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For sparse graphs, that is, graphs with far fewer than | | edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a self-balancing binary search tree, binary heap, pairing heap, Fibonacci heap or a priority heap as a priority queue to implement extracting minimum efficiently.
A Fibonacci heap is a collection of trees satisfying the minimum-heap property, that is, the key of a child is always greater than or equal to the key of the parent. This implies that the minimum key is always at the root of one of the trees. Compared with binomial heaps, the structure of a Fibonacci heap is more flexible.
A pairing heap is a type of heap data structure with relatively simple implementation and excellent practical amortized performance, introduced by Michael Fredman, Robert Sedgewick, Daniel Sleator, and Robert Tarjan in 1986. [1] Pairing heaps are heap-ordered multiway tree structures, and can be considered simplified Fibonacci heaps.
The first three stages of Johnson's algorithm are depicted in the illustration below. The graph on the left of the illustration has two negative edges, but no negative cycles. The center graph shows the new vertex q, a shortest path tree as computed by the Bellman–Ford algorithm with q as starting vertex, and the values h(v) computed at each other node as the length of the shortest path from ...
A strict Fibonacci heap is a single tree satisfying the minimum-heap property. That is, the key of a node is always smaller than or equal to its children. As a direct consequence, the node with the minimum key always lies at the root. Like ordinary Fibonacci heaps, [4] strict Fibonacci heaps possess substructures similar to binomial heaps. To ...
For graphs of even greater density (having at least |V| c edges for some c > 1), Prim's algorithm can be made to run in linear time even more simply, by using a d-ary heap in place of a Fibonacci heap. [10] [11] Demonstration of proof. In this case, the graph Y 1 = Y − f + e is already equal to Y. In general, the process may need to be repeated.
Dijkstra's algorithm. Open Shortest Path First; Flooding algorithm; Route inspection problem; ... Fibonacci heap; 2-3 heap; Kd-tree; Cover tree; Decision tree; Empty ...
Dijkstra's algorithm has a worse case time complexity of (), but using a Fibonacci heap it becomes (+ ), [3] where is the number of edges in the graph. Since Yen's algorithm makes K l {\displaystyle Kl} calls to the Dijkstra in computing the spur paths, where l {\displaystyle l} is the length of spur paths.