Comments on: Computer Algorithms: Kruskal’s Minimum Spanning Tree

By: Akshay Wagh

Akshay Wagh — Fri, 07 Apr 2017 14:15:31 +0000

I don’t want implementation,just give me an algorithm.

By: Victor

Victor — Tue, 28 Feb 2017 07:37:00 +0000

Just managed to implement it after almost four years. Took some time, but I managed to do it in O(N^2).

By: Preeti

Preeti — Fri, 22 Apr 2016 12:28:50 +0000

I am not able to understand the pseudocode.Please help me.

By: xavier

xavier — Thu, 21 Jan 2016 18:39:40 +0000

would this work with a loop-tree, one where an edge is added between exactly one of the leaves and another node in the tree?

By: Boody

Boody — Thu, 31 Dec 2015 04:41:15 +0000

Good morning ,

Could anyone solve this question ??
or explain it for me !!

(Kruskal)
Give an example of a family of graphs with n nodes and O(n) edges such that a naive implementation
of the union-find data structure without union-by-rank and path compression
leads to quadratic running time for Kruskal’s algorithm.

By: nora

nora — Thu, 04 Dec 2014 09:56:26 +0000

Hi!
Is it possible to let the user chose the graphe by puting the value in the form?
and How can we do it ?And how can we store the graph that we have to use to find the tree whith kruskal algorithm?

By: Victor

Victor — Mon, 06 May 2013 21:12:13 +0000

I’m triying to program this algorithm with C++
I need to keep the size of the distance matrix variable…
Any suggestions?
Thanks

By: Martin

Martin — Wed, 05 Dec 2012 10:44:01 +0000

Hi, indeed a great work here in the first place. Keep going!

In step 2, you are saying to make empty set out of each vertex of G. In my opinion, it shall be 1-element set containing exactly the given vertex. After in the loop, if the vertices are in different components, the sets containing u and v are joined.

By: Stoimen

Stoimen — Tue, 13 Nov 2012 14:33:38 +0000

For the curious readers, here's the PHP implementation of Kruskal's algorithm.

// the graph
$G = array(
    0 => array( 0,  4,  0,  0,  0,  0,  0,  0,  8),
    1 => array( 4,  0,  8,  0,  0,  0,  0,  0,  11),
    2 => array( 0,  8,  0,  7,  0,  4,  2,  0,  0),
    3 => array( 0,  0,  7,  0,  9,  14,  0,  0,  0),
    4 => array( 0,  0,  0,  9,  0,  10,  0,  0,  0),
    5 => array( 0,  0,  4,  14,  10,  0,  0,  2,  0),
    6 => array( 0,  0,  2,  0,  0,  0,  0,  6,  7),
    7 => array( 0,  0,  0,  0,  0,  2,  6,  0,  1),
    8 => array( 8,  11,  0,  0,  0,  0,  7,  1,  0),
);

function Kruskal(&$G)
{
    $len = count($G);
    
    // 1. Make T the empty tree (we'll modify the array G to keep only MST
    $T = array();
    
    // 2. Make a single node trees (sets) out of each vertex
    $S = array();
    foreach (array_keys($G) as $k) {
        $S[$k] = array($k);
    }
    
    // 3. Sort the edges
    $weights = array();
    for ($i = 0; $i < $len; $i++) {
        for ($j = 0; $j < $i; $j++) {
            if (!$G[$i][$j]) continue;
            
            $weights[$i . ' ' . $j] = $G[$i][$j];
        }
    }
    asort($weights);
    
    foreach ($weights as $k => $w) {
        list($i, $j) = explode(' ', $k);
        
        $iSet = find_set($S, $i);
        $jSet = find_set($S, $j);
        if ($iSet != $jSet) {
            $T[] = "Edge: ($i, $j)";
            union_sets($S, $iSet, $jSet);
        }
    }
    
    return $T;
}

function find_set(&$set, $index)
{
    foreach ($set as $k => $v) {
        if (in_array($index, $v)) {
            return $k;
        }
    }
    
    return false;
}

function union_sets(&$set, $i, $j)
{
    $a = $set[$i];
    $b = $set[$j];
    unset($set[$i], $set[$j]);
    $set[] = array_merge($a, $b);
}

$mst = Kruskal($G);

//Edge: (8, 7)
//Edge: (6, 2)
//Edge: (7, 5)
//Edge: (1, 0)
//Edge: (5, 2)
//Edge: (3, 2)
//Edge: (2, 1)
//Edge: (4, 3)
foreach ($mst as $v) {
    echo $v . PHP_EOL;
}