return true;
}
/**
* @brief check whether one board is lexicographically smaller
* @returns `true` if this->state is lexicographically smaller than
* `check.state`, else `false`
*/
bool operator<(const EightPuzzle<N> &check) const {
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
if (board[i][j] != check.board[i][j]) {
return board[i][j] < check.board[i][j];
}
}
}
return false;
}
/**
* @brief check whether one board is lexicographically smaller or equal
* @returns `true` if this->state is lexicographically smaller than
* `check.state` or same, else `false`
*/
bool operator<=(const EightPuzzle<N> &check) const {
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
if (board[i][j] != check.board[i][j]) {
return board[i][j] < check.board[i][j];
}
}
}
return true;
}
/**
* @brief friend operator to display EightPuzzle<>
* @param op ostream object
* @param SomeState a certain state.
* @returns ostream operator op
*/
friend std::ostream &operator<<(std::ostream &op,
const EightPuzzle<N> &SomeState) {
for (size_t i = 0; i < N; ++i) {
for (size_t j = 0; j < N; ++j) {
op << SomeState.board[i][j] << " ";
}
op << "\n";
}
return op;
}
};
/**
* @class AyStarSearch
* @brief A class defining [A* search
* algorithm](https://en.wikipedia.org/wiki/A*_search_algorithm). for some
* initial state and final state
* @details AyStarSearch class is defined as the informed search algorithm
* that is formulated in terms of weighted graphs: starting from a specific
* starting node of a graph (initial state), it aims to find a path to the given
* goal node having the smallest cost (least distance travelled, shortest time,
* etc.)
* The weighted edges (or cost) is evaluated on two factors, G score
* (cost required from starting node or initial state to current state) and H
* score (cost required from current state to final state). The `F(state)`, then
* is evaluated as:
* `F(state) = G(state) + H(state)`.
* The best search would be the final state having minimum `F(state)` value
* @tparam Puzzle denotes the puzzle or problem involving initial state and
* final state to be solved by A* search.
* @note 1. The algorithm is referred from pesudocode from
* [Wikipedia page](https://en.wikipedia.org/wiki/A*_search_algorithm)
* as is.
* 2. For `AyStarSearch` to work, the definitions for template Puzzle is
* compulsory.
* a. Comparison operator for template Puzzle (`<`, `==`, and `<=`)
* b. `generate_possible_moves()`
*/
template <typename Puzzle>
class AyStarSearch {
/**
* @brief Struct that handles all the information related to the current
* state.
*/
typedef struct Info {
Puzzle state; /// Holds the current state.
size_t heuristic_value = 0; /// stores h score
size_t depth = 0; /// stores g score
/**
* @brief Default constructor
*/
Info() = default;
/**
* @brief constructor having Puzzle as parameter
* @param A a puzzle object
*/
explicit Info(const Puzzle &A) : state(std::move(A)) {}
/**
* @brief constructor having three parameters
* @param A a puzzle object
* @param h_value heuristic value of this puzzle object
* @param depth the depth at which this node was found during traversal
*/
Info(const Puzzle &A, size_t h_value, size_t d)
: state(std::move(A)), heuristic_value(h_value), depth(d) {}
/**
* @brief Copy constructor
* @param A Info object reference
*/
Info(const Info &A)
: state(A.state),
heuristic_value(A.heuristic_value),
depth(A.depth) {}
/**
* @brief Move constructor
* @param A Info object reference
*/
Info(const Info &&A) noexcept
: state(std::move(A.state)),
heuristic_value(std::move(A.heuristic_value)),
depth(std::move(A.depth)) {}
/**
* @brief copy assignment operator
* @param A Info object reference
*/
Info &operator=(const Info &A) {
state = A.state;
heuristic_value = A.heuristic_value;
depth = A.depth;
return *this;
}
/**
* @brief move assignment operator
* @param A Info object reference
*/
Info &operator=(Info &&A) noexcept {
state = std::move(A.state);
heuristic_value = std::move(A.heuristic_value);
depth = std::move(A.depth);
return *this;
}
/**
* @brief Destructor for Info
*/
~Info() = default;
} Info;
Info Initial; // Initial state of the AyStarSearch
Info Final; // Final state of the AyStarSearch
/**
* @brief Custom comparator for open_list
*/
struct comparison_operator {
bool operator()(const Info &a, const Info &b) const {
return a.state < b.state;
}
};
public:
/**
* @brief Parameterized constructor for AyStarSearch
* @param initial denoting initial state of the puzzle
* @param final denoting final state of the puzzle
*/
AyStarSearch(const Puzzle &initial, const Puzzle &final) {
Initial = Info(initial);
Final = Info(final);
}
/**
* @brief A helper solution: launches when a solution for AyStarSearch
* is found
* @param FinalState the pointer to the obtained final state
* @param parent_of the list of all parents of nodes stored during A*
* search
* @returns the list of moves denoting moves from final state to initial
* state (in reverse)
*/
std::vector<Puzzle> Solution(
Info *FinalState,
const std::map<Info, Info *, comparison_operator> &parent_of) {
// Useful for traversing from final state to current state.
Info *current_state = FinalState;
/*
* For storing the solution tree starting from initial state to
* final state
*/
std::vector<Puzzle> answer;
while (current_state != nullptr) {
answer.emplace_back(current_state->state);
current_state = parent_of.find(*current_state)->second;
}
return answer;
}
/**
* Main algorithm for finding `FinalState`, given the `InitialState`
* @param dist the heuristic finction, defined by the user
* @param permissible_depth the depth at which the A* search discards
* searching for solution
* @returns List of moves from Final state to initial state, if
* evaluated, else returns an empty array
*/
std::vector<Puzzle> a_star_search(
const std::function<uint32_t(const Puzzle &, const Puzzle &)> &dist,
const uint32_t permissible_depth = 30) {
std::map<Info, Info *, comparison_operator>
parent_of; /// Stores the parent of the states
std::map<Info, uint32_t, comparison_operator>
g_score; /// Stores the g_score
std::set<Info, comparison_operator>
open_list; /// Stores the list to explore
std::set<Info, comparison_operator>
closed_list; /// Stores the list that are explored
// Before starting the AyStartSearch, initialize the set and maps
open_list.emplace(Initial);
parent_of[Initial] = nullptr;
g_score[Initial] = 0;
while (!open_list.empty()) {
// Iterator for state having having lowest f_score.
typename std::set<Info, comparison_operator>::iterator
it_low_f_score;
uint32_t min_f_score = 1e9;
for (auto iter = open_list.begin(); iter != open_list.end();
++iter) {
// f score here is evaluated by g score (depth) and h score
// (distance between current state and final state)
uint32_t f_score = iter->heuristic_value + iter->depth;
if (f_score < min_f_score) {
min_f_score = f_score;
it_low_f_score = iter;
}
}
// current_state, stores lowest f score so far for this state.
Info *current_state = new Info(*it_low_f_score);
// if this current state is equal to final, return
if (current_state->state == Final.state) {
return Solution(current_state, parent_of);
}
// else remove from open list as visited.
open_list.erase(it_low_f_score);
// if current_state has exceeded the allowed depth, skip
// neighbor checking
if (current_state->depth >= permissible_depth) {
continue;
}
// Generate all possible moves (neighbors) given the current
// state
std::vector<Puzzle> total_possible_moves =
current_state->state.generate_possible_moves();
for (Puzzle &neighbor : total_possible_moves) {
// calculate score of neighbors with respect to
// current_state
Info Neighbor = {neighbor, dist(neighbor, Final.state),
current_state->depth + 1};
uint32_t temp_g_score = Neighbor.depth;
// Check whether this state is explored.
// If this state is discovered at greater depth, then discard,
// else remove from closed list and explore the node
auto closed_list_iter = closed_list.find(Neighbor);
if (closed_list_iter != closed_list.end()) {
// 1. If state in closed list has higher depth, then remove
// from list since we have found better option,
// 2. Else don't explore this state.
if (Neighbor.depth < closed_list_iter->depth) {
closed_list.erase(closed_list_iter);
} else {
continue;
}
}
auto neighbor_g_score_iter = g_score.find(Neighbor);
// if the neighbor is already created and has minimum
// g_score, then update g_score and f_score else insert new
if (neighbor_g_score_iter != g_score.end()) {
if (neighbor_g_score_iter->second > temp_g_score) {
neighbor_g_score_iter->second = temp_g_score;
parent_of[Neighbor] = current_state;
}
} else {
g_score[Neighbor] = temp_g_score;
parent_of[Neighbor] = current_state;
}
// If this is a new state, insert into open_list
// else update if the this state has better g score than
// existing one.
auto iter = open_list.find(Neighbor);
if (iter == open_list.end()) {
open_list.emplace(Neighbor);
} else if (iter->depth > Neighbor.depth) {
open_list.erase(iter);
open_list.emplace(Neighbor);
}
}
closed_list.emplace(*current_state);
}
// Cannot find the solution, return empty vector
return std::vector<Puzzle>(0);
}
};
} // namespace aystar_search
} // namespace machine_learning
/**
* @brief Self test-implementations
* @returns void
*/
static void test() {
// Renaming for simplicity
using matrix3 = std::array<std::array<uint32_t, 3>, 3>;
using row3 = std::array<uint32_t, 3>;
using matrix4 = std::array<std::array<uint32_t, 4>, 4>;
using row4 = std::array<uint32_t, 4>;
// 1st test: A* search for simple EightPuzzle problem
matrix3 puzzle;
puzzle[0] = row3({0, 2, 3});
puzzle[1] = row3({1, 5, 6});
puzzle[2] = row3({4, 7, 8});
matrix3 ideal;
ideal[0] = row3({1, 2, 3});
ideal[1] = row3({4, 5, 6});
ideal[2] = row3({7, 8, 0});
/*
* Heuristic function: Manhattan distance
*/
auto manhattan_distance =
[](const machine_learning::aystar_search::EightPuzzle<> &first,
const machine_learning::aystar_search::EightPuzzle<> &second) {
uint32_t ret = 0;
for (int i = 0; i < first.get_size(); ++i) {
for (int j = 0; j < first.get_size(); ++j) {
uint32_t find = first.get(i, j);
int m = -1, n = -1;
for (int k = 0; k < second.get_size(); ++k) {
for (int l = 0; l < second.get_size(); ++l) {
if (find == second.get(k, l)) {
std::tie(m, n) = std::make_pair(k, l);
break;
}
}
if (m != -1) {
break;
}
}
if (m != -1) {
ret += abs(m - i) + abs(n - j);
}
}
}
return ret;
};
machine_learning::aystar_search::EightPuzzle<> Puzzle(puzzle);
machine_learning::aystar_search::EightPuzzle<> Ideal(ideal);
machine_learning::aystar_search::AyStarSearch<
machine_learning::aystar_search::EightPuzzle<3>>
search(Puzzle, Ideal); /// Search object
std::vector<matrix3> answer; /// Array that validates the answer
answer.push_back(
matrix3({row3({0, 2, 3}), row3({1, 5, 6}), row3({4, 7, 8})}));
answer.push_back(
matrix3({row3({1, 2, 3}), row3({0, 5, 6}), row3({4, 7, 8})}));
answer.push_back(
matrix3({row3({1, 2, 3}), row3({4, 5, 6}), row3({0, 7, 8})}));
answer.push_back(
matrix3({row3({1, 2, 3}), row3({4, 5, 6}), row3({7, 0, 8})}));
answer.push_back(
matrix3({row3({1, 2, 3}), row3({4, 5, 6}), row3({7, 8, 0})}));
auto Solution = search.a_star_search(manhattan_distance);
std::cout << Solution.size() << std::endl;
assert(Solution.size() == answer.size());
uint32_t i = 0;
for (auto it = Solution.rbegin(); it != Solution.rend(); ++it) {
assert(it->get_state() == answer[i]);
++i;
}
// 2nd test: A* search for complicated EightPuzzle problem
// Initial state
puzzle[0] = row3({5, 7, 3});
puzzle[1] = row3({2, 0, 6});
puzzle[2] = row3({1, 4, 8});
// Final state
ideal[0] = row3({1, 2, 3});
ideal[1] = row3({4, 5, 6});
ideal[2] = row3({7, 8, 0});
Puzzle = machine_learning::aystar_search::EightPuzzle<>(puzzle);
Ideal = machine_learning::aystar_search::EightPuzzle<>(ideal);
// Initialize the search object
search = machine_learning::aystar_search::AyStarSearch<
machine_learning::aystar_search::EightPuzzle<3>>(Puzzle, Ideal);
Solution = search.a_star_search(manhattan_distance);
std::cout << Solution.size() << std::endl;
// Static assertion due to large solution
assert(13 == Solution.size());
// Check whether the final state is equal to expected one
assert(Solution[0].get_state() == ideal);
for (auto it = Solution.rbegin(); it != Solution.rend(); ++it) {
std::cout << *it << std::endl;
}
// 3rd test: A* search for 15-Puzzle
// Initial State of the puzzle
matrix4 puzzle2;
puzzle2[0] = row4({5, 1, 2, 3});
puzzle2[1] = row4({9, 6, 8, 4});
puzzle2[2] = row4({13, 10, 7, 11});
puzzle2[3] = row4({14, 15, 12, 0});
// Final state of the puzzle
matrix4 ideal2;
ideal2[0] = row4({1, 2, 3, 4});
ideal2[1] = row4({5, 6, 7, 8});
ideal2[2] = row4({9, 10, 11, 12});
ideal2[3] = row4({13, 14, 15, 0});
// Instantiate states for a*, initial state and final states
machine_learning::aystar_search::EightPuzzle<4> Puzzle2(puzzle2),
Ideal2(ideal2);
// Initialize the search object
machine_learning::aystar_search::AyStarSearch<
machine_learning::aystar_search::EightPuzzle<4>>
search2(Puzzle2, Ideal2);
/**
* Heuristic function: Manhattan distance
*/
auto manhattan_distance2 =
[](const machine_learning::aystar_search::EightPuzzle<4> &first,
const machine_learning::aystar_search::EightPuzzle<4> &second) {
uint32_t ret = 0;
for (int i = 0; i < first.get_size(); ++i) {
for (int j = 0; j < first.get_size(); ++j) {
uint32_t find = first.get(i, j);
int m = -1, n = -1;
for (int k = 0; k < second.get_size(); ++k) {
for (int l = 0; l < second.get_size(); ++l) {
if (find == second.get(k, l)) {
std::tie(m, n) = std::make_pair(k, l);
break;
}
}
if (m != -1) {
break;
}
}
if (m != -1) {
ret += abs(m - i) + abs(n - j);
}
}
}
return ret;
};
auto sol2 = search2.a_star_search(manhattan_distance2);
std::cout << sol2.size() << std::endl;
// Static assertion due to large solution
assert(15 == sol2.size());
// Check whether the final state is equal to expected one
assert(sol2[0].get_state() == ideal2);
for (auto it = sol2.rbegin(); it != sol2.rend(); ++it) {
std::cout << *it << std::endl;
}
}
/**
* @brief Main function
* @returns 0 on exit
*/
int main() {
test(); // run self-test implementations
return 0;
}
A Star Search
D
A
