Merge pull request xtaci#60 from Hamada14/master

Add Lowest common ancestor algorithm

Author: xtaci

include/LCA.h

@@ -0,0 +1,38 @@
+/*******************************************************************************
+ *
+ *
+ *  /\  |  _   _  ._ o _|_ |_  ._ _   _
+ * /--\ | (_| (_) |  |  |_ | | | | | _>
+ *         _|
+ *
+ * LCA Finding using Binary Lifting and Dynamic Programming
+ *
+ * Features:
+ * 1. Answers Query about LCA of two nodes in O(log N)
+ *          where N is the total number of nodes in a tree.
+ *
+ * https://en.wikipedia.org/wiki/Lowest_common_ancestor
+ * http://www.csegeek.com/csegeek/view/tutorials/algorithms/trees/tree_part12.php
+ ******************************************************************************/
+
+#ifndef LCA_H
+#define LCA_H
+#include <vector>
+
+class LCA
+{
+    public:
+        LCA(std::vector< std::pair<int,int> > edges);
+        int lcaQuery(int a, int b);
+
+    private:
+        int getMaxLog();
+        void initDP();
+        void dfs(int currentNode, int currentParent);
+        std::vector< std::vector<int> > adjList, binaryLiftDp;
+        std::vector<int> parent, nodeHeight;
+        std::vector<bool> visited;
+        int _numberOfNodes, _maxLog;
+};
+
+#endif // LCA_H

src/lca_demo.cpp

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+#include "LCA.h"
+#include <cstdio>
+#include <vector>
+#include <iostream>
+/**
+*Constructor is initialized with a Adjacency List that
+*describe a tree and If It doesn't describe a tree it asserts failure.
+*/
+
+LCA::LCA(std::vector< std::pair<int,int> > edges): _numberOfNodes(edges.size() + 1), _maxLog(getMaxLog())
+{
+    //First we initialize the needed vectors
+    parent.resize(_numberOfNodes);
+    nodeHeight.resize(_numberOfNodes);
+    visited.resize(_numberOfNodes);
+    adjList.resize(_numberOfNodes);
+    binaryLiftDp = std::vector< std::vector<int> >(_numberOfNodes, std::vector<int>(_maxLog));
+    /**Construction of the Adjacency List to increase
+    *The efficiency of the tree traversal to O(V + E).
+    */
+    for(auto edge : edges){
+        adjList[edge.first].push_back(edge.second);
+        adjList[edge.second].push_back(edge.first);
+    }
+    //Initialize the Dynamic programming Vector.
+    initDP();
+}
+
+/**
+*DFS is used to find the parent and the height of each node
+*allowing the use of Binary Lifting.
+*/
+void LCA::dfs(int currentNode, int currentParent)
+{
+    visited[currentNode] = true;
+    parent[currentNode] = currentParent;
+    nodeHeight[currentNode] = nodeHeight[currentParent] + 1;
+    int adjacencySize = adjList[currentNode].size();
+    for(int idx = 0; idx < adjacencySize; idx++){
+        int nextNode = adjList[currentNode][idx];
+        if(!visited[nextNode])
+        {
+            dfs(nextNode, currentNode);
+        }
+    }
+}
+
+/**
+*Used to Calculate the Log to the base of two
+*for the number of the nodes to create the sparse table
+*used in binary Lifting.
+*/
+int LCA::getMaxLog(){
+    int curValue = 1;
+    int curLog = 1;
+    while(curValue < _numberOfNodes) curValue *= 2, curLog++;
+    return curLog;
+}
+
+void LCA::initDP()
+{
+    dfs(0, -1);
+    for(int i = 0; i < _numberOfNodes; i++) binaryLiftDp[i][0] = parent[i];
+    for(int i = 1; i <= _maxLog; i++)
+    {
+        for(int j = 0; j < _numberOfNodes; j++)
+        {
+            /**
+            * Since the ith parent of the current node is equal to
+            * the ith / 2 parent to the ith /2 parent of the current node
+            * That's why the Recurrence relation is described as follow
+            */
+            if(binaryLiftDp[j][i - 1] != -1)
+                binaryLiftDp[j][i] = binaryLiftDp[binaryLiftDp[j][i - 1]][i - 1];
+            else binaryLiftDp[j][i] = -1;
+        }
+    }
+}
+
+int LCA::lcaQuery(int a, int b)
+{
+    /**
+    * First Both nodes must have same height
+    * So we will rise the node with the deeper height up in
+    * the tree to where they're equal.
+    */
+    if(nodeHeight[a] < nodeHeight[b]) std::swap(a,b);
+    for(int i = _maxLog; i >= 0; i--)
+    {
+        if(binaryLiftDp[a][i] + 1 && nodeHeight[binaryLiftDp[a][i]] >= nodeHeight[b])
+            a = binaryLiftDp[a][i];
+    }
+    /**
+    * If the node Lower is the LCA then return it.
+    * Else keep moving both nodes up as much as they aren't the same
+    * until it's only 1 node left which is the direct parent of both of them
+    */
+    if(a == b) return a;
+    for(int i = _maxLog; i >= 0; i--)
+    {
+        if(binaryLiftDp[a][i] + 1 && binaryLiftDp[a][i] - binaryLiftDp[b][i])
+            a = binaryLiftDp[a][i], b = binaryLiftDp[b][i];
+    }
+    return parent[a];
+}
+
+int main(){
+    std::vector< std::pair<int,int> > edges;
+    edges.push_back({0,1});
+    edges.push_back({1,2});
+    edges.push_back({2,3});
+    edges.push_back({1,4});
+    LCA* l = new LCA(v);
+    std::cout << l->lcaQuery(0,1) << endl;
+    std::cout << l->lcaQuery(3,4) << endl;
+    std::cout << l->lcaQuery(3,2) << endl;
+}