I'm reading about Dijkstra's algorithm in CLRS, Third Edition (p. 662). Here is a part from the book I don't understand:
If the graph is sufficiently sparse — in particular,
E = o(V^2/lg V)— we can improve the algorithm by implementing the min-priority queue with a binary min-heap.
Why should the graph be sparse?
Here is another part:
Each DECREASE-KEY operation takes time
O(log V), and there are still at most E such operations.
Suppose my graph looks like this:
I'd like to calculate the shortest path from 1 to 6 and use the min-heap approach. So first off, I add all my nodes to a min priority queue. After building a min heap, the min node is the source node (since its distance to itself is 0). I extract it and update distances of all its neighbors.
Then I need to call decreaseKey on the node with the lowest distance to make a new minimum of the heap. But how do I know its index in constant time?
Node
private static class Node implements Comparable<Node> {
final int key;
int distance = Integer.MAX_VALUE;
Node prev = null;
public Node(int key) {
this.key = key;
}
@Override
public int compareTo(Node o) {
if (distance < o.distance) {
return -1;
} else if (distance > o.distance) {
return 1;
} else {
return 0;
}
}
@Override
public String toString() {
return "key=" + key + " distance=" + distance;
}
@Override
public int hashCode() {
return key;
}
@Override
public boolean equals(Object obj) {
if (this == obj) {
return true;
}
if (!(obj instanceof Node)) {
return false;
}
Node other = (Node) obj;
return key == other.key;
}
}
MinPriorityQueue
public static class MinPriorityQueue {
private Node[] array;
private int heapSize;
public MinPriorityQueue(Node[] array) {
this.array = array;
this.heapSize = this.array.length;
}
public Node extractMin() {
Node temp = array[0];
swap(0, heapSize - 1, array);
heapSize--;
sink(0);
return temp;
}
public boolean isEmpty() {
return heapSize == 0;
}
public void buildMinHeap() {
for (int i = heapSize / 2 - 1; i >= 0; i--) {
sink(i);
}
}
public void decreaseKey(int index, Node key) {
if (key.compareTo(array[index]) >= 0) {
throw new IllegalArgumentException("the new key must be greater than the current key");
}
array[index] = key;
while (index > 0 && array[index].compareTo(array[parentIndex(index)]) < 0) {
swap(index, parentIndex(index), array);
index = parentIndex(index);
}
}
private int parentIndex(int index) {
return (index - 1) / 2;
}
private int left(int index) {
return 2 * index + 1;
}
private int right(int index) {
return 2 * index + 2;
}
private void sink(int index) {
int smallestIndex = index;
int left = left(index);
int right = right(index);
if (left < heapSize && array[left].compareTo(array[smallestIndex]) < 0) {
smallestIndex = left;
}
if (right < heapSize && array[right].compareTo(array[smallestIndex]) < 0) {
smallestIndex = right;
}
if (index != smallestIndex) {
swap(smallestIndex, index, array);
sink(smallestIndex);
}
}
public Node min() {
return array[0];
}
private void swap(int i, int j, Node[] array) {
Node temp = array[i];
array[i] = array[j];
array[j] = temp;
}
}