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Dijkstra

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Title: Dijkstra


1
Dijkstras Algorithm Fibonacci Heap
Implementation
  • by Amber McKenzie
  • and Laura Boccanfuso

2
Dijkstras Algorithm
  • Question How do you know that Dijkstras
    algorithm finds the shortest path and is optimal
    when implemented with the Fibonacci heap?

3
Single-Source Shortest Path
  • For a given vertex, determine the shortest path
    between that vertex and every other vertex, i.e.
    minimum spanning tree.

4
Premise of Dijkstras Algorithm
  • First, finds the shortest path from the vertex to
    the nearest vertex.
  • Then, finds the shortest path to the next nearest
    vertex, and so on.
  • These vertices, for which the shortest paths have
    been found, form a subtree.
  • Thus, the next nearest vertex must be among the
    vertices that are adjacent to those in the
    subtree these next nearest vertices are called
    fringe vertices.

5
Premise cont.
  • The fringe vertices are maintained in a priority
    queue which is updated with new distances from
    the source vertex at every iteration.
  • A vertex is removed from the priority queue when
    it is the vertex with the shortest distance from
    the source vertex of those fringe vertices that
    are left.

6
Pseudocode
  • for every vertex v in V do
  • dv ? 8 pv ? null
  • Insert(Q, v, dv) //initialize vertex priority
    in the priority queue
  • ds ? 0 Decrease(Q, s, ds) //update priority of
    s with ds
  • VT ? Ø
  • for i ? 0 to V - 1 do
  • u ? DeleteMin(Q) //delete the minimum priority
    element
  • VT ? Vt U u
  • for every vertex u in V VT that is adjacent
    to u do
  • if du w(u, u) lt du
  • du ? du w(u, u) pu ? u
  • Decrease(Q, u, du)

7
Dijkstras Algorithm
8
Dijkstras Algorithm
2
b
a
5
c
8
f
d
e
Tree vertices Remaining vertices
a(-, 0)
b(a, 2) c(a, 5) d(a, 8) e(-, 8) f(-, 8)
9
Dijkstras Algorithm
2
b
a
5
6
2
c
8
f
d
e
Tree vertices Remaining vertices
b(a, 2)
c(b, 22) d(a, 8) e(-, 8 ) f(b, 26)
10
Dijkstras Algorithm
2
b
a
5
6
2
c
8
f
3
1
d
e
Tree vertices Remaining vertices
c(b, 4)
d(a, 8) e(c, 41) f(b, 8)
11
Dijkstras Algorithm
2
b
a
5
6
2
c
8
f
3
1
d
4
7
e
Tree vertices Remaining vertices
e(c, 5)
d(a, 8) f(b, 8)
12
Dijkstras Algorithm
2
b
a
5
6
2
c
8
f
3
1
d
4
7
e
Tree vertices Remaining vertices
d(a, 8)
f(b, 8)
13
Dijkstras Algorithm
14
Dijkstras Algorithm Priority Queue
  • Tree vertices Remaining vertices
  • a(-, 0) b(a, 2) c(a, 5) d(a, 8) e(-, 8) f(-,
    8)
  • b(a, 2) c(b, 22) d(a, 8) e(-, 8 ) f(b, 26)
  • c(b, 4) d(a, 8) e(c, 41) f(b, 8)
  • e(c, 5) d(a, 8) f(b, 8)
  • d(a, 8) f(b, 8)
  • f(b, 8)

15
Fibonacci Heap Implementation
  • What makes the Fibonacci Heap optimally suited
  • for implementing the Dijkstra algorithm?
  • Manipulation of heap/queue
  • Time complexity efficiency

http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fibnat.htmlRabbits
16
Fibonacci Heap Implementation
  • Manipulation of heap/queue
  • Insert operation creates a new heap with one
    element then performs a merge

4
2
  • Merge operation concatenate the lists of
  • tree roots of the two heaps
  • Decrease_key take the node, decrease the
  • key and reorder nodes if necessary, mark node
  • or cut (if smaller than parent)
  • Delete_min take root of min element and
    remove decrease
  • number of roots by linking together ones
    with same degree,
  • check each remaining node to find minimum
    and delete

7
5
9
17
Fibonacci Heap Implementation
Time Complexity Efficiency
Operation USDL List 2-3 tree Heap Binomial Fibonacci
make O(1) O(1) O(1) O(1) O(1)
empty O(1) O(1) O(1) O(1) O(1)
insert O(1) O(logn) O(logn) O(logn) O(1)
find_min O(n) O(logn) O(1) O(logn) O(1)
delete_min O(n) O(logn) O(logn) O(logn) O(logn)
delete O(1) O(logn) O(logn) O(logn) O(logn)
merge O(1) O(n) O(n) O(logn) O(1)
decrease_key O(1) O(logn) O(logn) O(logn) O(1)
USDL list Unsorted Doubly Linked list
18
Worst-case complexity
  • Formula to discover the worst-case complexity for
    Dijkstras algorithm
  • W(n,m) O(n cost of insert
  • n cost of delete_min
  • m
    cost of decrease_key)
  • (Where n maximum size of
    priority queue
  • m number of times inner loop is performed)

19
Worst-case complexity (cont.)
  • Unsorted linked list
  • W(n,m) O(n 1 n n m 1) O(n2)
  • 2-3 Tree
  • W(n,m) O(n logn n logn m
    logn) O(mlogn)
  • Fibonacci Heap
  • W(n,m) O(n 1 n logn m 1)
    O(nlogn m)

20
Optimality of Dijkstras Algorithm
  • Adversary argument
  • In this case, it is the argument that there
    exists a path between the source vertex s and the
    target vertex t that is shorter than the path
    already determined by the algorithm.

21
Adversary Argument
  • Since we have already determined the shortest
    paths to all the previous vertices that are now
    in the tree, this must mean that the path from s
    to t goes through some other vertex v whose
    distance from s has yet to be determined (meaning
    it is still in the priority queue).

v
22
Adversary Argument Cont.
  • The catch is that if this other vertex v through
    which t passes is still in the priority queue,
    then its distance to s is longer than that of all
    other vertices already in the tree.
  • Thus it cannot be a shorter distance than that
    which is already determined between s and t.

v
23
References
  • Algorithms and Data Structures
  • Design, Correctness and Analysis
  • Jeffrey H. Kingston
  • A Result on the Computational Complexity of
  • Heuristic Estimates for the A
    Algorithm
  • Marco Valtorta
  • The Design Analysis of Algorithms
  • Anany Levitin
  • Animation
  • http//www.cs.auckland.ac.nz/software/A
    lgAnim/dijkstra.htmldijkstra_anim
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