The Strassen’s method of matrix multiplication is a typical divide and conquer algorithm. We’ve seen so far some divide and conquer algorithms like merge sort and the Karatsuba’s fast multiplication of large numbers. However let’s get again on what’s behind the divide and conquer approach.

Unlike the dynamic programming where we “expand” the solutions of sub-problems in order to get the final solution, here we are talking more on joining sub-solutions together. These solutions of some sub-problems of the general problem are equal and their merge is somehow well defined.

A typical example is the merge sort algorithm. In merge sort we have two sorted arrays and all we want is to get the array representing their union again sorted. Of course, the tricky part in merge sort is the merging itself. That’s because we’ve to pass through the two arrays, A and B, and we’ve to compare each “pair” of items representing an item from A and from B. A bit off topic, but this is the weak point of merge sort and although its worst-case time complexity is O(n.log(n)), quicksort is often preferred in practice because there’s no “merge”. Quicksort just concatenates the two sub-arrays. Note that in quicksort the sub-arrays aren’t with an equal length in general and although its worst-case time complexity is O(n^2) it often outperforms merge sort.

This simple example from the paragraph above shows us how sometimes merging the solutions of two sub-problems actually isn’t a trivial task to do. Thus we must be careful when applying any divide and conquer approach.

History

Volker Strassen is a German mathematician born in 1936. He is well known for his works on probability, but in the computer science and algorithms he’s mostly recognized because of his algorithm for matrix multiplication that’s still one of the main methods that outperforms the general matrix multiplication algorithm.

Strassen firstly published this algorithm in 1969 and proved that the n^3 algorithm isn’t the optimal one. Actually the given solution by Strassen is slightly better, but his contribution is enormous because this resulted in many more researches about matrix multiplication that led to some faster approaches, i.e. the Coppersmith-Winograd algorithm with O(n^2,3737).

Overview

The general algorithm on multiplying two matrices A[NxN] and B[NxN] is fairly simple. Although it’s more difficult than multiplying two numbers and also it is not commutative it’s still very simple – but slow.

Let’s first define what’s a matrix A[NxN]. As we speak about matrices NxN we usually think of a square grid with N rows and N columns. In each row and column A[i][j] we’ve a value.

Of course, as developers, we can think of a matrix as a two-dimensional array.

// PHP two-dimensional array
$a = array(
    0 => array($v1, $v2, $v3, $v4),
    1 => array($v5, $v6, $v7, $v8),
    2 => array($v9, $v10, $v11, $v12),
); 

Don’t forget that a NxN matrix is just a private case for a matrix. We can equally likely have any other size of a matrix NxM (N <> M).

However the size of a matrix is crucial in order to multiply it with another matrix. Why is that?

As I said above multiplying matrices isn’t the same as multiplying numbers. First of all this operation isn’t commutative.

And the second problem is the way you multiply two matrices A with B.

Just because this works with NxN matrices we can see the problem with multiplying rectangular matrices. Indeed, this wouldn’t be possible unless the second dimension of A isn’t exactly equal to the first dimension of B.

Hopefully we are now talking about square matrices with exactly the same dimensions.

OK, so now we know how to multiply two square matrices (with the same dimensions NxN) and now let’s evaluate the time complexity for the general purpose algorithm.

As we know A.B = C only when:

C[i][j] = sum(A[i][k] * B[k][j]) for k = 0 .. n

Thus we have n^3 operations. Let’s try to find out a divide and conquer approach.

Indeed this isn’t difficult in case of matrices because as we know we can divide in matrix in smaller sub-matrices.

Now what do we have?

Again – the same complexity – we have 8 products and 4 sums. Where’s the catch?

Of course in order to get faster solution we’ve to be looking as Strassen did in 1969. He defined P1, P2, P3, P4, P5, P6 and P7 as defined on the image below.

Complexity

As I mentioned above the Strassen’s algorithm is slightly faster than the general matrix multiplication algorithm. The general algorithm’s time complexity is O(n^3), while the Strassen’s algorithm is O(n^2.80).

You can see on the chart below how slightly faster is this even for large n.

Application

Although this algorithm seems to be more close to pure mathematics than to computer practically everywhere we use NxN arrays we can benefit from matrix multiplication.

In the other hand the algorithm of Strassen is not much faster than the general n^3 matrix multiplication algorithm. That’s very important because for small n (usually n < 45) the general algorithm is practically a better choice. However as you can see from the chart above for n > 100 the difference can be very big.

In the same time typically NxN arrays are used always when we talk about adjacency matrix of graphs |V| = n and some graph algorithms practically depend on matrix multiplication.

Along with the Kruskal’s minimum spanning tree algorithm, there’s another general algorithm that solves the problem. The algorithm of Prim.

As we already know the algorithm of Kruskal works in a pretty natural and logical way. Since we’re trying to build a MST, which is naturally build by the minimal edges of the graph (G), we sort them in a non-descending order and we start building the tree.

During the whole process of building the final minimum spanning tree Kruskal’s algorithm keeps a forest of trees. The number of trees in that forest decreases on each step and finally we get the minimum weight spanning tree.

A key point in the Kruskal’s approach is the way we get the “next” edge from G that should be added to one of the trees of the forest (or to connect two trees from the forest). The only thing we should be aware of is to choose an edge that’s connecting two vertices – u and v and these two shouldn’t be in the same tree. That’s all.

An important feature of the Kruskal’s algorithm is that it builds the MST just by sorting the edges by their weight and doesn’t care about a particular starting vertex.

In the same time there’s another algorithm that builds a MST – the algorithm of Prim designed by Robert Prim in 1957.

Overview

The idea behind the Prim’s algorithm is rather different from Kruskal’s approach. During the process of building the MST this algorithm keeps a single tree, which is finally sub-tree of the final minimum weight spanning tree.

On each step we chose an edge which we add to the growing tree that finally forms the MST.

It is somehow unnatural approach! We start from a given vertex and initially we don’t choose the lightest edge. Thus during the whole process the tree grows, but outside the tree (T) there might be edges that are lighter than those in the tree (i.e. the edge (5, 1) from the tree above is lighter than (2, 5) but (2, 5) is added to the growing tree before the edge (5, 1)).

Compared to the Kruskal’s algorithm this time everything seems to be really unnatural. How we should be sure the final tree (T) will be a minimum spanning tree since we don’t get the lightest edge on each step?

Actually we are sure that the final tree is a MST because of another obvious feature of the minimum spanning trees. They should “connect” all the vertices of G, thus somehow at least one edge reaching each vertex will appear in the MST. Thus we shouldn’t care where do we start, the only important thing is to choose the lightest edge that’s visible so far.

This algorithm looks much like Dijkstra’s shortest path in a graph, because we start from a vertex, we push all the edges starting from this node to a priority queue and we chose the lightest edge. Going to the next node connected by this edge we append to the queue all the edges that aren’t in the queue.

That way the queue grows and we get always the lightest edge – thus forming a priority queue.

Now let’s summarize the algorithm of Prim

Pseudo Code

As an initial input we have the graph (G) and a starting vertex (s).

1.  Make a queue (Q) with all the vertices of G (V);
2.  For each member of Q set the priority to INFINITY;
3.  Only for the starting vertex (s) set the priority to 0;
4.  The parent of (s) should be NULL;
5.  While Q isn’t empty
6.     Get the minimum from Q – let’s say (u); (priority queue);
7.     For each adjacent vertex to (v) to (u)
8.        If (v) is in Q and weight of (u, v) < priority of (v) then
9.           The parent of (v) is set to be (u)
10.          The priority of (v) is the weight of (u, v)

Indeed it looks much like the Dijkstra’s algorithm.

Code

Here’s a PHP implementation of the algorithm of Prim, which directly follows the pseudo code.

// Prim's algorithm

define('INFINITY', 100000000);

// 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 prim(&$graph, $start)
{
    $q = array(); // queue
    $p = array(); // parent
    
    foreach (array_keys($graph) as $k) {
        $q[$k] = INFINITY;
    }
    
    $q[$start] = 0;
    $p[$start] = NULL;
    
    asort($q);
    
    while ($q) {
        // get the minimum value
        $keys = array_keys($q);
        $u = $keys[0];
        
        foreach ($graph[$u] as $v => $weight) {
            if ($weight > 0 && in_array($v, $keys) && $weight < $q[$v]) {
                $p[$v] = $u;
                $q[$v] = $weight;
            }
        }
        
        unset($q[$u]);
        asort($q);
    }
    
    return $p;
}

prim($G, 5);

History

It’s curious to say that the algorithm developed by Robert Prim isn’t developed by him. It’s considered that a Czech mathematician Vojtech Jarnik discovered back in 1930. However now we know this algorithm as the algorithm of Prim, which independently discovered it in 1957 as I said above, and finally Edsger Dijkstra described it in 1959. That’s why his algorithm on finding the single-source shortest paths in a graph looks so much to this algorithm. Perhaps by finding this algorithm on minimum spanning tree Dijkstra discovered how we can find the shortest paths to all vertices using a priority queue. Indeed the paths to all other vertices use the edges of the minimum spanning tree.

Just because Jarnik found and described this algorithm 27 years earlier than Robert Prim, today it’s more convenient to call this algorithm the Prim-Jarnik algorithm.