Dijkstra? ????? ??? ???? ???? ?? ?????? ?? ?? ????? ?? ??? ?? ? ???? ???? ?? ?? ???????. ? ????? ????? ??? ??? ???? JavaScript? ???? ??? ?????.

Dijkstra? ?????? ??????

Dijkstra? ????? ?? ?? ?? ???? ?? ??? ????? ?? ?? ????? ?? ??? ??? ??? ????? ???????. 1956? Edsger W. Dijkstra? ??? ? ????? ??? ???? ?? ?? ???? ???? ? ??? ?? ????.

?? ? ??

• ??: ??? $?=(?,E)$G=(V,E) , ?? $?$? ? ??? ????, $E$E ? ?? ???? ?? ?????. $s\in Vs$s∈?
• . ??: ?? ?? ?? $??$? ?? ?? ??? $?$?

?? ??
??:
2. ??? ????? ?? ??? ?????? ???????.
???? ?:
3. ?? ??? ?? ?? ??? ????? ?????.
??? ?? ??:

????

1. 
   

   ?? ???:
   $$\text{??}()=0,\text{??}(v)=\mathrm{??}\text{??}\mathrm{?}v\mathrm{\ne }s$$??(s)=0,??(v)=?v=s

2. ???? ???? ???? ??? ?? ??? ?????.

3. ??? ?? ?? ??? ????? ???? ?? ??? ??????.

?? - ??? ??

• ???: $\text{??}(s)=0,\text{??}(v)=\mathrm{}\text{?}\text{??}\text{?}v\mathrm{\ne }s$??(s)=0,??(v)= llv=

?? $(s)$(?) ?? ????, $(v)$(v) ?? ??? ?????.

• ?? ??: ? ????? ?? $(u,v)$(u,v) ??? $w(u,v)$w(u,v) : ??? $\text{??}(v)>\text{??}(?)w(u,v)$dist(v)>dist (?) w(u,v) , ????:
  $$\text{??}(v)=\text{??}(?)w(u,v),\text{??}(v)=u$$??(v)=??(u) w(u,v),prev(v)=u

?? ??: ??? ? ?? ??? ???? ??? ????? ?????? ?? ????? ?? ??? ??? ?????.

???? ??? - ??? ??

• ? ??:

  • ???? ???? ?? ??? ????? ?????. $(u)$(?) ?? ??? ?? ?? ??:
    $$?=\mathrm{a\; rg}?\underset{v\in Q}{??}\text{??}(v)$$u=arg v∈Q ????(v)
  • ?? ??: ?? ?? ??? ???? $(\text{??}(v))$(dist(v)) , ??? ???? ????? ?? ??? ?????. $(u)$(?) .

??? ??

?? ??? ???? Dijkstra ????? ???? ?????.

?????? ??????

????? ??? ???? ????, ??? ???? ?? $(?(n))$(P(n)) , ??? $(?(1),?(2),\dots ,?(k))$(P(1),P(2),…,P(k)) ???? $(?(k1))$( P(k 1)) . ?? ?? ???? ????. $(?(k))$(P(k)) ???? $(?(k1))$( P(k 1)) . ?? ????? ? ??? ?????.

????? ????? ???(??? ??)

1. ?? ??:
   
   ?? ?? $(s)$(?) ? ?????? $\text{??}(s)=0$??(s)=0 , ????.

2. ??? ??:
   
   ???? ??? ?? ??? ?? ?? ??? ????? ?????.

3. ?? ??:
   
   ?? ?? $(u)$(?) ???? ??? ?????. ?? $\text{dist}(u)$??(u) ? ?? ?? ???? ?? ?? ??? ??? ?????. $\text{dist}(u)$??(u) ? ????.

?????? ??

?? ??(???? ???):

// Simplified Queue using Sorting
// Use Binary Heap (good)
// or  Binomial Heap (better) or Pairing Heap (best) 
class PriorityQueue {
  constructor() {
    this.queue = [];
  }

  enqueue(node, priority) {
    this.queue.push({ node, priority });
    this.queue.sort((a, b) => a.priority - b.priority);
  }

  dequeue() {
    return this.queue.shift();
  }

  isEmpty() {
    return this.queue.length === 0;
  }
}

??? ???? ???? ???? Dijkstra ????? JavaScript ?????.

function dijkstra(graph, start) {
  const distances = {}; // hold the shortest distance from the start node to all other nodes
  const previous = {}; // Stores the previous node for each node in the shortest path (used to reconstruct the path later).
  const pq = new PriorityQueue(); // Used to efficiently retrieve the node with the smallest tentative distance.

  // Initialize distances and previous
  for (let node in graph) {
    distances[node] = Infinity; // Start with infinite distances
    previous[node] = null; // No previous nodes at the start
  }
  distances[start] = 0; // Distance to the start node is 0

  pq.enqueue(start, 0);

  while (!pq.isEmpty()) {
    const { node } = pq.dequeue(); // Get the node with the smallest tentative distance

    for (let neighbor in graph[node]) {
      const distance = graph[node][neighbor]; // The edge weight
      const newDist = distances[node] + distance;

      // Relaxation Step
      if (newDist < distances[neighbor]) {
        distances[neighbor] = newDist; // Update the shortest distance to the neighbor
        previous[neighbor] = node; // Update the previous node
        pq.enqueue(neighbor, newDist); // Enqueue the neighbor with the updated distance
      }
    }
  }

  return { distances, previous };
}

// Example usage
const graph = {
  A: { B: 1, C: 4 },
  B: { A: 1, C: 2, D: 5 },
  C: { A: 4, B: 2, D: 1 },
  D: { B: 5, C: 1 }
};

const result = dijkstra(graph, 'A'); // start node 'A'
console.log(result);

?? ???

// Simplified Queue using Sorting
// Use Binary Heap (good)
// or  Binomial Heap (better) or Pairing Heap (best) 
class PriorityQueue {
  constructor() {
    this.queue = [];
  }

  enqueue(node, priority) {
    this.queue.push({ node, priority });
    this.queue.sort((a, b) => a.priority - b.priority);
  }

  dequeue() {
    return this.queue.shift();
  }

  isEmpty() {
    return this.queue.length === 0;
  }
}

?? ??

??? ??

• ??: $A,B,C,D$A,B,C,D
• ????:
  • $A\to B=(1),A\to C=(4)$A→B=(1),A→C=(4)
  • $B\to C=(2),B\to D=(5)$B→C=(2),B→D=(5)
  • $?\to ?=(1)$C→D=(1)

??? ??

1. ?? ???:
   

   $$\text{??}(A)=0,\text{??}\text{??}(B)=\mathrm{},\text{??}\text{??}(C)=\mathrm{??},\text{??}\text{??}(D)=\mathrm{}$$??(A)=0,??(B)= ,??(C)=,??(D)=???

2. ???? A:

   • ???? ??: $A\to B,A\to C\mathrm{.}$??(B
     )=$$\text{,}??\mathrm{??}(C\mathrm{)}=\text{4}\text{???{??}(B) = 1, ; ???{??}(C) = 4}??(B)=1,\mathrm{??}(C)=$$ ???? B: ???? ??:
   B
3. →
   C
   • ,B$D.BB\to C,B\to D.\mathrm{}$??(?
     )=$$\text{,}??\mathrm{??}(D\mathrm{)}=\text{6}\text{???{??}(C) = 3, ; ???{??}(D) = 6}??(C)=3,\mathrm{??}(D)=$$ ???? C: ??? ????:
   C
4. →
   D
   • .C~D. $C\to D.\mathrm{}$??(?
     )=$$\text{???{dist}(D) = 4}??(D)=4\mathrm{}$$

   • ???? D:

     • ?? ????? ????.

   ?? ?? ? ??

   $$\text{??}(A)=0,\text{??}\text{??}(B)=1,\text{??}\text{??}(C)=3,\text{??}\text{??}(D)=4$$??(A)=0,??(B)= 1,??(C)=3,??(D)=4
   
   
   $$A\to B\to ?\to D$$A→B→C→D

   ??? ? ?? ???

   ??? ???? ??? ??? ?? Dijkstra ????? ?? ??? ??:

   Priority Queue Type	Insert (M)	Extract Min	Decrease Key	Overall Time Complexity
   Simple Array	O(1)	O(V)	O(V)	O(V^2)
   Binary Heap	O(log V)	O(log V)	O(log V)	O((V E) log V)
   Binomial Heap	O(log V)	O(log V)	O(log V)	O((V E) log V)
   Fibonacci Heap	O(1)	O(log V)	O(1)	O(V log V E)
   Pairing Heap	O(1)	O(log V)	O(log V)	O(V log V E) (practical)

   ?? ???:

   1. ??? ??:
      • extract-min? ?? ?? ???? ?? ? ????? ???????.
   2. ???? ?:
      • ???? ???? ???? ?? ???? ????? ?????.
   3. ?? ?:
      • ??? ??? ?? ? ????? ??? ? ?????.
   4. ???? ?:
      • ( O(1) ) ?? ?? ?? ???? ??? ??? ??? ????? ????? ? ?????.
   5. ??? ?:
      • ???? ??? ???? ?? ??? ??? ?????.

   ??

   Dijkstra? ????? ??? ?? ???? ?? ????? ?? ??? ?? ???? ???? ?????. ?? ??(?: ?? ?? ???? ??? ? ??)? ??? ????, ??? ? ?? ???????? ?? ?????.

   • ??? ??? ????? ?????? ?? ??? ?????.
   • ???? ???? ?? ?? ??? ??? ???? ???? ????? ?????.
   • ???? ??? ?? ?????. ??? ??? ???? ?? ??? ?????.

   ---

   ??? ??? ?? ? ??? ?? Dijkstra? ????? ??? ? ?? ? ?? ??? ??????.

   • ????? ???? PDF
   • SlideShare? ?? ?? ????

   ?? Wikipedia? ?? ??? ?? ??? ??? ?????.

   ??:
   [1] https://www.fuhuthu.com/CPSC420F2019/dijkstra.pdf

   ??? ???? ???? ?? ??? ???? ??? ???!