## Monday, December 29, 2014

### UVa Problem 10603 - Fill

Problem:

Solution:

Model the state space as a graph, using the amount of water poured as the edge weight. With that, we see that is a single source shortest path problem, with one caveat that we do not know if the destination state is reachable.

More precisely - note that given the rule, there is water is neither created nor destroyed, so the amount of water is always the same, the amount of water in the first jug and the amount of water in the second jug determines the current state (i.e. we can always recover how much water is in the last jug using subtraction). To ease programming, such a state is represented as a single base 201 number with two digits.

Using the water pouring rule we can generate the neighbors of a particular state. To be efficient, we don't want to generate all reachable states upfront, we could just explore the graph using the Dijkstra's algorithm. To do that, I adapted the Dijkstra implementation to take an implicit graph as an argument and generates the neighbor on the fly.

When the distance of a node is confirmed, it is passed through a call back to the implicit graph. The call back will record the current best, and if the current best is the goal, we stop the search. That allow us to record both the reaching distance, and in case it is not reachable, the best possible outcome.

As an aside, in the coding I had a bug that was hard to find, turn out I found the bug by trying out boundary values. As a reflection, if I could treat my code as a black box, I might be able to come up with the testing plan faster, turn out the input space of this code is not that large.

Code:

#include "stdafx.h"

// http://uva.onlinejudge.org/index.php?option=onlinejudge&page=show_problem&problem=1544

// #define LOG

#include "UVa10603.h"

#include <iostream>
#include <vector>
#include <map>

using namespace std;

class UVa10603_implicit_graph;

bool UVa10603_less(vector<int>& distances, vector<bool>& reachable, int one, int two);
void UVa10603_bubble_down(vector<int>& dijkstra_queue, vector<int>& distances, vector<bool>& reachable, vector<int>& node_index, int& queue_size, int parent_index);
int UVa10603_delete_min(vector<int>& dijkstra_queue, vector<int>& distances, vector<bool>& reachable, vector<int>& node_index, int& queue_size);
void UVa10603_change_key(vector<int>& dijkstra_queue, vector<int>& distances, vector<bool>& reachable, vector<int>& node_index, int& queue_size, int changed_node);
void UVa10063_solve(int a, int b, int c, int d);
void UVa10603_dijkstra(int src, UVa10603_implicit_graph* implicit_graph);

int UVa10603()
{
int number_of_test_cases;
cin >> number_of_test_cases;
for (int test_case = 1; test_case <= number_of_test_cases; test_case++)
{
int a, b, c, d;
cin >> a;
cin >> b;
cin >> c;
cin >> d;
UVa10063_solve(a, b, c, d);
}

return 0;
}

bool UVa10603_less(vector<int>& distances, vector<bool>& reachable, int one, int two)
{
if (reachable[one])
{
if (reachable[two])
{
return distances[one] < distances[two];
}
else
{
return true;
}
}
else
{
return false;
}
}

void UVa10603_bubble_down(vector<int>& dijkstra_queue, vector<int>& distances, vector<bool>& reachable, vector<int>& node_index, int& queue_size, int parent_index)
{
while (true)
{
int child2_index = (parent_index + 1) * 2;
int child1_index = child2_index - 1;

if (child2_index < queue_size)
{
int parent_value = dijkstra_queue[parent_index];
int child1_value = dijkstra_queue[child1_index];
int child2_value = dijkstra_queue[child2_index];

if (UVa10603_less(distances, reachable, child1_value, child2_value))
{
if (UVa10603_less(distances, reachable, child1_value, parent_value))
{
dijkstra_queue[parent_index] = child1_value;
dijkstra_queue[child1_index] = parent_value;
node_index[parent_value] = child1_index;
node_index[child1_value] = parent_index;
parent_index = child1_index;
}
else
{
break;
}
}
else
{
if (UVa10603_less(distances, reachable, child2_value, parent_value))
{
dijkstra_queue[parent_index] = child2_value;
dijkstra_queue[child2_index] = parent_value;
node_index[parent_value] = child2_index;
node_index[child2_value] = parent_index;
parent_index = child2_index;
}
else
{
break;
}
}
}
else if (child1_index < queue_size)
{
int parent_value = dijkstra_queue[parent_index];
int child1_value = dijkstra_queue[child1_index];

if (UVa10603_less(distances, reachable, child1_value, parent_value))
{
dijkstra_queue[parent_index] = child1_value;
dijkstra_queue[child1_index] = parent_value;
node_index[parent_value] = child1_index;
node_index[child1_value] = parent_index;
parent_index = child1_index;
}
else
{
break;
}
}
else
{
break;
}
}
}

// In order to avoid building the graph, an implicit graph is used
// The graph, however, knows the maximum number of nodes, which fit this particular case
// General implicit graph might not have this privilege
class UVa10603_implicit_graph
{
public:
UVa10603_implicit_graph(int _a, int _b, int _c, int _d) : a(_a), b(_b), c(_c), d(_d), best_volume_achieved(0), archieving_distance(0)
{
}

int size()
{
return 201 * 201;
}

// A node is presented as a base 201 number
// representing the volume of water in a and b respectively
vector<pair<int, int> > get_neighbors(int node)
{
int a_curr_volume = node / 201;
int b_curr_volume = node % 201;
int c_curr_volume = c - a_curr_volume - b_curr_volume;

vector<pair<int, int> > result;

int a_next_volume = -1;
int b_next_volume = -1;
int c_next_volume = -1;
int poured_volume = -1;
int neighbor = -1;

if (a_curr_volume > 0)
{
pour(a_curr_volume, b_curr_volume, b, &a_next_volume, &b_next_volume, &poured_volume);
c_next_volume = c - a_next_volume - b_next_volume;
neighbor = a_next_volume * 201 + b_next_volume;
#ifdef LOG
log_neighbor(node, neighbor, poured_volume);
#endif
result.push_back(pair<int, int>(neighbor, poured_volume));

pour(a_curr_volume, c_curr_volume, c, &a_next_volume, &c_next_volume, &poured_volume);
b_next_volume = c - a_next_volume - c_next_volume;
neighbor = a_next_volume * 201 + b_next_volume;
#ifdef LOG
log_neighbor(node, neighbor, poured_volume);
#endif
result.push_back(pair<int, int>(neighbor, poured_volume));
}

if (b_curr_volume > 0)
{
pour(b_curr_volume, a_curr_volume, a, &b_next_volume, &a_next_volume, &poured_volume);
c_next_volume = c - a_next_volume - b_next_volume;
neighbor = a_next_volume * 201 + b_next_volume;
#ifdef LOG
log_neighbor(node, neighbor, poured_volume);
#endif
result.push_back(pair<int, int>(neighbor, poured_volume));

pour(b_curr_volume, c_curr_volume, c, &b_next_volume, &c_next_volume, &poured_volume);
a_next_volume = c - b_next_volume - c_next_volume;
neighbor = a_next_volume * 201 + b_next_volume;
#ifdef LOG
log_neighbor(node, neighbor, poured_volume);
#endif
result.push_back(pair<int, int>(neighbor, poured_volume));
}

if (c_curr_volume > 0)
{
pour(c_curr_volume, a_curr_volume, a, &c_next_volume, &a_next_volume, &poured_volume);
b_next_volume = c - a_next_volume - c_next_volume;
neighbor = a_next_volume * 201 + b_next_volume;
#ifdef LOG
log_neighbor(node, neighbor, poured_volume);
#endif
result.push_back(pair<int, int>(neighbor, poured_volume));

pour(c_curr_volume, b_curr_volume, b, &c_next_volume, &b_next_volume, &poured_volume);
a_next_volume = c - b_next_volume - c_next_volume;
neighbor = a_next_volume * 201 + b_next_volume;
#ifdef LOG
log_neighbor(node, neighbor, poured_volume);
#endif
result.push_back(pair<int, int>(neighbor, poured_volume));
}
#ifdef LOG
cout << endl;
#endif
return result;
}

bool on_node_explored(int node, int distance)
{
int a_curr_volume = node / 201;
int b_curr_volume = node % 201;
int c_curr_volume = c - a_curr_volume - b_curr_volume;

if (a_curr_volume == d || b_curr_volume == d || c_curr_volume == d)
{
best_volume_achieved = d;
archieving_distance = distance;
return true;
}
else
{
if (a_curr_volume < d && a_curr_volume > best_volume_achieved)
{
best_volume_achieved = a_curr_volume;
archieving_distance = distance;
}
if (b_curr_volume < d && b_curr_volume > best_volume_achieved)
{
best_volume_achieved = b_curr_volume;
archieving_distance = distance;
}

if (c_curr_volume < d && c_curr_volume > best_volume_achieved)
{
best_volume_achieved = c_curr_volume;
archieving_distance = distance;
}
return false;
}
}

void conclude()
{
cout << archieving_distance << " " << best_volume_achieved << endl;
}

#ifdef LOG
void log_neighbor(int from, int to, int cost)
{
int from_a = from / 201;
int from_b = from % 201;
int from_c = c - from_a - from_b;

int to_a = to / 201;
int to_b = to % 201;
int to_c = c - to_a - to_b;

cout << "(" << from_a << ", " << from_b << ", " << from_c << "):" << from << " -" << cost << "-> (" << to_a << ", " << to_b << ", " << to_c << "):" << to << endl;
}
#endif

private:
int a;
int b;
int c;
int d;

int best_volume_achieved;
int archieving_distance;

void pour(int src_cur_volume, int dst_cur_volume, int dst_max_volume, int* src_result_volume, int* dst_result_volume, int* poured_volume)
{
int dst_avail_volume = dst_max_volume - dst_cur_volume;
if (src_cur_volume <= dst_avail_volume)
{
// dst have enough space to hold them all
*src_result_volume = 0;
*dst_result_volume = dst_cur_volume + src_cur_volume;
*poured_volume = src_cur_volume;
}
else
{
// dst don't have enought space to hold them all
*src_result_volume = src_cur_volume - dst_avail_volume;
*dst_result_volume = dst_cur_volume + dst_avail_volume;
*poured_volume = dst_avail_volume;
}
}
};

int UVa10603_delete_min(vector<int>& dijkstra_queue, vector<int>& distances, vector<bool>& reachable, vector<int>& node_index, int& queue_size)
{
int u = dijkstra_queue[0];
node_index[u] = -1;
if (queue_size > 1)
{
dijkstra_queue[0] = dijkstra_queue[queue_size - 1];
dijkstra_queue[queue_size - 1] = -1;
node_index[dijkstra_queue[0]] = 0;
queue_size--;
UVa10603_bubble_down(dijkstra_queue, distances, reachable, node_index, queue_size, 0);
}
else
{
dijkstra_queue[queue_size - 1] = -1;
queue_size--;
}
return u;
}

void UVa10603_change_key(vector<int>& dijkstra_queue, vector<int>& distances, vector<bool>& reachable, vector<int>& node_index, int& queue_size, int changed_node)
{
int child_index = node_index[changed_node];
while (child_index != 0)
{
int parent_index = (child_index + 1) / 2 - 1;

int child_value = dijkstra_queue[child_index];
int parent_value = dijkstra_queue[parent_index];

if (UVa10603_less(distances, reachable, child_value, parent_value))
{
dijkstra_queue[child_index] = parent_value;
dijkstra_queue[parent_index] = child_value;
node_index[parent_value] = child_index;
node_index[child_value] = parent_index;
child_index = parent_index;
}
else
{
break;
}
}
}

void UVa10603_dijkstra(int src, UVa10603_implicit_graph* implicit_graph)
{
vector<int> dijkstra_queue; // containing the node numbers
vector<bool> reachable;     // false if the node is unreachable
vector<int> node_index;     // the priority queue position of the node in the queue
vector<int> distances;
int queue_size;
int num_nodes = implicit_graph->size();
dijkstra_queue.resize(num_nodes);
distances.resize(num_nodes);
reachable.resize(num_nodes);
node_index.resize(num_nodes);
queue_size = num_nodes;

// Step 1: Initialize the nodes and get them into the queue
int j = 1;
for (int i = 0; i < num_nodes; i++)
{
if (i == src)
{
distances[i] = 0;
reachable[i] = true;
dijkstra_queue[0] = i;
node_index[i] = 0;
}
else
{
distances[i] = -1;
reachable[i] = false;
dijkstra_queue[j] = i;
node_index[i] = j;
j++;
}
}

// Step 2: Main Loop
while (queue_size > 0)
{
int explored = UVa10603_delete_min(dijkstra_queue, distances, reachable, node_index, queue_size);

if (reachable[explored])
{
if(implicit_graph->on_node_explored(explored, distances[explored]))
{
break;
};

vector<pair<int, int> > neighbors = implicit_graph->get_neighbors(explored);
for (unsigned int ni = 0; ni < neighbors.size(); ni++)
{
pair<int, int> neighbor_edge = neighbors[ni];
int neighbor = neighbor_edge.first;
int new_neighbor_distance = distances[explored] + neighbor_edge.second;
if (!reachable[neighbor] || new_neighbor_distance < distances[neighbor])
{
distances[neighbor] = new_neighbor_distance;
reachable[neighbor] = true;
UVa10603_change_key(dijkstra_queue, distances, reachable, node_index, queue_size, neighbor);
}
}
}
else
{
break;
}
}
}

void UVa10063_solve(int a, int b, int c, int d)
{
UVa10603_implicit_graph* graph = new UVa10603_implicit_graph(a, b, c, d);
UVa10603_dijkstra(0, graph);
graph->conclude();
delete graph;
}