how to save shortest path in dijkstra algorithm - algorithm

So first let's define Dijkstra algorithm:
Dijkstra's algorithm finds single-source shortest paths in a directed graph with non-negative edge weights.
I want to know how can I save the shortest path form s to t with Dijkstra algorithm.
I searched on google, but I couldn't find anything particular; I also changed Dijkstra algorithm, but I could't get any answer. How can I save the shortest path from s to t with Dijkstra?
I know my question is basic and unprofessional, but any help would be appreciated. Thanks for considering my question.

If you look at the pseudocode from the Wikipedia link you gave, you'll see an array in there called prev[]. This array contains, for each node v in the graph, the previous node u in the shortest path between the source node s and v. (This array is also called the predecessor or parent array.)
In other words, the shortest path between s and v is:
s -> u -> v
where u = prev[v]
The path from s to u might have several nodes in between, so to reconstruct the path from s to v, you just walk back along the path defined by the prev[] array using the code snippet below the main pseudocode (target is v):
1 S ← empty sequence
2 u ← target
3 while prev[u] is defined: // Construct the shortest path with a stack S
4 insert u at the beginning of S // Push the vertex onto the stack
5 u ← prev[u] // Traverse from target to source
6 end while

One extremely short way to do so is to use recursion and a "parent array."
If you initialize all of the points' parents to -1, and then as you complete dijkstra's, update the parent array, you are able to recurse back from any point till you get to the source and print out the path. Here is a very short and easy to understand recursion snippet:
// Function to print shortest path from source to j using parent array
void path(parent array, int j)
{
    // Base Case : If j is source
    if (jth element of parent is -1) return;
 
    path(parent, jth element of parent);
  print j;
}
Note that instead of printing "j" out, you can store it in a global vector (or other datatype for languages that are not C-related) for later use.

just a modify form there
# define INF 0x3f3f3f3f
// iPair ==> Integer Pair
typedef pair<int, int> iPair;
void addEdge(vector <pair<int, int> > adj[], int u, int v, int wt)
{
adj[u].push_back(make_pair(v, wt));
adj[v].push_back(make_pair(u, wt));
}
void shortestPath(vector<pair<int, int> > adj[], int V, int src, int target)
{
priority_queue< iPair, vector <iPair>, greater<iPair> > pq;
vector<int> dist(V, INF);
vector<bool> visited(V, false);
vector<int> prev(V, -1);
pq.push(make_pair(0, src));
dist[src] = 0;
while (!pq.empty() && !visited[target])
{
int u = pq.top().second;
pq.pop();
if (visited[u]) {
continue;
}
visited[u] = true;
for (auto x : adj[u])
{
int v = x.first;
int weight = x.second;
if (dist[v] > dist[u] + weight)
{
//relax
dist[v] = dist[u] + weight;
pq.push(make_pair(dist[v], v));
prev[v] = u;
}
}
}
vector<int> res;
res.push_back(target);
int temp = target;
while (temp != 0)
{
temp = prev[temp];
res.push_back(temp);
}
//cout << res;
}
int main()
{
const int V = 9;
vector<iPair > adj[V];
addEdge(adj, 0, 1, 4);
addEdge(adj, 0, 7, 8);
addEdge(adj, 1, 2, 8);
addEdge(adj, 1, 7, 11);
addEdge(adj, 2, 3, 7);
addEdge(adj, 2, 8, 2);
addEdge(adj, 2, 5, 4);
addEdge(adj, 3, 4, 9);
addEdge(adj, 3, 5, 14);
addEdge(adj, 4, 5, 10);
addEdge(adj, 5, 6, 2);
addEdge(adj, 6, 7, 1);
addEdge(adj, 6, 8, 6);
addEdge(adj, 7, 8, 7);
shortestPath(adj, V, 0, 6); //the last one means target
return 0;
}

Most libraries that use this algorithm will likely give you a way to do this. But in general just keep track of the path to each node. Ie give each node an attrribute shortestPathToNode in which you store the list of nodes

Related

How to solve the knapsack Problem when more than two properties of the Item are given (3 variables)

In the knapsack Problem how to solve it if one more property other than that of weight and value of the Item is given? Like recently I was asked a question where I was given 3 properties of the same items, Weight, Value and Type:
int weights[5] = {1, 3, 5, 9, 10};
int values[5] = {11, 13, 1, 19, 9};
int types[5] = {1,2,1,3,4};
int capacity = 26;
We can choose items of each type only once, e.g. we can have only one item of type = 1 in our final knapsack, so we either choose the item with weight 1 or the item with weight 5 both cannot be present. We have to maximize the profit we get.
I was thinking along the lines of a 3D matrix, but I'm not able to think of the proper solution, if I couldn't even think of the base case in this situation for the types, how would the recursive solution look like in this case?
Here is what I tried, I choose each element, the type of the element is put in a set, then we recurse. If the next item type is already present in the set, then we remove it and then recalculate the max value for that particular type among all of them, but if the item type is not present in the set, then we have not considered it, and we add it to the set and proceed further.
I also have a lingering doubt about the base case. Since we can have a different item of the same type which gives a more profit, in my base case, if the value for that particular item type is already considered, then it will return the same output without maximizing it for the second item of the same type. But then I countered myself that the n and w values are different for the same type so it will in fact maximize this too. Is this correct?
Related Questions that I found but weren't answered or Did not have a Pseudo code on which I could build upon:
Solving the knapsack problem with special constraint
How can I solve knapsack problems with 3 variables?
After reading the second link, I'm also curious on how we could apply the constraint of the Volume, how would the code actually look like with the base cases?
I was also suggested to use a map of (string, int) values to store the unique values from the recursion, which method would be better? to use a map or to use 3D table?
Here is my code:
set<int> ttypes;
int static t2[6][26][5];
int knapSack_Memo_Modified(int weights[], int values[], int types[], int w, int n){
if(w <= 0 || n <= 0){
return 0;
}
if(t2[n][w][types[n-1]] != -1){
return t2[n][w][types[n-1]];
}
if(ttypes.count(types[n-1])){
ttypes.erase(types[n-1]);
return t2[n][w][types[n-1]] = max(t2[n][w][types[n-1]], knapSack_Memo_Modified(weights, values, types, w, n-1));
}
else{
ttypes.insert(types[n-1]);
if(weights[n-1] <= w){
return t2[n][w][types[n-1]] = max((values[n-1]+knapSack_Memo_Modified(weights, values, types, w-weights[n-1], n-1)), (knapSack_Memo_Modified(weights, values, types, w, n-1)));
}
else if(weights[n-1] > w){
return t2[n][w][types[n-1]] = knapSack_Memo_Modified(weights, values, types, w, n-1);
}
}
}
int main(){
int weights[5] = {1, 3, 5, 9, 10};
int values[5] = {11, 13, 1, 19, 9};
int types[5] = {1,2,1,3,4};
int capacity = 25;
memset(t, -1, sizeof(t));
int ans = knapSack_Memo_Modified(weights, values, types, capacity, 5);
int mx = 0;
for(int i = 0; i < 6; i++){
for(int j = 0; j < 26; j++){
for(int k = 0; k < 5; k++){
mx = max(mx, t2[i][j][k]);
}
}
}
cout << mx << endl;
}

Boost Graph max-flow algorithm to find out the arcs on the minimal S/T cut

I have an application where for a given fixed number of vertices, there is a need to solve large number of different max-flow algorithms from a given fixed source (S) to a given fixed sink (T). Each max-flow problem differs in that the directed arcs themselves change along with their capacities. As an example, see below.
The number of vertices remains fixed, but the actual arcs and their capacities differ from one problem to the next.
I have the following code that solves the max-flow problem iteratively for Graph 1 and Graph 2 in the figure above using boost thus (apologies for the wall of text, I have tried to make it as minimal as possible. The code below fully compiles on g++ on my linux box, but I am unable to have this correcly compile on online compilers such as wandbox, etc.):
#include <boost/config.hpp>
#include <iostream>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/boykov_kolmogorov_max_flow.hpp>
using namespace boost;
typedef adjacency_list_traits<vecS, vecS, directedS> Traits;
typedef adjacency_list<
vecS, vecS, directedS,
property<
vertex_name_t, std::string,
property<vertex_index_t, int,
property<vertex_color_t, boost::default_color_type,
property<vertex_distance_t, double,
property<vertex_predecessor_t, Traits::edge_descriptor>
> > > >,
property<
edge_index_t, int,
property<edge_capacity_t, double,
property<edge_weight_t, double,
property<edge_residual_capacity_t, double,
property<edge_reverse_t, Traits::edge_descriptor>
> > > > >
Graph;
Graph g;
property_map<Graph, edge_index_t>::type e;
property_map<Graph, edge_capacity_t>::type cap;
property_map<Graph, edge_weight_t>::type cost;
property_map<Graph, edge_residual_capacity_t>::type rescap;
property_map<Graph, edge_reverse_t>::type rev;
property_map<Graph, vertex_color_t>::type colors;
void initialize(int nnodes) {
e = get(edge_index, g);
cap = get(edge_capacity, g);
cost = get(edge_weight, g);
rescap = get(edge_residual_capacity, g);
rev = get(edge_reverse, g);
colors = get(vertex_color, g);
for(int i = 0; i < nnodes; i++)
add_vertex(g);
}
void clearedges() {
Graph::vertex_iterator v, vend;
for (boost::tie(v, vend) = vertices(g); v != vend; ++v)
boost::clear_out_edges(*v, g);
}
void createedges(std::vector<std::pair<int, int>>& arcs, std::vector<double>& capacity) {
Traits::edge_descriptor edf, edr;//forward and reverse
for (int eindex = 0, sz = static_cast<int>(arcs.size()); eindex < sz; eindex++) {
int fr, to;
fr = arcs[eindex].first;
to = arcs[eindex].second;
edf = add_edge(fr, to, g).first;
edr = add_edge(to, fr, g).first;
e[edf] = 2 * eindex;
e[edr] = e[edf] + 1;
cap[edf] = capacity[eindex];
cap[edr] = capacity[eindex];
rev[edf] = edr;
rev[edr] = edf;
}
}
double solve_max_flow(int s, int t) {
double retval = boykov_kolmogorov_max_flow(g, s, t);
return retval;
}
bool is_part_of_source(int i) {
if (colors[i] == boost::black_color)
return true;
return false;
}
int main() {
initialize(6);
std::vector<std::pair<int, int>> arcs1 = { std::make_pair<int,int>(0,1),
std::make_pair<int,int>(0,2),
std::make_pair<int,int>(1,2),
std::make_pair<int,int>(1,3),
std::make_pair<int,int>(1,4),
std::make_pair<int,int>(2,4),
std::make_pair<int,int>(3,4),
std::make_pair<int,int>(3,5),
std::make_pair<int,int>(4,5)
};
std::vector<double> capacities1 = { 10, 10, 10, 10, 1, 4, 3, 2, 10 };
clearedges();
createedges(arcs1, capacities1);
double maxflow = solve_max_flow(0, 5);
printf("max flow is %f\n", maxflow);
for (int i = 0; i < 6; i++)
if (is_part_of_source(i))
printf("Node %d belongs to subset source is in\n", i);
Graph::edge_iterator e_, eend_;
int Eindex = 0;
for (boost::tie(e_, eend_) = edges(g); e_ != eend_; ++e_) {
int fr = source(*e_, g);
int to = target(*e_, g);
printf("(%d) Edge %d: (%d -> %d), capacity %f\n", Eindex, e[*e_], fr, to, cap[*e_]);
Eindex++;
if (is_part_of_source(fr) && is_part_of_source(to) == false)
printf("----is part of ST Cut-----\n");
else
printf("x\n");
}
std::vector<std::pair<int, int>> arcs2 = { std::make_pair<int,int>(0,1),
std::make_pair<int,int>(0,2),
std::make_pair<int,int>(1,3),
std::make_pair<int,int>(2,4),
std::make_pair<int,int>(3,5),
std::make_pair<int,int>(4,5)
};
std::vector<double> capacities2 = { 10, 10, 10, 4, 2, 0 };
clearedges();
createedges(arcs2, capacities2);
maxflow = solve_max_flow(0, 5);
printf("max flow is %f\n", maxflow);
for (int i = 0; i < 6; i++)
if (is_part_of_source(i))
printf("Node %d belongs to subset source is in\n", i);
Eindex = 0;
for (boost::tie(e_, eend_) = edges(g); e_ != eend_; ++e_) {
int fr = source(*e_, g);
int to = target(*e_, g);
printf("(%d) Edge %d: (%d -> %d), capacity %f\n", Eindex, e[*e_], fr, to, cap[*e_]);
Eindex++;
if (is_part_of_source(fr) && is_part_of_source(to) == false)
printf("----is part of ST Cut-----\n");
else
printf("x\n");
}
getchar();
}
I have the following questions.
(a) If the underlying vertices remain fixed, but only the arcs and their capacities change from iteration to iteration, is there anything faster than using clear_out_edges to clear the arcs and then using add_edge to add the new arcs with their new capacities? Also, does clear_out_edges correctly also clear the property map entries that may have the edge descriptor just deleted as key?
(b) Boost max-flow algorithms seem to want the explicit addition of reverse arcs. As of now, in function createedges I explicitly do this via a forward edge descriptor (edf) and a reverse edge descriptor (edr). Is there any performance penalty for this especially when the number of max flow problems that need to be solved is in the 1000s? Is there anything that is more efficient than this?
(c) I am able to correctly enumerate the arcs of the minimal S/T cut via the following portion of the code:
int Eindex = 0;
for (boost::tie(e_, eend_) = edges(g); e_ != eend_; ++e_) {
int fr = source(*e_, g);
int to = target(*e_, g);
printf("(%d) Edge %d: (%d -> %d), capacity %f\n", Eindex, e[*e_], fr, to, cap[*e_]);
Eindex++;
if (is_part_of_source(fr) && is_part_of_source(to) == false)
printf("----is part of ST Cut-----\n");
else
printf("x\n");
}
Is there any more efficient way or enumerating the arcs of the S/T cut than the above?
There's many issues. If you use modern C++ and compiler warnings, you can reduce the code and spot the bugs in printing vertex descriptors (printf is just not safe; use the diagnostics!).
Here's my take after review.
Notable changes:
bundled properties instead of separate interior properties
this implies passing named arguments (but see https://stackoverflow.com/a/64744086/85371)
no more global variables, no more loopy initialization if the simple constructor suffices
no more duplicated code (nothing invites error quite like having capacities1 and capacities2 lying around)
using clear_vertex instead of just clear_out_edges - this may not make a difference (?) but seems to express intent a bit better
no more printf (I'll use libfmt, which is also in c++23), so e.g.
fmt::print("Max flow {}\nNodes {} are in source subset\n", maxflow,
vertices(_g) | filtered(is_source));
prints
Max flow 10
Nodes {0, 1, 2, 3} are in source subset
all in one go
print what you think you are printing. In particular, use the library support for printing edges if you can
Live On Compiler Explorer
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/boykov_kolmogorov_max_flow.hpp>
#include <boost/range/adaptors.hpp>
#include <fmt/ostream.h>
#include <fmt/ranges.h>
using boost::adaptors::filtered;
using Traits = boost::adjacency_list_traits<boost::vecS, boost::vecS, boost::directedS>;
using V = Traits::vertex_descriptor;
using E = Traits::edge_descriptor;
using Capacity = double;
using Color = boost::default_color_type;
struct VertexProps {
// std::string name;
Color color;
Capacity distance;
E predecessor;
};
struct EdgeProps {
int id;
Capacity weight, residual;
E reverse;
};
using Graph = boost::adjacency_list<
boost::vecS, boost::vecS, boost::directedS,
VertexProps,
// see https://stackoverflow.com/a/64744086/85371 :(
boost::property<boost::edge_capacity_t, Capacity, EdgeProps>>;
struct MyGraph {
MyGraph(size_t nnodes) : _g(nnodes) {}
void runSimulation(auto const& arcs, auto const& capacities)
{
reconfigure(arcs, capacities);
Capacity maxflow = solve_max_flow(0, 5);
auto cap = get(boost::edge_capacity, _g);
auto is_source = [this](V v) { return _g[v].color == Color::black_color; };
fmt::print("Max flow {}\nNodes {} are in source subset\n", maxflow,
vertices(_g) | filtered(is_source));
for (E e : boost::make_iterator_range(edges(_g))) {
bool st_cut =
is_source(source(e, _g)) and
not is_source(target(e, _g));
fmt::print("Edge {} (id #{:2}), capacity {:3} {}\n", e, _g[e].id,
cap[e], st_cut ? "(ST Cut)" : "");
}
}
private:
Graph _g;
void reconfigure(auto const& arcs, auto const& capacities)
{
assert(arcs.size() == capacities.size());
for (auto v : boost::make_iterator_range(vertices(_g))) {
// boost::clear_out_edges(v, g);
boost::clear_vertex(v, _g);
}
auto cap = get(boost::edge_capacity, _g);
auto eidx = get(&EdgeProps::id, _g);
auto rev = get(&EdgeProps::reverse, _g);
auto eindex = 0;
for (auto [fr, to] : arcs) {
auto edf = add_edge(fr, to, _g).first;
auto edr = add_edge(to, fr, _g).first;
eidx[edf] = 2 * eindex;
eidx[edr] = eidx[edf] + 1;
cap[edf] = cap[edr] = capacities[eindex];
rev[edf] = edr;
rev[edr] = edf;
++eindex;
}
}
Capacity solve_max_flow(V src, V sink)
{
return boykov_kolmogorov_max_flow(
_g, src, sink,
// named arguments
boost::reverse_edge_map(get(&EdgeProps::reverse, _g))
.residual_capacity_map(get(&EdgeProps::residual, _g))
.vertex_color_map(get(&VertexProps::color, _g))
.predecessor_map(get(&VertexProps::predecessor, _g))
.distance_map(get(&VertexProps::distance, _g))
// end named arguments
);
}
};
int main() {
MyGraph g{6};
using namespace std;
for (auto&& [arcs, capacities] : { tuple
// 1
{vector{pair{0, 1}, {0, 2}, {1, 2}, {1, 3}, {1, 4},
{2, 4}, {3, 4}, {3, 5}, {4, 5}},
vector{10, 10, 10, 10, 1, 4, 3, 2, 10}},
// 2
{vector{pair{0, 1}, {0, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 5}},
vector{10, 10, 10, 4, 2, 0}},
})
{
g.runSimulation(arcs, capacities);
}
}
Prints
Max flow 10
Nodes {0, 1, 2, 3} are in source subset
Edge (0,1) (id # 0), capacity 10
Edge (0,2) (id # 2), capacity 10
Edge (1,0) (id # 1), capacity 10
Edge (1,2) (id # 4), capacity 10
Edge (1,3) (id # 6), capacity 10
Edge (1,4) (id # 8), capacity 1 (ST Cut)
Edge (2,0) (id # 3), capacity 10
Edge (2,1) (id # 5), capacity 10
Edge (2,4) (id #10), capacity 4 (ST Cut)
Edge (3,1) (id # 7), capacity 10
Edge (3,4) (id #12), capacity 3 (ST Cut)
Edge (3,5) (id #14), capacity 2 (ST Cut)
Edge (4,1) (id # 9), capacity 1
Edge (4,2) (id #11), capacity 4
Edge (4,3) (id #13), capacity 3
Edge (4,5) (id #16), capacity 10
Edge (5,3) (id #15), capacity 2
Edge (5,4) (id #17), capacity 10
Max flow 2
Nodes {0, 1, 2, 3, 4} are in source subset
Edge (0,1) (id # 0), capacity 10
Edge (0,2) (id # 2), capacity 10
Edge (1,0) (id # 1), capacity 10
Edge (1,3) (id # 4), capacity 10
Edge (2,0) (id # 3), capacity 10
Edge (2,4) (id # 6), capacity 4
Edge (3,1) (id # 5), capacity 10
Edge (3,5) (id # 8), capacity 2 (ST Cut)
Edge (4,2) (id # 7), capacity 4
Edge (4,5) (id #10), capacity 0 (ST Cut)
Edge (5,3) (id # 9), capacity 2
Edge (5,4) (id #11), capacity 0
Side Note
If you think main is overcomplicated, here's another way to write it for just the two invocations:
Live On Compiler Explorer
g.runSimulation({{0, 1}, {0, 2}, {1, 2}, {1, 3}, {1, 4}, {2, 4}, {3, 4}, {3, 5}, {4, 5}},
{10, 10, 10, 10, 1, 4, 3, 2, 10});
g.runSimulation({{0, 1}, {0, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 5}},
{10, 10, 10, 4, 2, 0});

Approximation Algorithm for non-intersecting paths in a grid

I recently came across this question and thought I could share it here, since I wasn't able to get it.
We are given a 5*5 grid numbered from 1-25, and a set of 5 pairs of points,that are start and end points of a path on the grid.
Now we need to find 5 corresponding paths for the 5 pairs of points, such that no two paths should overlap. Also note that only vertical and horizontal moves are allowed. Also the combined 5 path should cover the entire grid.
For example we are given the pair of points as:
P={1,22},{4,17},{5,18},{9,13},{20,23}
Then the corresponding paths will be
1-6-11-16-21-22
4-3-2-7-12-17
5-10-15-14-19-18
9-8-13
20-25-24-23
What I have thought of so far:
Maybe i can compute all paths from source to destination for all pairs of points and then check if there's no common point in the paths. However this seems to be of higher time complexity.
Can anyone propose a better algorithm? I would be glad if one could explain through a pseudo code.Thanks
This problem is essentially the Hamiltonian path/cycle problem problem (since you can connect the end of one path to the start of another, and consider all the five paths as a part of one big cycle). There are no known efficient algorithms for this, as the problem is NP-complete, so you do essentially need to try all possible paths with backtracking (there are fancier algorithms, but they're not much faster).
Your title asks for an approximation algorithm, but this is not an optimization problem - it's not the case that some solutions are better than others; all correct solutions are equally good, and if it isn't correct, then it's completely wrong - so there is no possibility for approximation.
Edit: The below is a solution to the original problem posted by the OP, which did not include the "all cells must be covered" constraint. I'm leaving it up for those that might face the original problem.
This can be solved with a maximum flow algorithm, such as Edmonds-Karp.
The trick is to model the grid as a graph where there are two nodes per grid cell; one "outgoing" node and one "incoming" node. For each adjacent pair of cells, there are edges from the "outgoing" node in either cell to the "incoming" node in the other cell. Within each cell, there is also an edge from the "incoming" to the "outgoing" node. Each edge has the capacity 1. Create one global source node that has an edge to all the start nodes, and one global sink node to which all end nodes have an edge.
Then, run the flow algorithm; the resulting flow shows the non-intersecting paths.
This works because all flow coming in to a cell must pass through the "internal" edge from the "incoming" to the "ougoing" node, and as such, the flow through each cell is limited to 1 - therefore, no paths will intersect. Also, Edmonds-Karp (and all Floyd-Warshall based flow algorithms) will produce integer flows as long as all capacities are integers.
Here's a program written in Python that walks all potential paths. It uses recursion and backtracking to find the paths, and it marks a grid to see which locations are already being used.
One key optimization is that it marks the start and end points on the grid (10 of the 25 points).
Another optimization is that it generates all moves from each point before starting the "walk" across the grid. For example, from point 1 the moves are to points 2 & 6; from point 7, the moves are to points 2, 6, 8 & 12.
points = [(1,22), (4,17), (5,18), (9,13), (20,23)]
paths = []
# find all moves from each position 0-25
moves = [None] # set position 0 with None
for i in range(1,26):
m = []
if i % 5 != 0: # move right
m.append(i+1)
if i % 5 != 1: # move left
m.append(i-1)
if i > 5: # move up
m.append(i-5)
if i < 21: # move down
m.append(i+5)
moves.append(m)
# Recursive function to walk path 'p' from 'start' to 'end'
def walk(p, start, end):
for m in moves[start]: # try all moves from this point
paths[p].append(m) # keep track of our path
if m == end: # reached the end point for this path?
if p+1 == len(points): # no more paths?
if None not in grid[1:]: # full coverage?
print
for i,path in enumerate(paths):
print "%d." % (i+1), '-'.join(map(str, path))
else:
_start, _end = points[p+1] # now try to walk the next path
walk(p+1, _start, _end)
elif grid[m] is None: # can we walk onto the next grid spot?
grid[m] = p # mark this spot as taken
walk(p, m, end)
grid[m] = None # unmark this spot
paths[p].pop() # backtrack on this path
grid = [None for i in range(26)] # initialize the grid as empty points
for p in range(len(points)):
start, end = points[p]
paths.append([start]) # initialize path with its starting point
grid[start] = grid[end] = p # optimization: pre-set the known points
start, end = points[0]
walk(0, start, end)
Well, I started out thinking about a brute force algorithm, and I left that below, but it turns out it's actually simpler to search for all answers rather than generate all configurations and test for valid answers. Here's the search code, which ended up looking much like #Brent Washburne's. It runs in 53 milliseconds on my laptop.
import java.util.Arrays;
class Puzzle {
final int path[][];
final int grid[] = new int[25];
Puzzle(int[][] path) {
// Make the path endpoints 0-based for Java arrays.
this.path = Arrays.asList(path).stream().map(pair -> {
return new int[] { pair[0] - 1, pair[1] - 1 };
}).toArray(int[][]::new);
}
void print() {
System.out.println();
for (int i = 0; i < grid.length; i += 5)
System.out.println(
Arrays.toString(Arrays.copyOfRange(grid, i, i + 5)));
}
void findPaths(int ip, int i) {
if (grid[i] != -1) return; // backtrack
grid[i] = ip; // mark visited
if(i == path[ip][1]) // path complete
if (ip < path.length - 1) findPaths(ip + 1, path[ip + 1][0]); // find next path
else print(); // solution complete
else { // continue with current path
if (i < 20) findPaths(ip, i + 5);
if (i > 4) findPaths(ip, i - 5);
if (i % 5 < 4) findPaths(ip, i + 1);
if (i % 5 > 0) findPaths(ip, i - 1);
}
grid[i] = -1; // unmark
}
void solve() {
Arrays.fill(grid, -1);
findPaths(0, path[0][0]);
}
public static void main(String[] args) {
new Puzzle(new int[][]{{1, 22}, {4, 17}, {5, 18}, {9, 13}, {20, 23}}).solve();
}
}
Old, bad answer
This problem is doable by brute force if you think about it "backward:" assign all the grid squares to paths and test to see if the assignment is valid. There are 25 grid squares and you need to construct 5 paths, each with 2 endpoints. So you know the paths these 10 points lie on. All that's left is to label the remaining 15 squares with the paths they lie on. There are 5 possibilities for each, so 5^15 in all. That's about 30 billion. All that's left is to build an efficient checker that says whether a given assignment is a set of 5 valid paths. This is simple to do by linear time search. The code below finds your solution in about 2 minutes and takes a bit under 11 minutes to test exhaustively on my MacBook:
import java.util.Arrays;
public class Hacking {
static class Puzzle {
final int path[][];
final int grid[] = new int[25];
Puzzle(int[][] path) { this.path = path; }
void print() {
System.out.println();
for (int i = 0; i < grid.length; i += 5)
System.out.println(
Arrays.toString(Arrays.copyOfRange(grid, i, i + 5)));
}
boolean trace(int p, int i, int goal) {
if (grid[i] != p) return false;
grid[i] = -1; // mark visited
boolean rtn =
i == goal ? !Arrays.asList(grid).contains(p) : nsew(p, i, goal);
grid[i] = p; // unmark
return rtn;
}
boolean nsew(int p, int i, int goal) {
if (i < 20 && trace(p, i + 5, goal)) return true;
if (i > 4 && trace(p, i - 5, goal)) return true;
if (i % 5 < 4 && trace(p, i + 1, goal)) return true;
if (i % 5 > 0 && trace(p, i - 1, goal)) return true;
return false;
}
void test() {
for (int ip = 0; ip < path.length; ip++)
if (!trace(ip, path[ip][0] - 1, path[ip][1] - 1)) return;
print();
}
void enumerate(int i) {
if (i == grid.length) test();
else if (grid[i] != -1) enumerate(i + 1); // already known
else {
for (int ip = 0; ip < 5; ip++) {
grid[i] = ip;
enumerate(i + 1);
}
grid[i] = -1;
}
}
void solve() {
Arrays.fill(grid, -1);
for (int ip = 0; ip < path.length; ip++)
grid[path[ip][0] - 1] = grid[path[ip][1] - 1] = ip;
enumerate(0);
}
}
public static void main(String[] args) {
new Puzzle(new int[][]{{1, 22}, {4, 17}, {5, 18}, {9, 13}, {20, 23}}).solve();
}
}
The starting array:
[ 0, -1, -1, 1, 2]
[-1, -1, -1, 3, -1]
[-1, -1, 3, -1, -1]
[-1, 1, 2, -1, 4]
[-1, 0, 4, -1, -1]
The result:
[ 0, 1, 1, 1, 2]
[ 0, 1, 3, 3, 2]
[ 0, 1, 3, 2, 2]
[ 0, 1, 2, 2, 4]
[ 0, 0, 4, 4, 4]

using binary search to count occurrences

assume I have sorted array A in length n so 1
I need to write pseuodocode of a program that give output of all occurrences of each element.
the algorithm runtime has to be maximum k(c1+c2*log(n)).
example - A=[1,1,2,2,2,5,5,5,5] ----> (1,2)(2,3)(5,4)
I thought about using binary search when the first element I want to count is A[1] and I need to find his last occurrence.
then the next element is A[last occurrence index + 1] and so on.
I have a bit difficult with the idea and writig it down as pseuodocode.
tnx
Recursive algorithm, it gets left and right position and calculates middle position. Going deeper if there is number change, en edge. Up to here it is simple binary search. But once it detects (on distance=1) an edge, change of numbers, it will return it in 4 values: 'what number sequence ended', 'on what position', 'what started', 'on what position'. Parent node then merges these 4 values from left and right side and if it detects complete sequence 'in the middle', it immediately prints it and pass just ending edge (from left side) and starting edge (from right).
It is not possible to achieve that asymptotic complexity.
The reason is no matter what algorithm it is, When all the n elements are distinct, It has to return all the elements.That implies it has to read all of them.Of course, this operation takes O(n).
You can count number of occurrences for one entry in O(log(n))
static int count(int[] array, int target, int start, int end, Func<int, int, bool> compare)
{
if (end < start) { return start; }
int m = (start + end) / 2;
if (compare(target, array[m])) { return count(array, target, start, m - 1, compare); }
else { return count(array, target, m + 1, end, compare); }
}
static void Main(string[] args)
{
int[] a = { 1, 3, 8, 12, 12, 12, 25, 88 };
int r1 = count(a, 12, 0, a.Length - 1, (x1, x2) =>
{
return x1 < x2;
});
int r2 = count(a, 12, 0, a.Length - 1, (x1, x2) =>
{
return x1 <= x2;
});
Console.Out.WriteLine("count=" + (r1 - r2).ToString());
}

Find minimal elements of set of vectors

I have a set of vectors, and I need to write algorithm in java, to find minimal elements of this set. Problem is, that there are elements which are incomparable. E.g. minset{(1,4,6),(3,2,5),(2,3,4),(5,4,6)} = {(1,4,6),(3,2,5),(2,3,4)}. For set of minimal element "minset" following holds: every vector from the original set is either in "minset" or >= than some vector in the new set in every component. E.g. minset{(2,3,4),(2,3,5)} = {(2,3,4)}. I've already have algorithm for this, but I think it can be done with better computional complexity. My algorithm takes one element, mark it as minimal, then take other element, compare them, if there are incomparable, mark both as minimal, if second is smaller then mark only it as minimal etc... Is it possible to use mergesort or heapsort to optimize this algorithm? Thank for all responses.
I created a runnable example. It uses Java's builtin Arrays.sort() and a Comparator which compares two vectors of size L. You could do something similar using Collections.sort() if you prefer to use List data structures over arrays.
From the Java API Specification:
This algorithm offers guaranteed n*log(n) performance.
import java.util.*;
import java.lang.*;
class Main
{
public static void main (String[] args) throws java.lang.Exception
{
final int L = 3;
int[][] vectors = {
{1, 4, 6},
{3, 2, 5},
{2, 3, 4},
{5, 4, 6},
{7, 7, 7},
{3, 3, 5},
{8, 8, 8},
};
Comparator<int[]> cmp = new Comparator<int[]>() {
public int compare(int[] v1, int[] v2) {
int cmp0 = Integer.signum(v1[0] - v2[0]);
for (int i = 0; i < L; i++) {
int cmp1 = Integer.signum(v1[i] - v2[i]);
if (cmp1 != 0) {
if (cmp1 != cmp0) {
return 0;
}
cmp0 = cmp1;
}
}
return cmp0;
}
};
Arrays.sort(vectors, cmp);
System.out.println("minset:");
int i = 0;
int[] vPref = vectors[0];
while (cmp.compare(vectors[i], vPref) == 0) {
for (int x : vectors[i]){
System.out.print(x + ", ");
}
System.out.println();
vPref = vectors[i];
i++;
}
}
}
Pseudocode:
foreach inputVector in vectors
foreach minVector in minSet
if (all components of inputVector <= components of minVector
delete minVector
elseif (all components of inputVector >= components of minVector)
skip to next inputVector
if inputVector made it through the entire minSet, then add it to minSet
I found solution for my problem in article http://repository.cmu.edu/cgi/viewcontent.cgi?article=2758&context=compsci, where is algorithm for finding maximal elements of set of vectors which is similar problem. It works in much more better computional complexity that my firts intuitive algorithm.

Resources