<< problem 105 - Special subset sums: testing | Diophantine reciprocals I - problem 108 >> |

# Problem 107: Minimal network

(see projecteuler.net/problem=107)

The following undirected network consists of seven vertices and twelve edges with a total weight of 243.

The same network can be represented by the matrix below.

ABCDEFG

A 161221

B16 1720

C12 28 31

D211728 181923

E 20 18 11

F 3119 27

G 231127

However, it is possible to optimise the network by removing some edges and still ensure that all points on the network remain connected.

The network which achieves the maximum saving is shown below. It has a weight of 93, representing a saving of 243 − 93 = 150 from the original network.

Using network.txt (right click and 'Save Link/Target As...'), a 6K text file containing a network with forty vertices,

and given in matrix form, find the maximum saving which can be achieved by removing redundant edges whilst ensuring that the network remains connected.

# Algorithm

I use Prim's algorithm (see en.wikipedia.org/wiki/Prim's_algorithm):

- `done`

contains these nodes where all connections are optimized

- `done`

contains initially an arbitrarily chosen node

- algorithm is finished when `done`

contains all nodes

- in each iteration, one node is added to `done`

One iteration performs these tasks:

- store all edges in `next`

where one node is in `done`

and the other node isn't

- sort `next`

by the weight

- pick the lowest

- add that edge to the optimized graph `minimal`

- add the other edge's node to `done`

, too

There will be 40-1=39 iterations. The most time-consuming part is locating all potential edges.

The "true" algorithm of Prim uses a priority queue quite efficiently whereas I rebuild it from scratch every time.

Nevertheless, my code solves the 2500x2500 networks of Hackerrank within less than a second.

## Modifications by HackerRank

- the input format is completely different

- two nodes may be connected via multiple edges

- print the optimized weight instead of the gain

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, as well as the input data, too.

The code contains `#ifdef`

s to switch between the original problem and the Hackerrank version.

Enable `#ifdef ORIGINAL`

to produce the result for the original problem (default setting for most problems).

//#define ORIGINAL

#include <iostream>
#include <set>
#include <map>
#include <queue>
typedef unsigned int Node;
typedef unsigned int Weight;
// connect two nodes and assign a weight

struct Edge
{
Node from;
Node to;
Weight weight;
// for sorting
bool operator<(const Edge& other) const
{
// STL's priority queue returns the __largest__ element first, therefore I invert the comparison sign
if (weight != other.weight)
return weight > other.weight;
// sort by nodes, too, to avoid collisions of nodes with the same weight
// the order doesn't matter, these nodes just "have to be different"
if (from != other.from)
return from < other.from;
return to < other.to;
}
};
// all edges (full network)

std::map<std::pair<Node, Node>, Weight> network;
// all nodes

std::set<Node> nodes;
// weight of the full network, including duplicate connections

unsigned int initialSum = 0;
// insert an edge into the network, true if already existing

void addEdge(Node from, Node to, Weight weight)
{
// count all originals weights
initialSum += weight;
// all edges have the lower ID first (I can do that because it's an undirected graph)
if (from > to)
std::swap(from, to);
// already existing edge ?
auto fromTo = std::make_pair(from, to);
if (network.count(fromTo) != 0)
// has the existing edge a lower or the same weight ? => we're done
if (network[fromTo] <= weight)
return;
// store new (or improved) connection between two nodes
network[fromTo] = weight;
nodes.insert(from);
nodes.insert(to);
}
int main()
{
#ifdef ORIGINAL
// read Project Euler's weird file format
for (unsigned int i = 0; i < 40; i++)
for (unsigned int j = 0; j < 40; j++)
{
char c = 0;
unsigned int weight = 0;
// CSV-format
while (c != ',' && c != '\n')
{
c = std::cin.get();
// ignore dashes
if (c >= '0' && c <= '9')
{
weight *= 10;
weight += c - '0';
}
}
// only valid weights
if (weight != 0 && i < j)
addEdge(i, j, weight);
}
#else
// read Hackerrank input
unsigned int numNodes, numEdges;
std::cin >> numNodes >> numEdges;
for (unsigned int i = 0; i < numEdges; i++)
{
Node from, to;
Weight weight;
std::cin >> from >> to >> weight;
addEdge(from, to, weight);
}
#endif
// optimized graph, initially empty
std::set<Edge> minimal;
// start with a random node
std::set<Node> done;
done.insert(*nodes.begin());
// not all nodes optimized yet ?
while (done.size() < nodes.size())
{
// add all edges where one node is part of tree and the other isn't
std::priority_queue<Edge> next;
for (auto e : network)
{
auto fromTo = e.first;
bool hasFrom = done.count(fromTo.first) != 0;
bool hasTo = done.count(fromTo.second) != 0;
if (hasFrom == hasTo) // both nodes are already optimized or both aren't ?
continue;
next.push({fromTo.first, fromTo.second, e.second});
}
// get edge with minimal weight
auto add = next.top();
// add new edge to tree
done.insert(add.from); // one insert is redundant but set::set throws it away
done.insert(add.to);
// add edge to the optimized graph
minimal.insert(add);
}
// "measure" both graphs
unsigned int optimizedSum = 0;
for (auto i : minimal)
optimizedSum += i.weight;
#ifdef ORIGINAL
// difference
auto gain = initialSum - optimizedSum;
std::cout << gain << std::endl;
#else
std::cout << optimizedSum << std::endl;
#endif
return 0;
}

This solution contains 25 empty lines, 28 comments and 10 preprocessor commands.

# Interactive test

*This feature is not available for the current problem.*

# Benchmark

The correct solution to the original Project Euler problem was found in **less than 0.01** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

May 11, 2017 submitted solution

May 11, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler107

My code solved **8** out of **8** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **35%** (out of 100%).

Hackerrank describes this problem as **easy**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Links

projecteuler.net/thread=107 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-107-efficient-network/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p107.java (written by Nayuki)

Scala: github.com/samskivert/euler-scala/blob/master/Euler107.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

*Please click on a problem's number to open my solution to that problem:*

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<< problem 105 - Special subset sums: testing | Diophantine reciprocals I - problem 108 >> |