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Algorithms: Shortest Path in Graphs - Dijkstra Algorithm ( with C Program source code)

Dijkstra’s Algorithm

Dijkstra’s algorithm solves the single source shortest path problem on a weighted, directed graph only when all edge-weights are non-negative. It maintains a set S of vertices whose final shortest path from the source has already been determined and it repeatedly selects the left vertices with the minimum shortest-path estimate, inserts them into S, and relaxes all edges leaving that edge. In this we maintain a priority-Queue which is implemented via heap.

Algorithm:

Input Format: Graph is directed and weighted. First two integers must be number of vertices and edges which must be followed by pairs of vertices which has an edge between them.

Analysis:

The running time of dijkstra’s algorithm using priority Queue implemented via binary-heap is O((|V| + |E|)log|V|) as in binary-heap DeleteMin function takes O(log|V|) time and Insert function takes O(log|V|) time.

Algorithm (described in detail within the document for this tutorial)

Input Format: Graph is directed and weighted. First two integers must be number of vertices and edges which must be followed by pairs of vertices which has an edge between them.

Dijkstra Algorithm - C Program Source Code

`#include<stdio.h>#include<limits.h>#include<assert.h>#define maxVertices 100#define  infinity 1000010000/*Declaring heap globally so that we do not need to pass it as an argument every time*//* Heap used here is Min Heap */typedef struct Node{        int vertex,distance;}Node;Node heap;int seen[maxVertices];int heapSize;/*Initialize Heap*/void Init(){        heapSize = 0;        heap.distance = -INT_MAX;        heap.vertex  = -1;}/*Insert an element into the heap */void Insert(Node element){        heapSize++;        heap[heapSize] = element; /*Insert in the last place*/        /*Adjust its position*/        int now = heapSize;        while(heap[now/2].distance > element.distance)         {                heap[now] = heap[now/2];                now /= 2;        }        heap[now] = element;}Node DeleteMin(){        /* heap is the minimum element. So we remove heap. Size of the heap is decreased.            Now heap has to be filled. We put the last element in its place and see if it fits.           If it does not fit, take minimum element among both its children and replaces parent with it.           Again See if the last element fits in that place.*/        Node minElement,lastElement;        int child,now;        minElement = heap;        lastElement = heap[heapSize--];        /* now refers to the index at which we are now */        for(now = 1; now*2 <= heapSize ;now = child)        {                /* child is the index of the element which is minimum among both the children */                 /* Indexes of children are i*2 and i*2 + 1*/                child = now*2;                /*child!=heapSize beacuse heap[heapSize+1] does not exist, which means it has only one                   child */                if(child != heapSize && heap[child+1].distance < heap[child].distance )                 {                        child++;                }                /* To check if the last element fits ot not it suffices to check if the last element                   is less than the minimum element among both the children*/                if(lastElement.distance > heap[child].distance)                {                        heap[now] = heap[child];                }                else /* It fits there */                {                        break;                }        }        heap[now] = lastElement;        return minElement;}int main(){        int graph[maxVertices][maxVertices],size[maxVertices]={0},distance[maxVertices]={0},cost[maxVertices][maxVertices];        int vertices,edges,weight;        int iter;        /* vertices represent number of vertices and edges represent number of edges in the graph. */        scanf("%d%d",&vertices,&edges);        int from,to;        for(iter=0;iter<edges;iter++)        {                scanf("%d%d%d",&from,&to,&weight);                assert(from>=0 && from<vertices);                assert(to>=0 && to<vertices);                graph[from][size[from]] = to;                cost[from][size[from]] = weight;                size[from]++;        }        int source;        scanf("%d",&source);        Node temp;        for(iter=0;iter<vertices;iter++)        {                if(iter==source)                {                        temp.distance = 0;                        distance=0;                }                else                {                        temp.distance = infinity;                        distance[iter]= infinity;                }                temp.vertex = iter;                Insert(temp);        }        while(heapSize)        {                Node min = DeleteMin();                int presentVertex = min.vertex;                if(seen[presentVertex])                {                        /* This has already been processed */                        continue;                }                seen[presentVertex] = 1;                for(iter=0;iter<size[presentVertex];iter++)                {                        int to = graph[presentVertex][iter];                        if(distance[to] > distance[presentVertex] + cost[presentVertex][iter])                        {                                distance[to] = distance[presentVertex] + cost[presentVertex][iter];                                /* Instead of updating it in the queue, insert it again. This works because the updated                                   distance is less than previous distance which makes it to pop out of the queue early */                                temp.vertex = to;                                temp.distance = distance[to];                                Insert(temp);                        }                }        }        for(iter=0;iter<vertices;iter++)        {                printf("vertex is %d, its distance is %d\n",iter,distance[iter]);        }        return 0;}`

Some Important Data Structures and Algorithms, at a glance:

 Arrays : Popular Sorting and Searching Algorithms Bubble Sort Insertion Sort Selection Sort Shell Sort Merge Sort Quick Sort Heap Sort Binary Search Algorithm Basic Data Structures  and Operations on them Stacks Queues Single Linked List Double Linked List Circular Linked List
 Tree Data Structures Binary Search Trees Heaps Height Balanced Trees Graphs and Graph Algorithms Depth First Search Breadth First Search Minimum Spanning Trees: Kruskal Algorithm Minumum Spanning Trees: Prim's Algorithm Dijkstra Algorithm for Shortest Paths Floyd Warshall Algorithm for Shortest Paths Bellman Ford Algorithm Popular Algorithms in Dynamic Programming Dynamic Programming Integer Knapsack problem Matrix Chain Multiplication Longest Common Subsequence Greedy Algorithms Elementary cases : Fractional Knapsack Problem, Task Scheduling Data Compression using Huffman Trees