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Dijkstra’s algorithm

The document describes Dijkstra's algorithm for finding the shortest paths between nodes in a graph. It begins with an overview of how the algorithm works by solving subproblems to find the shortest path from the source to increasingly more nodes. It then provides pseudocode for the algorithm and analyzes its time complexity of O(E+V log V) when using a Fibonacci heap. Some applications of the algorithm include traffic information systems and routing protocols like OSPF.

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SCHOOL MANAGEMENT SYSTEM Submited BY: Faisal Patel 254 Parth Bharuch 257 Dhavan Shah 160
Introduction  Our project is about
Introduction Contd.  Greedy algorithms use problem solving methods based on actions to see if there’s a better long term strategy.  Dijkstra’s algorithm uses the greedy approach to solve the single source shortest problem. It repeatedly selects from the unselected vertices, vertex v nearest to source s and declares the distance to be the actual shortest distance from s to v.  The edges of v are then checked to see if their destination can be reached by v followed by the relevant outgoing edges.  For a given source node in the graph, the algorithm finds the shortest path between that node and every other.  It can also be 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.
How It Works??  Before going into details of the pseudo-code of the algorithm it is important to know how the algorithm works.  Dijkstra’s algorithm works by solving the sub-problem k, which computes the shortest path from the source to vertices among the k closest vertices to the source.  For the dijkstra’s algorithm to work it should be directed- weighted graph and the edges should be non-negative.  If the edges are negative then the actual shortest path cannot be obtained.
More Detailed Knowledge  At the kth round, there will be a set called Frontier of k vertices that will consist of the vertices closest to the source and the vertices that lie outside frontier are computed and put into New Frontier.  The shortest distance obtained is maintained in sDist[w].  It holds the estimate of the distance from s to w.  Dijkstra’s algorithm finds the next closest vertex by maintaining the New Frontier vertices in a priority-min queue.
6 Dijkstra's Shortest Path Algorithm  Find shortest path from s to t. s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6
7 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6        0 distance label S = { } PQ = { s, 2, 3, 4, 5, 6, 7, t }
8 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6        0 distance label S = { } PQ = { s, 2, 3, 4, 5, 6, 7, t } delmin
9 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9    14  0 distance label S = { s } PQ = { 2, 3, 4, 5, 6, 7, t } decrease key X  X X
10 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9    14  0 distance label S = { s } PQ = { 2, 3, 4, 5, 6, 7, t } X  X X delmin
11 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9    14  0 S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t } X  X X
12 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9    14  0 S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t } X  X X decrease key X 33
13 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9    14  0 S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t } X  X X X 33 delmin
14 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9    14  0 S = { s, 2, 6 } PQ = { 3, 4, 5, 7, t } X  X X X 33 44 X X 32
15 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 6 } PQ = { 3, 4, 5, 7, t } X  X X 44 X delmin X 33X 32
16 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 6, 7 } PQ = { 3, 4, 5, t } X  X X 44 X 35X 59 X 24 X 33X 32
17 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 6, 7 } PQ = { 3, 4, 5, t } X  X X 44 X 35X 59 X delmin X 33X 32
18 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 6, 7 } PQ = { 4, 5, t } X  X X 44 X 35X 59 XX51 X 34 X 33X 32
19 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 6, 7 } PQ = { 4, 5, t } X  X X 44 X 35X 59 XX51 X 34 delmin X 33X 32 24
20 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 5, 6, 7 } PQ = { 4, t } X  X X 44 X 35X 59 XX51 X 34 24 X50 X45 X 33X 32
21 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 5, 6, 7 } PQ = { 4, t } X  X X 44 X 35X 59 XX51 X 34 24 X50 X45 delmin X 33X 32
22 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 4, 5, 6, 7 } PQ = { t } X  X X 44 X 35X 59 XX51 X 34 24 X50 X45 X 33X 32
23 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 4, 5, 6, 7 } PQ = { t } X  X X 44 X 35X 59 XX51 X 34 X50 X45 delmin X 33X 32 24
24 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 4, 5, 6, 7, t } PQ = { } X  X X 44 X 35X 59 XX51 X 34 X50 X45 X 33X 32
25 Dijkstra's Shortest Path Algorithm s 3 t 2 6 7 4 5 24 18 2 9 14 15 5 30 20 44 16 11 6 19 6 15 9   14  0 S = { s, 2, 3, 4, 5, 6, 7, t } PQ = { } X  X X 44 X 35X 59 XX51 X 34 X50 X45 X 33X 32
ALgorithm function Dijkstra(Graph, source): dist[source] ← 0 // Distance from source to source prev[source] ← undefined // Previous node in optimal path initialization for each vertex v in Graph: // Initialization if v ≠ source // Where v has not yet been removed from Q (unvisited nodes) dist[v] ← infinity // Unknown distance function from source to v prev[v] ← undefined // Previous node in optimal path from source end if add v to Q // All nodes initially in Q (unvisited nodes) end for
while Q is not empty: u ← vertex in Q with min dist[u] // Source node in first case remove u from Q for each neighbor v of u: // where v is still in Q. alt ← dist[u] + length(u, v) if alt < dist[v]: // A shorter path to v has been found dist[v] ← alt prev[v] ← u end if end for end while return dist[], prev[] end function
EFFICIENCY  The complexity efficiency can be expressed in terms of Big-O Notation. Big-O gives another way of talking about the way input affects the algorithm’s running time. It gives an upper bound of the running time.  In Dijkstra’s algorithm, the efficiency varies depending on |V| and |E| updates for priority queues that were used.  If a Fibonacci heap was used then the complexity is O( | E | + | V | log | V | ) , which is the best bound.
DIS-ADVANTAGES  The major disadvantage of the algorithm is the fact that it does a blind search there by consuming a lot of time waste of necessary resources.  Another disadvantage is that it cannot handle negative edges. This leads to acyclic graphs and most often cannot obtain the right shortest path
APPLICATIONS  Traffic information systems use Dijkstra’s algorithm in order to track the source and destinations from a given particular source and destination .  OSPF- Open Shortest Path First, used in Internet routing.  It uses a link-state in the individual areas that make up the hierarchy.  The computation is based on Dijkstra's algorithm which is used to calculate the shortest path tree inside each area of the network.