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EagerPrimsAdjacencyList.java
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EagerPrimsAdjacencyList.java
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/**
* An implementation of the eager version of Prim's algorithm which relies on using an indexed
* priority queue data structure to query the next best edge.
*
* <p>Time Complexity: O(ElogV)
*
* @author William Fiset, william.alexandre.fiset@gmail.com
*/
package com.williamfiset.algorithms.graphtheory;
import static java.lang.Math.*;
import java.util.*;
public class EagerPrimsAdjacencyList {
static class Edge implements Comparable<Edge> {
int from, to, cost;
public Edge(int from, int to, int cost) {
this.from = from;
this.to = to;
this.cost = cost;
}
@Override
public int compareTo(Edge other) {
return cost - other.cost;
}
}
// Inputs
private final int n;
private final List<List<Edge>> graph;
// Internal
private boolean solved;
private boolean mstExists;
private boolean[] visited;
private MinIndexedDHeap<Edge> ipq;
// Outputs
private long minCostSum;
private Edge[] mstEdges;
public EagerPrimsAdjacencyList(List<List<Edge>> graph) {
if (graph == null || graph.isEmpty()) throw new IllegalArgumentException();
this.n = graph.size();
this.graph = graph;
}
// Returns the edges used in finding the minimum spanning tree,
// or returns null if no MST exists.
public Edge[] getMst() {
solve();
return mstExists ? mstEdges : null;
}
public Long getMstCost() {
solve();
return mstExists ? minCostSum : null;
}
private void relaxEdgesAtNode(int currentNodeIndex) {
visited[currentNodeIndex] = true;
// edges will never be null if the createEmptyGraph method was used to build the graph.
List<Edge> edges = graph.get(currentNodeIndex);
for (Edge edge : edges) {
int destNodeIndex = edge.to;
// Skip edges pointing to already visited nodes.
if (visited[destNodeIndex]) continue;
if (ipq.contains(destNodeIndex)) {
// Try and improve the cheapest edge at destNodeIndex with the current edge in the IPQ.
ipq.decrease(destNodeIndex, edge);
} else {
// Insert edge for the first time.
ipq.insert(destNodeIndex, edge);
}
}
}
// Computes the minimum spanning tree and minimum spanning tree cost.
private void solve() {
if (solved) return;
solved = true;
int m = n - 1, edgeCount = 0;
visited = new boolean[n];
mstEdges = new Edge[m];
// The degree of the d-ary heap supporting the IPQ can greatly impact performance, especially
// on dense graphs. The base 2 logarithm of n is a decent value based on my quick experiments
// (even better than E/V in many cases).
int degree = (int) Math.ceil(Math.log(n) / Math.log(2));
ipq = new MinIndexedDHeap<>(max(2, degree), n);
// Add initial set of edges to the priority queue starting at node 0.
relaxEdgesAtNode(0);
while (!ipq.isEmpty() && edgeCount != m) {
int destNodeIndex = ipq.peekMinKeyIndex(); // equivalently: edge.to
Edge edge = ipq.pollMinValue();
mstEdges[edgeCount++] = edge;
minCostSum += edge.cost;
relaxEdgesAtNode(destNodeIndex);
}
// Verify MST spans entire graph.
mstExists = (edgeCount == m);
}
/* Graph construction helpers. */
// Creates an empty adjacency list graph with n nodes.
static List<List<Edge>> createEmptyGraph(int n) {
List<List<Edge>> g = new ArrayList<>();
for (int i = 0; i < n; i++) g.add(new ArrayList<>());
return g;
}
static void addDirectedEdge(List<List<Edge>> g, int from, int to, int cost) {
g.get(from).add(new Edge(from, to, cost));
}
static void addUndirectedEdge(List<List<Edge>> g, int from, int to, int cost) {
addDirectedEdge(g, from, to, cost);
addDirectedEdge(g, to, from, cost);
}
/* Example usage. */
public static void main(String[] args) {
// example1();
// firstGraphFromSlides();
// squareGraphFromSlides();
// disjointOnFirstNode();
// disjointGraph();
eagerPrimsExampleFromSlides();
// lazyVsEagerAnalysis();
}
private static void example1() {
int n = 10;
List<List<Edge>> g = createEmptyGraph(n);
addUndirectedEdge(g, 0, 1, 5);
addUndirectedEdge(g, 1, 2, 4);
addUndirectedEdge(g, 2, 9, 2);
addUndirectedEdge(g, 0, 4, 1);
addUndirectedEdge(g, 0, 3, 4);
addUndirectedEdge(g, 1, 3, 2);
addUndirectedEdge(g, 2, 7, 4);
addUndirectedEdge(g, 2, 8, 1);
addUndirectedEdge(g, 9, 8, 0);
addUndirectedEdge(g, 4, 5, 1);
addUndirectedEdge(g, 5, 6, 7);
addUndirectedEdge(g, 6, 8, 4);
addUndirectedEdge(g, 4, 3, 2);
addUndirectedEdge(g, 5, 3, 5);
addUndirectedEdge(g, 3, 6, 11);
addUndirectedEdge(g, 6, 7, 1);
addUndirectedEdge(g, 3, 7, 2);
addUndirectedEdge(g, 7, 8, 6);
EagerPrimsAdjacencyList solver = new EagerPrimsAdjacencyList(g);
Long cost = solver.getMstCost();
if (cost == null) {
System.out.println("No MST does not exists");
} else {
System.out.println("MST cost: " + cost);
for (Edge e : solver.getMst()) {
System.out.println(String.format("from: %d, to: %d, cost: %d", e.from, e.to, e.cost));
}
}
// Output:
// MST cost: 14
// from: 0, to: 4, cost: 1
// from: 4, to: 5, cost: 1
// from: 4, to: 3, cost: 2
// from: 3, to: 1, cost: 2
// from: 3, to: 7, cost: 2
// from: 7, to: 6, cost: 1
// from: 6, to: 8, cost: 4
// from: 8, to: 9, cost: 0
// from: 8, to: 2, cost: 1
}
private static void firstGraphFromSlides() {
int n = 7;
List<List<Edge>> g = createEmptyGraph(n);
addUndirectedEdge(g, 0, 1, 9);
addUndirectedEdge(g, 0, 2, 0);
addUndirectedEdge(g, 0, 3, 5);
addUndirectedEdge(g, 0, 5, 7);
addUndirectedEdge(g, 1, 3, -2);
addUndirectedEdge(g, 1, 4, 3);
addUndirectedEdge(g, 1, 6, 4);
addUndirectedEdge(g, 2, 5, 6);
addUndirectedEdge(g, 3, 5, 2);
addUndirectedEdge(g, 3, 6, 3);
addUndirectedEdge(g, 4, 6, 6);
addUndirectedEdge(g, 5, 6, 1);
EagerPrimsAdjacencyList solver = new EagerPrimsAdjacencyList(g);
Long cost = solver.getMstCost();
if (cost == null) {
System.out.println("No MST does not exists");
} else {
System.out.println("MST cost: " + cost);
for (Edge e : solver.getMst()) {
System.out.println(String.format("from: %d, to: %d, cost: %d", e.from, e.to, e.cost));
}
}
}
private static void squareGraphFromSlides() {
int n = 9;
List<List<Edge>> g = createEmptyGraph(n);
addUndirectedEdge(g, 0, 1, 6);
addUndirectedEdge(g, 0, 3, 3);
addUndirectedEdge(g, 1, 2, 4);
addUndirectedEdge(g, 1, 4, 2);
addUndirectedEdge(g, 2, 5, 12);
addUndirectedEdge(g, 3, 4, 1);
addUndirectedEdge(g, 3, 6, 8);
addUndirectedEdge(g, 4, 5, 7);
addUndirectedEdge(g, 4, 7, 9);
addUndirectedEdge(g, 5, 8, 10);
addUndirectedEdge(g, 6, 7, 11);
addUndirectedEdge(g, 7, 8, 5);
EagerPrimsAdjacencyList solver = new EagerPrimsAdjacencyList(g);
Long cost = solver.getMstCost();
if (cost == null) {
System.out.println("No MST does not exists");
} else {
System.out.println("MST cost: " + cost);
for (Edge e : solver.getMst()) {
System.out.println(String.format("from: %d, to: %d, cost: %d", e.from, e.to, e.cost));
}
}
}
private static void disjointOnFirstNode() {
int n = 4;
List<List<Edge>> g = createEmptyGraph(n);
// Node edges connected to zero
addUndirectedEdge(g, 1, 2, 1);
addUndirectedEdge(g, 2, 3, 1);
addUndirectedEdge(g, 3, 1, 1);
EagerPrimsAdjacencyList solver = new EagerPrimsAdjacencyList(g);
Long cost = solver.getMstCost();
if (cost == null) {
System.out.println("No MST does not exists");
} else {
System.out.println("MST cost: " + cost);
for (Edge e : solver.getMst()) {
System.out.println(String.format("from: %d, to: %d, cost: %d", e.from, e.to, e.cost));
}
}
}
private static void disjointGraph() {
int n = 6;
List<List<Edge>> g = createEmptyGraph(n);
// Component 1
addUndirectedEdge(g, 0, 1, 1);
addUndirectedEdge(g, 1, 2, 1);
addUndirectedEdge(g, 2, 0, 1);
// Component 2
addUndirectedEdge(g, 3, 4, 1);
addUndirectedEdge(g, 4, 5, 1);
addUndirectedEdge(g, 5, 3, 1);
EagerPrimsAdjacencyList solver = new EagerPrimsAdjacencyList(g);
Long cost = solver.getMstCost();
if (cost == null) {
System.out.println("No MST does not exists");
} else {
System.out.println("MST cost: " + cost);
for (Edge e : solver.getMst()) {
System.out.println(String.format("from: %d, to: %d, cost: %d", e.from, e.to, e.cost));
}
}
}
private static void eagerPrimsExampleFromSlides() {
int n = 7;
List<List<Edge>> g = createEmptyGraph(n);
addDirectedEdge(g, 0, 2, 0);
addDirectedEdge(g, 0, 5, 7);
addDirectedEdge(g, 0, 3, 5);
addDirectedEdge(g, 0, 1, 9);
addDirectedEdge(g, 2, 0, 0);
addDirectedEdge(g, 2, 5, 6);
addDirectedEdge(g, 3, 0, 5);
addDirectedEdge(g, 3, 1, -2);
addDirectedEdge(g, 3, 6, 3);
addDirectedEdge(g, 3, 5, 2);
addDirectedEdge(g, 1, 0, 9);
addDirectedEdge(g, 1, 3, -2);
addDirectedEdge(g, 1, 6, 4);
addDirectedEdge(g, 1, 4, 3);
addDirectedEdge(g, 5, 2, 6);
addDirectedEdge(g, 5, 0, 7);
addDirectedEdge(g, 5, 3, 2);
addDirectedEdge(g, 5, 6, 1);
addDirectedEdge(g, 6, 5, 1);
addDirectedEdge(g, 6, 3, 3);
addDirectedEdge(g, 6, 1, 4);
addDirectedEdge(g, 6, 4, 6);
addDirectedEdge(g, 4, 1, 3);
addDirectedEdge(g, 4, 6, 6);
EagerPrimsAdjacencyList solver = new EagerPrimsAdjacencyList(g);
Long cost = solver.getMstCost();
if (cost == null) {
System.out.println("No MST does not exists");
} else {
System.out.println("MST cost: " + cost);
for (Edge e : solver.getMst()) {
System.out.println(String.format("from: %d, to: %d, cost: %d", e.from, e.to, e.cost));
}
}
}
static Random random = new Random();
private static void lazyVsEagerAnalysis() {
int n = 5000;
List<List<EagerPrimsAdjacencyList.Edge>> g1 = EagerPrimsAdjacencyList.createEmptyGraph(n);
List<List<LazyPrimsAdjacencyList.Edge>> g2 = LazyPrimsAdjacencyList.createEmptyGraph(n);
for (int i = 0; i < n; i++) {
for (int j = i + 1; j < n; j++) {
int r = random.nextInt() % 10;
EagerPrimsAdjacencyList.addUndirectedEdge(g1, i, j, r);
LazyPrimsAdjacencyList.addUndirectedEdge(g2, i, j, r);
}
}
EagerPrimsAdjacencyList eagerSolver = new EagerPrimsAdjacencyList(g1);
LazyPrimsAdjacencyList lazySolver = new LazyPrimsAdjacencyList(g2);
long startTime = System.nanoTime();
Long eagerCost = eagerSolver.getMstCost();
long endTime = System.nanoTime();
System.out.println("Eager: " + (endTime - startTime));
startTime = System.nanoTime();
Long lazyCost = lazySolver.getMstCost();
endTime = System.nanoTime();
System.out.println("Lazy: " + (endTime - startTime));
if (eagerCost.longValue() != lazyCost.longValue()) {
System.out.println("Oh dear. " + eagerCost + " != " + lazyCost);
}
}
/* Supporting indexed priority queue implementation. */
private static class MinIndexedDHeap<T extends Comparable<T>> {
// Current number of elements in the heap.
private int sz;
// Maximum number of elements in the heap.
private final int N;
// The degree of every node in the heap.
private final int D;
// Lookup arrays to track the child/parent indexes of each node.
private final int[] child, parent;
// The Position Map (pm) maps Key Indexes (ki) to where the position of that
// key is represented in the priority queue in the domain [0, sz).
public final int[] pm;
// The Inverse Map (im) stores the indexes of the keys in the range
// [0, sz) which make up the priority queue. It should be noted that
// 'im' and 'pm' are inverses of each other, so: pm[im[i]] = im[pm[i]] = i
public final int[] im;
// The values associated with the keys. It is very important to note
// that this array is indexed by the key indexes (aka 'ki').
public final Object[] values;
// Initializes a D-ary heap with a maximum capacity of maxSize.
public MinIndexedDHeap(int degree, int maxSize) {
if (maxSize <= 0) throw new IllegalArgumentException("maxSize <= 0");
D = max(2, degree);
N = max(D + 1, maxSize);
im = new int[N];
pm = new int[N];
child = new int[N];
parent = new int[N];
values = new Object[N];
for (int i = 0; i < N; i++) {
parent[i] = (i - 1) / D;
child[i] = i * D + 1;
pm[i] = im[i] = -1;
}
}
public int size() {
return sz;
}
public boolean isEmpty() {
return sz == 0;
}
public boolean contains(int ki) {
keyInBoundsOrThrow(ki);
return pm[ki] != -1;
}
public int peekMinKeyIndex() {
isNotEmptyOrThrow();
return im[0];
}
public int pollMinKeyIndex() {
int minki = peekMinKeyIndex();
delete(minki);
return minki;
}
@SuppressWarnings("unchecked")
public T peekMinValue() {
isNotEmptyOrThrow();
return (T) values[im[0]];
}
public T pollMinValue() {
T minValue = peekMinValue();
delete(peekMinKeyIndex());
return minValue;
}
public void insert(int ki, T value) {
if (contains(ki)) throw new IllegalArgumentException("index already exists; received: " + ki);
valueNotNullOrThrow(value);
pm[ki] = sz;
im[sz] = ki;
values[ki] = value;
swim(sz++);
}
@SuppressWarnings("unchecked")
public T valueOf(int ki) {
keyExistsOrThrow(ki);
return (T) values[ki];
}
@SuppressWarnings("unchecked")
public T delete(int ki) {
keyExistsOrThrow(ki);
final int i = pm[ki];
swap(i, --sz);
sink(i);
swim(i);
T value = (T) values[ki];
values[ki] = null;
pm[ki] = -1;
im[sz] = -1;
return value;
}
@SuppressWarnings("unchecked")
public T update(int ki, T value) {
keyExistsAndValueNotNullOrThrow(ki, value);
final int i = pm[ki];
T oldValue = (T) values[ki];
values[ki] = value;
sink(i);
swim(i);
return oldValue;
}
// Strictly decreases the value associated with 'ki' to 'value'
public void decrease(int ki, T value) {
keyExistsAndValueNotNullOrThrow(ki, value);
if (less(value, values[ki])) {
values[ki] = value;
swim(pm[ki]);
}
}
// Strictly increases the value associated with 'ki' to 'value'
public void increase(int ki, T value) {
keyExistsAndValueNotNullOrThrow(ki, value);
if (less(values[ki], value)) {
values[ki] = value;
sink(pm[ki]);
}
}
/* Helper functions */
private void sink(int i) {
for (int j = minChild(i); j != -1; ) {
swap(i, j);
i = j;
j = minChild(i);
}
}
private void swim(int i) {
while (less(i, parent[i])) {
swap(i, parent[i]);
i = parent[i];
}
}
// From the parent node at index i find the minimum child below it
private int minChild(int i) {
int index = -1, from = child[i], to = min(sz, from + D);
for (int j = from; j < to; j++) if (less(j, i)) index = i = j;
return index;
}
private void swap(int i, int j) {
pm[im[j]] = i;
pm[im[i]] = j;
int tmp = im[i];
im[i] = im[j];
im[j] = tmp;
}
// Tests if the value of node i < node j
@SuppressWarnings("unchecked")
private boolean less(int i, int j) {
return ((Comparable<? super T>) values[im[i]]).compareTo((T) values[im[j]]) < 0;
}
@SuppressWarnings("unchecked")
private boolean less(Object obj1, Object obj2) {
return ((Comparable<? super T>) obj1).compareTo((T) obj2) < 0;
}
@Override
public String toString() {
List<Integer> lst = new ArrayList<>(sz);
for (int i = 0; i < sz; i++) lst.add(im[i]);
return lst.toString();
}
/* Helper functions to make the code more readable. */
private void isNotEmptyOrThrow() {
if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
}
private void keyExistsAndValueNotNullOrThrow(int ki, Object value) {
keyExistsOrThrow(ki);
valueNotNullOrThrow(value);
}
private void keyExistsOrThrow(int ki) {
if (!contains(ki)) throw new NoSuchElementException("Index does not exist; received: " + ki);
}
private void valueNotNullOrThrow(Object value) {
if (value == null) throw new IllegalArgumentException("value cannot be null");
}
private void keyInBoundsOrThrow(int ki) {
if (ki < 0 || ki >= N)
throw new IllegalArgumentException("Key index out of bounds; received: " + ki);
}
/* Test functions */
// Recursively checks if this heap is a min heap. This method is used
// for testing purposes to validate the heap invariant.
public boolean isMinHeap() {
return isMinHeap(0);
}
private boolean isMinHeap(int i) {
int from = child[i], to = min(sz, from + D);
for (int j = from; j < to; j++) {
if (!less(i, j)) return false;
if (!isMinHeap(j)) return false;
}
return true;
}
}
}