JAVA - SCRIPT-PRIM'S ALGORITHM - MINIMUM SPANNING TREE - CODE

JAVA PYTHON GO KOTLIN C# RUBY C++ Java Script

Example :

<html>
<body>    
<script> 

    class Edge {

    }

    function heapify(verticesArray, root, length) {
        var left = (2*root)+1
        var right = (2*root)+2
        var smallest = root

        if (left < length && right <= length && vertexVal[verticesArray[right]] < vertexVal[verticesArray[left]]) {
            smallest = right
      	}
        else if (left <= length) {
            smallest = left
      	}  

        if (vertexVal[verticesArray[root]] > vertexVal[verticesArray[smallest]]) {
            var temp = verticesArray[root]
            verticesArray[root] = verticesArray[smallest]
            verticesArray[smallest] = temp
            heapify(verticesArray, smallest, length)
        }	
    } 

    function buildHeap() {
        verticesArray = Object.keys(vertexVal)

        for (let parent = Math.trunc(((verticesArray.length-1)-1)/ 2); parent > 0; parent--) {
            heapify(verticesArray, parent, verticesArray.length-1)
        }	
        verticesKeyArray = verticesArray
    }	

    function updateHeap(vertex, length) {
        vertexVal[vertex] = length
        verticesArray = Object.keys(vertexVal)

        for (let parent = Math.trunc(((verticesArray.length-1)-1)/ 2); parent > 0; parent--) {
            heapify(verticesArray, parent, verticesArray.length-1)
        }		
        verticesKeyArray = verticesArray
    }	

    function containsVertex(vertex) {
        if (vertex in vertexVal)
            return true
        else
            return false
    }		

    function deleteMin() {
        var temp = verticesKeyArray[0]
        lengthArray = verticesKeyArray.length-1
        verticesKeyArray[0] = verticesKeyArray[lengthArray]
        delete vertexVal[temp]

        verticesKeyArray.length = lengthArray
        if (lengthArray>0) 
            heapify(verticesKeyArray, 0, lengthArray-1)
        return temp
    }	

    function getWeight( vertex) {
        return vertexVal[vertex]
    }

    function empty() {

        if (verticesKeyArray.length > 0) 
            return false
        else
            return true
    }		

    function primMST() {

        vertexVal = {}

        // Vertex to Edge Object
        vertexToEdge = {}

        // Assign all the initial values as infinity for all the Vertices.
        for (v of vertices)
            vertexVal[v] = Infinity

        // Call buildHeap() to create the MinHeap
        buildHeap()

      	// Replace the value of start vertex to 0.
        updateHeap("a",0)

        // Continue until the Min-Heap is not empty.
        while (empty() !== true) {
            // Extract minimum value vertex from Map in Heap
            currentVertex = deleteMin()
         	// Need to get the edge for the vertex and add it to the Minimum Spanning Tree..
         	// Just note, the edge for the source vertex will not be added.

            spanningTreeEdge = vertexToEdge[currentVertex]

            if(spanningTreeEdge != null) 
                result.push(spanningTreeEdge)

            // Get all the adjacent vertices and iterate through them.
            for (edge of getEdges(currentVertex)) {
                adjacentVertex = edge.endVertex

          		// We check if adjacent vertex exist in 'Map in Heap' and length of the edge is with this vertex
          		// is greater than this edge length.
                if(containsVertex(adjacentVertex) && getWeight(adjacentVertex) > edge.value) {
            		// Replace the edge length with this edge weight.
                    updateHeap(adjacentVertex, edge.value)
                    vertexToEdge[adjacentVertex] = edge
                }
            }
        }
    }		

    function getEdges(vertex) {
        var edgeList = []
        var i = vertices.indexOf(vertex)
        for (iter of adjcList[i])
            edgeList.push(iter)
        return edgeList
    }	

    function constructAdjacencyList(vertex1, vertex2, edgeVal) {
        edge = new Edge()
        edge.startVertex = vertex1
        edge.endVertex = vertex2
        edge.value = edgeVal

        adjcList.push([])
        adjcList[vertices.indexOf(vertex1)].push(edge)
    }	

    function insertVertex(vertex) {
        vertices.push(vertex)
    }

    function printEdgeList() {
        for (edge of result)
            document.write("The Edge between ", edge.startVertex ," and ", edge.endVertex, " is ", edge.value, "</br>")
    }	


    var verticesKeyArray = []
    var adjcList = [[]]
    var vertices = []
    var vertexVal = {}

    // Stores the Minimum spanning Tree
    var result = []

    // Insert Vertices
    insertVertex("a")
    insertVertex("b")
    insertVertex("c")
    insertVertex("d")
    insertVertex("e")

    // Create Adjacency List with Edges.
    constructAdjacencyList("a", "b",1)
    constructAdjacencyList("a", "c",5)
    constructAdjacencyList("a", "d",4)

    constructAdjacencyList("b", "a",1)
    constructAdjacencyList("b" ,"e",8)

    constructAdjacencyList("c", "a",5)
    constructAdjacencyList("c", "d",12)
    constructAdjacencyList("c", "e",9)

    constructAdjacencyList("d", "a", 4)
    constructAdjacencyList("d", "c", 12)
    constructAdjacencyList("d", "e", 3)

    constructAdjacencyList("e", "b", 8)
    constructAdjacencyList("e", "c", 9)
    constructAdjacencyList("e", "d", 3)

    primMST()

    printEdgeList()

</script>
</body>
</html>

Output :

The Edge between a and b is 1
The Edge between a and c is 5
The Edge between a and d is 4
The Edge between d and e is 3

Note : To understand, Prim's Algorithm for Minimum Spanning Tree. Prior knowledge of 'Heap' and 'Adjacency List' data structure is needed. But if you don't know, no worries. We will explain them in brief.

Let us recapitulate the 3 step process for Prim's Algorithm :

First step is, we select any vertex and start from it(We have selected the vertex a in this case).

Then, we try finding the adjacent Edges of that Vertex(In this case, we try finding the adjacent edges of vertex a).

This is where, we will be using Adjacency List for finding the Adjacent Edges.

Next, we will take all the Adjacent edges from above and check, which edge has the smallest length/weight.

Again, this is where, we will be using Min-Heap for finding the Adjacent Edges.

Code explanation for Prim's Algorithm - Minimum Spanning Tree

Let us take the below graph, to understand the above code.

The first task is to construct the Graph. So, we need to store the Vertices first.

vertices = []

And also we know, we have to represent the edges as Adjacency List.

adjcList = [[]]

Eventually, in the main program. We have initialised the vertices,

// Insert Vertices
insertVertex("a")
insertVertex("b")
insertVertex("c")
insertVertex("d")
insertVertex("e")

And, created the Adjacency List with Edges.

// Create Adjacency List with Edges.
constructAdjacencyList("a", "b",1)
constructAdjacencyList("a", "c",5)
constructAdjacencyList("a", "d",4)

constructAdjacencyList("b", "a",1)
constructAdjacencyList("b" ,"e",8)

constructAdjacencyList("c", "a",5)
constructAdjacencyList("c", "d",12)
constructAdjacencyList("c", "e",9)

constructAdjacencyList("d", "a", 4)
constructAdjacencyList("d", "c", 12)
constructAdjacencyList("d", "e", 3)

constructAdjacencyList("e", "b", 8)
constructAdjacencyList("e", "c", 9)
constructAdjacencyList("e", "d", 3)

Let us understand creation process of Adjacency List. If you already know it, you can skip.

Click Here - For the creation process of Adjacency List.

Explanation of function constructAdjacencyList(vertex1, vertex2, edgeVal) method

function constructAdjacencyList(vertex1, vertex2, edgeVal) {
	edge = new Edge()
	edge.startVertex = vertex1
	edge.endVertex = vertex2
	edge.value = edgeVal

	adjcList.push([])
	adjcList[vertices.indexOf(vertex1)].push(edge)
}

Let us take the example of the Edge (a,b) with length 1 and understand the above code.

constructAdjacencyList("a", "b", 1)

Now, if we look at the method definition,

function constructAdjacencyList(vertex1, vertex2, edgeVal)

vertex1 is initialised with a.

vertex2 is initialised with b.

edgeVal is initialised with 1.

The next thing we do is, create an Edge object.

edge = new Edge()

And initialise the Edge edge object with the start vertex, end vertex and the length of the Edge.

edge.startVertex = vertex1
edge.endVertex = vertex2
edge.value = edgeVal

And the Edge object looks somewhat like,

Now, since we have the Edge (a,b) initialised. The next task would be, to add this Edge to the AdjacencyList.

And as we have seen, there is a 2D LinkedList to store the Adjacency List.

adjcList = [[]]

Now, we will be creating the first row of LinkedList,

adjcList.push([])

And then we will try finding out, for which vertex we are going to insert the Edge.

adjcList[vertices.indexOf(vertex1)].push(edge)

In simple words, vertices.indexOf(vertex1) will tell us about the position of the start vertex.

In this case, the start vertex is a. And vertices.indexOf(vertex1) will tell us about its position, i.e. 0.

So, if we substitute the line,

adjcList[vertices.indexOf(vertex1)].push(edge)

With the value of vertices.indexOf(vertex1),

adjcList[0].push(edge)

The edge would be added to the 0th location of the 2d LinkedList.

This is how a 2D LinkedList looks like. Just remember, the empty blocks are not yet created, but will be created eventually.

For now, only the first row is created with the first column where the edge

is inserted.

Next, when we try to insert the second adjacent edge of a i.e. a,c.

constructAdjacencyList("a", "c", 5)

An edge object will be created

edge.startVertex = vertex1
edge.endVertex = vertex2
edge.value = edgeVal

And the Edge object looks somewhat like,

Once again the start vertex is a. And vertices.indexOf(vertex1) will tell us about its position, i.e. 0.

So, if we substitute the line,

adjcList[vertices.indexOf(vertex1)].push(edge)

With the value of vertices.indexOf(vertex1),

adjcList[0].push(edge)

The edge would be added to the 1st row of the 2d LinkedList, just next to the first edge.

Similarly, we insert the third adjacent edge of a i.e. a,d.

constructAdjacencyList("a", "d", 4)

An edge object will be created

edge.startVertex = vertex1
edge.endVertex = vertex2
edge.value = edgeVal

And the Edge object looks somewhat like,

Once again the start vertex is a. And vertices.indexOf(vertex1) will tell us about its position, i.e. 0.

So, if we substitute the line,

adjcList[vertices.indexOf(vertex1)].push(edge)

With the value of vertices.indexOf(vertex1),

adjcList[0].push(edge)

The edge would be added to the 1st row of the 2d LinkedList, just next to the first edge.

Similarly, we insert the adjacent edge of b i.e. b,a.

constructAdjacencyList("b", "a", 1);

An edge object will be created

edge.startVertex = vertex1
edge.endVertex = vertex2
edge.value = edgeVal

And the Edge object looks somewhat like,

Now, the start vertex is b. And vertices.indexOf(vertex1) will tell us about its position, i.e. 1.

So, if we substitute the line,

adjcList[vertices.indexOf(vertex1)].push(edge)

With the value of vertices.indexOf(vertex1),

adjcList[1].push(edge)

And this time, the edge would be added to the 2nd row of the 2d LinkedList.

And eventually the AdjacencyList with Edges would be created with the 2D LinkedList.

Once we are done creating the AdjacencyList with Edges. The next task would be to call the actual method for Prim's Algorithm,

primMST()

Before we understand primMST() method in detail. Let us understand the Min-Heap Data Structure.

If you already know Min-Heap Data Structure, you can skip the below part.

Click Here - For Min-Heap Data Structure Recap.

There are four important method in the MinHeap class that are quite important.

They are :

function deleteMin()

function updateHeap(vertex, length)

function buildHeap()

function heapify(verticesArray, root, length)

Now, if we look at the primMST() method.

The Object vertexVal is getting initialised here.

vertexVal = {}

Next, we initialise the Object with the vertices as key and assign the value as Infinity.

// Assign all the initial values as infinity for all the Vertices.
for (v of vertices)
	vertexVal[v] = Infinity

So, the Object vertexVal is loaded with values.

Next, we call the buildHeap() method to create the MinHeap, with the values of the Object vertexVal.

buildHeap()

Explanation of function buildHeap() method :

function buildHeap() {
	verticesArray = Object.keys(vertexVal)

	for (let parent = Math.trunc(((verticesArray.length-1)-1)/ 2); parent > 0; parent--) {
		heapify(verticesArray, parent, verticesArray.length-1)
	}
	verticesKeyArray = verticesArray
}

Just remember, Object is not a good Data Structure for Heap. Whereas array is an excellent Data Structure for Heap.

So, we take the keys from the Object and create an Array out of it.

To do this, there is a two step process involved.

First, we extract the key from the Object and put it into an Array type variable.

verticesArray = Object.keys(vertexVal)

Then, we divide the Array into two parts and run a for loop. Then call the Heapify method.

for (let parent = Math.trunc(((verticesArray.length-1)-1)/ 2); parent > 0; parent--) {
	heapify(verticesArray, parent, verticesArray.length-1)
}

So, what does the heapify(..) method do?

Let us see.

Explanation of function heapify(verticesArray, root, length) method :

function heapify(verticesArray, root, length) {
	var left = (2*root)+1
	var right = (2*root)+2
	var smallest = root

	if (left < length && right <= length && vertexVal[verticesArray[right]] < vertexVal[verticesArray[left]]) {
		smallest = right
}
	else if (left <= length) {
		smallest = left
}

	if (vertexVal[verticesArray[root]] > vertexVal[verticesArray[smallest]]) {
		var temp = verticesArray[root]
		verticesArray[root] = verticesArray[smallest]
		verticesArray[smallest] = temp
		heapify(verticesArray, smallest, length)
	}
}

Since, all the values of the Object vertexVal are infinity now. So, it doesn't matter how the values will be inserted in the Min-Heap.

For the sake of understanding, let us understand the functionality of heapify(...) method.

So, we have passed the verticesArray, root element and the length of the Array to heapify(...) method.

Then, we have calculated the left, right and root element of the Heap.

left = (2*root)+1
right = (2*root)+2
smallest = root

Since, this is a Min-Heap. The smallest element would be at the root of the Heap.

So, we try initialising the smallest element with its right value.

And the below if condition checks, if the left and right element is less than the length of the Array.

if (left < length && right <= length && vertexVal[verticesArray[right]] < vertexVal[verticesArray[left]]) {
	smallest = right
}
else if (left <= length) {
	smallest = left
}

And, if the element on the right side of the Object vertexVal is less than the element on the right side.

vertexVal[verticesArray[right]] < vertexVal[verticesArray[left]]

Then element on the right side of the Object is the smallest element.

smallest = right

And in the else part, we can assume that the element on the left side of the Object has the smallest element.

smallest = left

Now, since we got the smallest element. We need to check if the actual root element is greater than the smallest element or not.

if (vertexVal[verticesArray[root]] > vertexVal[verticesArray[smallest]]) {
	var temp = verticesArray[root]
	verticesArray[root] = verticesArray[smallest]
	verticesArray[smallest] = temp
	heapify(verticesArray, smallest, length)
}

And this is where, we swap the contents of the root element and the smallest element. Assuming the element in the root is greater than the smallest element.

Which shouldn't be. As the root element should always be the smallest element.

Then a recursive call is made to heapify(...) method.

heapify(verticesArray, smallest, length)

And the recursion continues until all the elements are arranged in MinHeap.

In the next line, we would update the value of source element a to 0.

// Replace the value of start vertex to 0.
updateHeap("a",0)

Now, if we see the updateHeap(...) method,

Explanation of function updateHeap(vertex, length) method :

function updateHeap(vertex, length) {
	vertexVal[vertex] = length
	verticesArray = Object.keys(vertexVal)

	for (let parent = Math.trunc(((verticesArray.length-1)-1)/ 2); parent > 0; parent--) {
		heapify(verticesArray, parent, verticesArray.length-1)
	}
	verticesKeyArray = verticesArray
}

The first thing we do is, replace the value in the Object vertexVal.

vertexVal[vertex] = length

Then we follow the same process we followed above. i.e. Extract the keys from the Object. And convert into an Array.

verticesArray = Object.keys(vertexVal)

Then we call the heapify(...) method. Because we have updated a new value and in that case the Heap has to be rearranged to form a Min-Heap.

Now, that we have understood Min-Heap. Let us understand the details about the actual method that calculates the Minimum Spanning Tree using Prim's Algorithm.

Explanation of function primMST() method :

function primMST() {

	vertexVal = {}

	// Vertex to Edge Object
	vertexToEdge = {}

	// Assign all the initial values as infinity for all the Vertices.
	for (v of vertices)
		vertexVal[v] = Infinity

	// Call buildHeap() to create the MinHeap
	buildHeap()

// Replace the value of start vertex to 0.
	updateHeap("a",0)

	// Continue until the Min-Heap is not empty.
	while (empty() !== true) {
		// Extract minimum value vertex from Map in Heap
		currentVertex = deleteMin()
	// Need to get the edge for the vertex and add it to the Minimum Spanning Tree..
	// Just note, the edge for the source vertex will not be added.

		spanningTreeEdge = vertexToEdge[currentVertex]

		if(spanningTreeEdge != null)
			result.push(spanningTreeEdge)

	// Get all the adjacent vertices and iterate through them.
		for (edge of getEdges(currentVertex)) {
			adjacentVertex = edge.endVertex

	// We check if adjacent vertex exist in 'Map in Heap' and length of the edge is with this vertex
	// is greater than this edge length.
			if(containsVertex(adjacentVertex) && getWeight(adjacentVertex) > edge.value) {
		// Replace the edge length with this edge weight.
				updateHeap(adjacentVertex, edge.value)
				vertexToEdge[adjacentVertex] = edge
			}
		}
	}
}

Three things to remember :

vertexVal;

The Object with key as vertex and value as the Edge length. We put this in the Heap. Also we will call the Object vertexVal as Map in the Heap.

vertexToEdge

The Object with key as vertex and value as the actual Edge.

result

The List that stores the final result. i.e. All the edges to be included in the Minimum Spanning Tree.

Now, let us look at the steps involved :

So, firstly we initialise the Object vertexVal. Which we are going to place in the Heap. Also, we will call the Object vertexVal as Map in the Heap.
```
vertexVal = {}
```

Then we assign the Object vertexToEdge, where we will be storing the vertex as key and Edge as value.
```
vertexToEdge = {}
```

The next task is to assign the value infinity to all the vertices in the Object vertexToEdge.
```
for (v of vertices)
	vertexVal[v] = Infinity
```
It can be said as infinity as it has a very large value that can be compared with infinity.

Next, we call the buildHeap(...) method . That will create the Heap for us, by placing the elements in the right order.
```
buildHeap()
```
Just remember, in a Min-Heap the root element is always smaller than its left and right child.

Then we replace the value of the start element to 0 by calling the updateHeap(...) method.
```
updateHeap("a",0)
```

Then we come to the while loop, that continues until the Min-Heap is empty.

while (empty() !== true) {
	// Extract minimum value vertex from Map in Heap
	currentVertex = deleteMin()
// Need to get the edge for the vertex and add it to the Minimum Spanning Tree..
// Just note, the edge for the source vertex will not be added.
	spanningTreeEdge = vertexToEdge[currentVertex]
	if(spanningTreeEdge != null)
		result.push(spanningTreeEdge)
// Get all the adjacent vertices and iterate through them.
	for (edge of getEdges(currentVertex)) {
		adjacentVertex = edge.endVertex
// We check if adjacent vertex exist in 'Map in Heap' and length of the edge is with this vertex
// is greater than this edge length.
		if(containsVertex(adjacentVertex) && getWeight(adjacentVertex) > edge.value) {
	// Replace the edge length with this edge weight.
			updateHeap(adjacentVertex, edge.value)
			vertexToEdge[adjacentVertex] = edge
		}
	}
}

Inside the while loop, the first thing we do is, get the smallest element from the Heap. That is the root element.

Also we delete the root element from the Heap at the same time.

And the deleteMin() method of the Heap does the work for us.
```
currentVertex = deleteMin()
```
Since, the vertex a has the smallest value i.e. 0. It is deleted from vertexVal. i.e. Heap in the Map.

Then we go to the Object vertexToEdge which has the key as vertex and the value as actual Edge. And try getting the Edge for that corresponding vertex (i.e. a).

But in this case, the Object vertexToEdge has not yet being initialised.

Why is it so?

Because a is the starting vertex and we won't be adding the starting vertex to the Minimum Spanning Tree.

So, the Edge spanningTreeEdge gets null and is not added to the Mimimim Spanning Tree.
```
spanningTreeEdge = vertexToEdge[currentVertex]
if(spanningTreeEdge != null)
		result.push(spanningTreeEdge)
```

The next task we have done is getting the adjacent vertices and iterate through and get the corresponding Edges for the vertices.

And the getEdges() method helps us getting the Adjacent Edges of the vertex a.

for (edge of getEdges(currentVertex)) {
	adjacentVertex = edge.endVertex
	// We check if adjacent vertex exist in 'Map in Heap' and length of the edge is with this vertex
	// is greater than this edge length.
	if(containsVertex(adjacentVertex) && getWeight(adjacentVertex) > edge.value) {
		// Replace the edge length with this edge weight.
		updateHeap(adjacentVertex, edge.value)
		vertexToEdge[adjacentVertex] = edge
	}
}

Once we have the first Adjacent Edge i.e. a,b,

We would only take the end vertex from it. And edge object already has the endVertex.
```
adjacentVertex = edge.endVertex
```

Now, that we got the adjacentVertex i.e. b. The next thing we check is, if this adjacentVertex i.e. b, exist in Map in Heap and also check, if the value of the edge length associated with this vertex i.e. The value associated with vertex b is greater than this edge length i.e. The length of edge a,b. Which is 1.

So, we find that vertex b is present in vertexVal i.e. Map in heap.

And the value of b in vertexVal i.e. Map in heap is ∞. Which is greater that the value of the edge i.e. 1.

So, the if condition is satisfied and we get into the if block.
```
if(containsVertex(adjacentVertex) && getWeight(adjacentVertex) > edge.value) {
	// Replace the edge length with this edge weight.
	updateHeap(adjacentVertex, edge.value)
	vertexToEdge[adjacentVertex] = edge
}
```

Now, since we are in the blocks of if statement, we had updated the vertex with the new value of the Edge i.e. 1.
```
updateHeap(adjacentVertex, edge.value)
```
Similarly, we put the actual edge i.e. a,b to the value of vertex b in the Object vertexToEdge that stores vertex as key and the Edge associated to it as value.
```
vertexToEdge[adjacentVertex] = edge
```

The similar process gets repeated and we get the Minimum Spanning Tree using Prim's Algorithm.

Note : Have you noticed the link between the Object vertexToEdge and vertexVal(i.e. Map in Heap)? The Edge length is stored in the Object vertexVal(i.e. Map in Heap) and the actual Edge associated to it is stored in the Object vertexToEdge.

Say vertex b has the Edge length 1 that is stored in the Object vertexVal(i.e. Map in Heap) and the actual Edge a,b is stored in the Object vertexToEdge.

Time Complexity of Prim's Algorithm for Minimum Spanning Tree

The time complexity of Prim's Algorithm for Minimum Spanning Tree is : O(E logV)

Where E is the Edge

And V is the Vertex

The time complexity O(E logV) is only when we will be using the above combination of MIn-Heap and Adjacency List.

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