## Friday, January 2, 2015

### UVa Problem 563 - Crimewave

Problem:

Solution:

Without competitive programming putting it under the heading of maximum flow, I wouldn't have thought about using maximum flow for this problem at all.

Look like we should consider maximum flow when we are asked to find multiple paths, after all maximum flow is about finding multiple augmenting paths.

The modeling to this problem is easy once it is told. Split all the vertex and make them have capacity 1, have a master source to all banks being rob, and make outside of the grid a master sink, that's all.

Code:

#include "stdafx.h"

// http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=504

#include "UVa563.h"

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

using namespace std;

// #define LOG

int UVa563_get_incoming_node_id(int row, int col, int number_of_rows, int number_of_cols, int number_of_nodes);
int UVa563_get_outgoing_node_id(int row, int col, int number_of_rows, int number_of_cols, int number_of_nodes);
int UVa563_Edmonds_Karps(vector<vector<int>>& capacities, vector<vector<int>>& adjacency_list, int src, int dst);

int UVa563()
{
int number_of_test_case;
cin >> number_of_test_case;
for (int test_case = 1; test_case <= number_of_test_case; test_case++)
{
int number_of_rows;
int number_of_cols;
int number_of_robs;
cin >> number_of_rows;
cin >> number_of_cols;
cin >> number_of_robs;
vector<pair<int, int>> robs;
robs.resize(number_of_robs);
for (int r = 0; r < number_of_robs; r++)
{
int rob_row;
int rob_col;
cin >> rob_row;
cin >> rob_col;
rob_row--;
rob_col--;
robs[r] = pair<int, int>(rob_row, rob_col);
}

// Step 2: Graph modeling
// for each crossing, model as a splitted vertex (row * num_col + col) * 2 and (row * num_col + col) * 2 + 1
// one master source, one master sink (number_of_node - 2, number_of_nodes - 1)
// all cost is 1, cost does not matter really
int number_of_nodes = number_of_rows * number_of_cols * 2 + 2;

vector<vector<int>> capacities;
capacities.resize(number_of_nodes);
for (int src = 0; src < number_of_nodes; src++)
{
capacities[src].resize(number_of_nodes);
for (int dst = 0; dst < number_of_nodes; dst++)
{
capacities[src][dst] = 0;
}
}

for (int row = 0; row < number_of_rows; row++)
{
for (int col = 0; col < number_of_cols; col++)
{
int source_node_1 = UVa563_get_incoming_node_id(row, col, number_of_rows, number_of_cols, number_of_nodes);
int source_node_2 = UVa563_get_outgoing_node_id(row, col, number_of_rows, number_of_cols, number_of_nodes);
int target_node_1 = UVa563_get_incoming_node_id(row - 1, col, number_of_rows, number_of_cols, number_of_nodes);
int target_node_2 = UVa563_get_incoming_node_id(row + 1, col, number_of_rows, number_of_cols, number_of_nodes);
int target_node_3 = UVa563_get_incoming_node_id(row, col - 1, number_of_rows, number_of_cols, number_of_nodes);
int target_node_4 = UVa563_get_incoming_node_id(row, col + 1, number_of_rows, number_of_cols, number_of_nodes);

capacities[source_node_1][source_node_2] = 1;
capacities[source_node_2][target_node_1] = 1;
capacities[source_node_2][target_node_2] = 1;
capacities[source_node_2][target_node_3] = 1;
capacities[source_node_2][target_node_4] = 1;

}
}

int master_source = number_of_nodes - 2;
int master_sink = number_of_nodes - 1;
for (vector<pair<int, int>>::iterator ri = robs.begin(); ri != robs.end(); ri++)
{
int rob_row = ri->first;
int rob_col = ri->second;

int rob_target = UVa563_get_incoming_node_id(rob_row, rob_col, number_of_rows, number_of_cols, number_of_nodes);
capacities[master_source][rob_target] = 1;
}

int total_flow = UVa563_Edmonds_Karps(capacities, adjacency_list, master_source, master_sink);
if (total_flow == robs.size())
{
cout << "possible" << endl;
}
else
{
cout << "not possible" << endl;
}
}
return 0;
}

int UVa563_get_incoming_node_id(int row, int col, int number_of_rows, int number_of_cols, int number_of_nodes)
{
if (row < 0 || row >= number_of_rows || col < 0 || col >= number_of_cols)
{
return number_of_nodes - 1;
}
else
{
return (row * number_of_cols + col) * 2;
}
}

int UVa563_get_outgoing_node_id(int row, int col, int number_of_rows, int number_of_cols, int number_of_nodes)
{
if (row < 0 || row >= number_of_rows || col < 0 || col >= number_of_cols)
{
return number_of_nodes - 1;
}
else
{
return (row * number_of_cols + col) * 2 + 1;
}
}

int UVa563_Edmonds_Karps(vector<vector<int>>& capacities, vector<vector<int>>& adjacency_list, int src, int dst)
{
int total_flow = 0;
// Step 2: Edmonds Karp's
vector<int> parents; // Allow back-tracking the path found from bfs
int number_of_nodes = capacities.size();
parents.resize(number_of_nodes); // avoid reallocation
while (true)
{
// Step 2.1: Use BFS to find an augmenting flow
queue<int> bfs_queue;
for (int n = 0; n < number_of_nodes; n++)
{
parents[n] = -1; // indicating the node is not enqueued
}

parents[src] = -2; // indicating the node is enqueued but no actual parent because this is the root
bfs_queue.push(src);
while (bfs_queue.size() > 0)
{
int current = bfs_queue.front();
bfs_queue.pop();
{
int neighbor = *ni;
if (parents[neighbor] == -1 && capacities[current][neighbor] > 0)
{
parents[neighbor] = current;
bfs_queue.push(neighbor);

if (neighbor == dst)
{
break;
}
}
}
if (parents[dst] != -1)
{
break;
}
}

if (parents[dst] == -1)
{
break;
}
else
{
// We have found an augmenting path, go through the path and find the max flow through this path
int cur = dst;
bool first = true;
int max_flow_through_path = 0;
while (true)
{
int src = parents[cur];
if (src != -2)
{
int dst = cur;
int available = capacities[src][dst];
#ifdef LOG
cout << src << "--" << available << "->" << dst << endl;
#endif
cur = parents[cur];
if (first)
{
max_flow_through_path = available;
first = false;
}
else
{
max_flow_through_path = min(max_flow_through_path, available);
}
}
else
{
break;
}
}
#ifdef LOG
cout << "flowing " << max_flow_through_path << endl << endl;
#endif
total_flow += max_flow_through_path;
// Flow the max flow through the augmenting path
cur = dst;
while (true)
{
int src = parents[cur];
if (src != -2)
{
capacities[src][cur] -= max_flow_through_path;
capacities[cur][src] += max_flow_through_path;
cur = parents[cur];
}
else
{
break;
}
}
}
}

}