Dijkstra algorithm: directed, shortest path

다익스트라 알고리즘은 간선에 가중치가 있는 방향 그래프에서 최단거리 경로를 구하기 위해 사용한다. 따라서 만약 방향 그래프가 아니라면 하나의 간선에 대해 양방향으로 이어지는 방향 그래프로 변환해야 한다. 즉, 무방향 그래프로 데이터가 주어지는 경우 그래프를 인접 리스트 형태로 만들어줄 필요가 있다.

Dijkstra

Prim vs Dijkstra

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최단경로 알고리즘 다익스트라 이해집주변이 최고야

Time: O(v^2 + e)

Space: O(V)

with out heap

function Node(v, predecessor, cost) {
  this.v = v;
  this.predecessor = predecessor;
  this.cost = cost;
}

function findMinimumCostVertex(nodes, visited) {
  let currentMinConst = Infinity;
  let here = -1;

  for (let vertex = 0; vertex < nodes.length; vertex++) {
    if (visited.has(vertex)) continue;

    const cost = nodes[vertex].cost;
    if (cost < currentMinConst) {
      currentMinConst = cost;
      here = vertex;
    }
  }

  return [here, currentMinConst];
}

function dijkstrasAlgorithm(start, edges) {

  const nodes = [];
  // initialize
  for (let i = 0; i < edges.length; i++) {
    const node = new Node(i, i, Infinity);
    nodes[i] = node;
  }

  nodes[start].cost = 0;
  const visited = new Set();

  while (visited.size !== edges.length) {
    const [here, currentMinCost] = findMinimumCostVertex(nodes, visited);
    // console.log(here, currentMinCost);

    // visited a vertex with Infinity > can not reach there
    if (currentMinCost === Infinity) {
      break;
    }
    
    const adj = edges[here] ?? [];
    for (const [there, cost] of adj) {
      if (visited.has(there)) continue;
      
      const prev = nodes[there].cost; // current min there cost
      const next = currentMinCost + cost;
      if (prev > next) {
        nodes[there].predecessor = here;
        nodes[there].cost = next;
      }
    }

    visited.add(here);

    // break;
  }

  const answer = [];
  for (let i = 0; i < edges.length; i++) {
     if (nodes[i].cost === Infinity) {
        answer[i] = -1;
     } else {
        answer[i] = nodes[i].cost;
     }

    console.log(nodes[i]);
  }

  return answer;
  
}

// Do not edit the line below.
exports.dijkstrasAlgorithm = dijkstrasAlgorithm;

With Priority Queue

With visited Array

Time: O((v + e)logv ⇒ elogV)

Space: O(v)

using visited array not to add visited vertex

function Node(vertex, predecessor, cost) {
  this.vertex = vertex;
  this.predecessor = predecessor;
  this.cost = cost;
}

// O((V + E)logV => ElogV, usually E > V)
function dijkstrasAlgorithm(start, edges) {

  const dist = [];
  for (let vertex = 0; vertex < edges.length; vertex++) {
    dist[vertex] = new Node(vertex, vertex, Infinity);
  }
  
  const visited = new Set();
  dist[start].cost = 0;
  const minHeap = new Heap([], (a, b) => a.cost <= b.cost);
  minHeap.insert(new Node(start, start, 0));

  while (minHeap.size > 0) {
    const {
      vertex: here,
      predecessor,
      cost: currentCost,
    } = minHeap.remove();

    const adj = edges[here];
    for (const [there, incommingCost] of adj) {
      if (visited.has(there)) continue;
      
      const prev = dist[there].cost;
      const next = currentCost + incommingCost;
      if (prev > next) {
        dist[there].cost = next;
        dist[there].predecessor = here;
        
        minHeap.insert(new Node(there, here, next));
      }
    }

    console.log(minHeap.heap);

    visited.add(here);
  }
  const answer = [];
  for (let vertex = 0; vertex < edges.length; vertex++) {
    const node = dist[vertex];
    answer[vertex] = node.cost === Infinity ? -1 : node.cost;
  }

  return answer;
}

class Heap {
    constructor(arr, predicate = (a, b) => a <= b) {
        this.predicate = predicate;
        this.size = arr.length;
        this.heap = this.buildHeap(arr);
    }

    buildHeap(arr) {
        let currentIndex = Math.floor((arr.length - 1 - 1) / 2);
        while (currentIndex >= 0) {
            this.siftDown(arr, currentIndex, arr.length - 1);
            currentIndex--;
        }

        return arr;
    }

    insert(v) {
        this.heap.push(v);
        this.size++;

        this.siftDown(this.heap, 0, this.heap.length - 1);

    }

    remove(v) {
        this.swap(this.heap, 0, this.heap.length - 1);
        const elementToRemove = this.heap.pop();
        this.size--;
        
        this.siftDown(this.heap, 0, this.heap.length - 1);
        return elementToRemove;
    }

    siftUp(heap, currentIndex) {
        let parentIndex = Math.floor((currentIndex - 1) / 2);
        while (parentIndex >= 0) {
            if (this.predicate(heap[parentIndex], heap[currentIndex])) {
                break;
            }

            this.swap(heap, parentIndex, currentIndex);
            currentIndex = parentIndex;
            parentIndex = Math.floor((currentIndex - 1) / 2);
        }
        return heap;
    }

    siftDown(heap, currentIndex, endIndex) {
        let left = currentIndex * 2 + 1;

        while (left <= endIndex) {
            let min = left;
            const right = currentIndex * 2 + 2;
            if (heap[right] !== undefined && this.predicate(heap[right], heap[min])) {
                min = right;
            }

            if (this.predicate(heap[currentIndex], heap[min])) break;
            this.swap(heap, currentIndex, min);
            currentIndex = min;
            left = currentIndex * 2 + 1;
        }

        return heap;
    }

    swap(arr, a, b) {
        [arr[b], arr[a]] = [arr[a], arr[b]];
    }

}

// Do not edit the line below.
exports.dijkstrasAlgorithm = dijkstrasAlgorithm;

without visited Array

function Node(vertex, predecessor, cost) {
  this.vertex = vertex;
  this.predecessor = predecessor;
  this.cost = cost;
}

// O((V + E)logV => ElogV, usually E > V)
function dijkstrasAlgorithm(start, edges) {

  const dist = [];
  for (let vertex = 0; vertex < edges.length; vertex++) {
    dist[vertex] = new Node(vertex, vertex, Infinity);
  }
  
  const minHeap = new Heap([], (a, b) => a.cost <= b.cost);
  dist[start].cost = 0;
  minHeap.insert(new Node(start, start, 0));

  while (minHeap.size > 0) {
    const {
      vertex: here,
      predecessor,
      cost: currentCost,
    } = minHeap.remove();

    // we pu every connected nodes into heap
    // so we filter smallest cost from heap is bigger than or equal to current distance min
    // without using visited array
    if (dist[here].cost < currentCost) 
      continue;

    const adj = edges[here];
    for (const [there, incommingCost] of adj) {
      
      const prev = dist[there].cost;
      const next = currentCost + incommingCost;
      if (prev > next) {
        dist[there].cost = next;
        dist[there].predecessor = here;
        
        minHeap.insert(new Node(there, here, next));
      }
    }

    console.log(minHeap.heap);
  }
  
  const answer = [];
  for (let vertex = 0; vertex < edges.length; vertex++) {
    const node = dist[vertex];
    answer[vertex] = node.cost === Infinity ? -1 : node.cost;
  }

  return answer;
}

class Heap {
    constructor(arr, predicate = (a, b) => a <= b) {
        this.predicate = predicate;
        this.size = arr.length;
        this.heap = this.buildHeap(arr);
    }

    buildHeap(arr) {
        let currentIndex = Math.floor((arr.length - 1 - 1) / 2);
        while (currentIndex >= 0) {
            this.siftDown(arr, currentIndex, arr.length - 1);
            currentIndex--;
        }

        return arr;
    }

    insert(v) {
        this.heap.push(v);
        this.size++;

        this.siftDown(this.heap, 0, this.heap.length - 1);

    }

    remove(v) {
        this.swap(this.heap, 0, this.heap.length - 1);
        const elementToRemove = this.heap.pop();
        this.size--;
        
        this.siftDown(this.heap, 0, this.heap.length - 1);
        return elementToRemove;
    }

    siftUp(heap, currentIndex) {
        let parentIndex = Math.floor((currentIndex - 1) / 2);
        while (parentIndex >= 0) {
            if (this.predicate(heap[parentIndex], heap[currentIndex])) {
                break;
            }

            this.swap(heap, parentIndex, currentIndex);
            currentIndex = parentIndex;
            parentIndex = Math.floor((currentIndex - 1) / 2);
        }
        return heap;
    }

    siftDown(heap, currentIndex, endIndex) {
        let left = currentIndex * 2 + 1;

        while (left <= endIndex) {
            let min = left;
            const right = currentIndex * 2 + 2;
            if (heap[right] !== undefined && this.predicate(heap[right], heap[min])) {
                min = right;
            }

            if (this.predicate(heap[currentIndex], heap[min])) break;
            this.swap(heap, currentIndex, min);
            currentIndex = min;
            left = currentIndex * 2 + 1;
        }

        return heap;
    }

    swap(arr, a, b) {
        [arr[b], arr[a]] = [arr[a], arr[b]];
    }

}

// Do not edit the line below.
exports.dijkstrasAlgorithm = dijkstrasAlgorithm;