Earlier we have seen the basics of Dijkstra algorithm. In this article, we will see its implementation using the adjacency list and Min Heap.

brief: **What is Dijkstra’s algorithm?**

- Dijkstra algorithm is a greedy algorithm.
- It finds a shortest-path tree for a weighted undirected graph.
- This means it finds the shortest paths between nodes in a graph, which may represent, for example, road networks
- For a given source node in the graph, the algorithm finds the shortest path between the source node and every other node.
- This algorithm also used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined.
- Dijkstra’s algorithm is very similar to Prim’s algorithm. In Prim’s algorithm, we create minimum spanning tree (MST) and in the Dijkstra algorithm, we create a
*shortest-path tree (SPT)*from the given source.

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**Implementation – **Adjacency List and Min Heap

- Create min Heap of size = no of vertices.
- Create a heapNode for each vertex which will store two pieces of information. a). vertex b). Distance from vertex from source vertex.
- Use spt[] to keep track of the vertices which are currently in min-heap.
- For each heapNode, initialize distance as +∞ except the heapNode for the source vertex for which distance will be 0.
- while minHeap is not empty
- Extract the min node from the heap, say it vertex
and add it to the SPT.*u,* - Decrease distance: For adjacent vertex v, if v is not in SPT[] and distance[v] > distance[u] + edge u-v
then update*weight**distance[v] = distance[u] + edge u-v weight*

- Extract the min node from the heap, say it vertex

**Time Complexity:**

Total vertices: V, Total Edges : E

- O(logV) – to extract each vertex from heap. So for V vertices – O(VlogV)
- O(logV) – each time decrease the distance of a vertex. Decrease distance will be called for at most once for each edge. So for total E edge – O(ElogV)
- So over all complexity: O(VlogV) + O(ElogV) = O((E+V)logV) =
**O(ElogV)**

See the animation below for more understanding

**Complete Code: **

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import java.util.LinkedList; | |

public class DijkstraUsingMinHeap { | |

static class Edge { | |

int source; | |

int destination; | |

int weight; | |

public Edge(int source, int destination, int weight) { | |

this.source = source; | |

this.destination = destination; | |

this.weight = weight; | |

} | |

} | |

static class HeapNode{ | |

int vertex; | |

int distance; | |

} | |

static class Graph { | |

int vertices; | |

LinkedList<Edge>[] adjacencylist; | |

Graph(int vertices) { | |

this.vertices = vertices; | |

adjacencylist = new LinkedList[vertices]; | |

//initialize adjacency lists for all the vertices | |

for (int i = 0; i <vertices ; i++) { | |

adjacencylist[i] = new LinkedList<>(); | |

} | |

} | |

public void addEdge(int source, int destination, int weight) { | |

Edge edge = new Edge(source, destination, weight); | |

adjacencylist[source].addFirst(edge); | |

edge = new Edge(destination, source, weight); | |

adjacencylist[destination].addFirst(edge); //for undirected graph | |

} | |

public void dijkstra_GetMinDistances(int sourceVertex){ | |

int INFINITY = Integer.MAX_VALUE; | |

boolean[] SPT = new boolean[vertices]; | |

// //create heapNode for all the vertices | |

HeapNode [] heapNodes = new HeapNode[vertices]; | |

for (int i = 0; i <vertices ; i++) { | |

heapNodes[i] = new HeapNode(); | |

heapNodes[i].vertex = i; | |

heapNodes[i].distance = INFINITY; | |

} | |

//decrease the distance for the first index | |

heapNodes[sourceVertex].distance = 0; | |

//add all the vertices to the MinHeap | |

MinHeap minHeap = new MinHeap(vertices); | |

for (int i = 0; i <vertices ; i++) { | |

minHeap.insert(heapNodes[i]); | |

} | |

//while minHeap is not empty | |

while(!minHeap.isEmpty()){ | |

//extract the min | |

HeapNode extractedNode = minHeap.extractMin(); | |

//extracted vertex | |

int extractedVertex = extractedNode.vertex; | |

SPT[extractedVertex] = true; | |

//iterate through all the adjacent vertices | |

LinkedList<Edge> list = adjacencylist[extractedVertex]; | |

for (int i = 0; i <list.size() ; i++) { | |

Edge edge = list.get(i); | |

int destination = edge.destination; | |

//only if destination vertex is not present in SPT | |

if(SPT[destination]==false ) { | |

///check if distance needs an update or not | |

//means check total weight from source to vertex_V is less than | |

//the current distance value, if yes then update the distance | |

int newKey = heapNodes[extractedVertex].distance + edge.weight ; | |

int currentKey = heapNodes[destination].distance; | |

if(currentKey>newKey){ | |

decreaseKey(minHeap, newKey, destination); | |

heapNodes[destination].distance = newKey; | |

} | |

} | |

} | |

} | |

//print SPT | |

printDijkstra(heapNodes, sourceVertex); | |

} | |

public void decreaseKey(MinHeap minHeap, int newKey, int vertex){ | |

//get the index which distance's needs a decrease; | |

int index = minHeap.indexes[vertex]; | |

//get the node and update its value | |

HeapNode node = minHeap.mH[index]; | |

node.distance = newKey; | |

minHeap.bubbleUp(index); | |

} | |

public void printDijkstra(HeapNode[] resultSet, int sourceVertex){ | |

System.out.println("Dijkstra Algorithm: (Adjacency List + Min Heap)"); | |

for (int i = 0; i <vertices ; i++) { | |

System.out.println("Source Vertex: " + sourceVertex + " to vertex " + + i + | |

" distance: " + resultSet[i].distance); | |

} | |

} | |

} | |

static class MinHeap{ | |

int capacity; | |

int currentSize; | |

HeapNode[] mH; | |

int [] indexes; //will be used to decrease the distance | |

public MinHeap(int capacity) { | |

this.capacity = capacity; | |

mH = new HeapNode[capacity + 1]; | |

indexes = new int[capacity]; | |

mH[0] = new HeapNode(); | |

mH[0].distance = Integer.MIN_VALUE; | |

mH[0].vertex=–1; | |

currentSize = 0; | |

} | |

public void display() { | |

for (int i = 0; i <=currentSize; i++) { | |

System.out.println(" " + mH[i].vertex + " distance " + mH[i].distance); | |

} | |

System.out.println("________________________"); | |

} | |

public void insert(HeapNode x) { | |

currentSize++; | |

int idx = currentSize; | |

mH[idx] = x; | |

indexes[x.vertex] = idx; | |

bubbleUp(idx); | |

} | |

public void bubbleUp(int pos) { | |

int parentIdx = pos/2; | |

int currentIdx = pos; | |

while (currentIdx > 0 && mH[parentIdx].distance > mH[currentIdx].distance) { | |

HeapNode currentNode = mH[currentIdx]; | |

HeapNode parentNode = mH[parentIdx]; | |

//swap the positions | |

indexes[currentNode.vertex] = parentIdx; | |

indexes[parentNode.vertex] = currentIdx; | |

swap(currentIdx,parentIdx); | |

currentIdx = parentIdx; | |

parentIdx = parentIdx/2; | |

} | |

} | |

public HeapNode extractMin() { | |

HeapNode min = mH[1]; | |

HeapNode lastNode = mH[currentSize]; | |

// update the indexes[] and move the last node to the top | |

indexes[lastNode.vertex] = 1; | |

mH[1] = lastNode; | |

mH[currentSize] = null; | |

sinkDown(1); | |

currentSize—; | |

return min; | |

} | |

public void sinkDown(int k) { | |

int smallest = k; | |

int leftChildIdx = 2 * k; | |

int rightChildIdx = 2 * k+1; | |

if (leftChildIdx < heapSize() && mH[smallest].distance > mH[leftChildIdx].distance) { | |

smallest = leftChildIdx; | |

} | |

if (rightChildIdx < heapSize() && mH[smallest].distance > mH[rightChildIdx].distance) { | |

smallest = rightChildIdx; | |

} | |

if (smallest != k) { | |

HeapNode smallestNode = mH[smallest]; | |

HeapNode kNode = mH[k]; | |

//swap the positions | |

indexes[smallestNode.vertex] = k; | |

indexes[kNode.vertex] = smallest; | |

swap(k, smallest); | |

sinkDown(smallest); | |

} | |

} | |

public void swap(int a, int b) { | |

HeapNode temp = mH[a]; | |

mH[a] = mH[b]; | |

mH[b] = temp; | |

} | |

public boolean isEmpty() { | |

return currentSize == 0; | |

} | |

public int heapSize(){ | |

return currentSize; | |

} | |

} | |

public static void main(String[] args) { | |

int vertices = 6; | |

Graph graph = new Graph(vertices); | |

int sourceVertex = 0; | |

graph.addEdge(0, 1, 4); | |

graph.addEdge(0, 2, 3); | |

graph.addEdge(1, 2, 1); | |

graph.addEdge(1, 3, 2); | |

graph.addEdge(2, 3, 4); | |

graph.addEdge(3, 4, 2); | |

graph.addEdge(4, 5, 6); | |

graph.dijkstra_GetMinDistances(sourceVertex); | |

} | |

} |

**Output**:

Dijkstra Algorithm: (Adjacency List + Min Heap) Source Vertex: 0 to vertex 0 distance: 0 Source Vertex: 0 to vertex 1 distance: 4 Source Vertex: 0 to vertex 2 distance: 3 Source Vertex: 0 to vertex 3 distance: 6 Source Vertex: 0 to vertex 4 distance: 8 Source Vertex: 0 to vertex 5 distance: 14