An Orthogonal Linked List is a data structure composed of fundamental elements called Nodes (similar to linked lists). Each node in an orthogonal Linked List points to 4 other nodes, namely up, down, left and right. In essence, just like a matrix is a 2D version of an array, an orthogonal linked list is a 2D version of a linear linked list.Â
Algorithm to convert a matrix into an Orthogonal Linked List :
- Create a node for each cell in the matrix. In a map, and also store the value of the cell and a pointer to the created node for the cell.
- If the current row (i) is not the 0th row of the matrix, set the current node’s up pointer to the node of the cell just above it (use the map to get the correct pointer) and set the node above’s down pointer to the current node.Â
- Similarly, if the current column (j) is not the 0the column of the matrix, set the current node’s left pointer to the node of the cell to the left of the current node and set the node to the left’s right pointer to the current node
- repeat process 1 to 3 for every cell in the matrix
- return map[(matrix[0][0])] to return the pointer to the top-left node of the orthogonal linked list
Below is an implementation of the algorithm above.
Input:
matrix = {
{1, 2, 3},
{4, 5, 6},
{7, 8, 9}
}
Output:
A Node pointing to the top-left corner of the orthogonal linked list.
^ ^ ^
| | |
<--1 <--> 2 <--> 3-->
^ ^ ^
| | |
v v v
<--4 <--> 5 <--> 6-->
^ ^ ^
| | |
v v v
<--7 <--> 8 <--> 9-->
| | |
v v v
C++
#include <bits/stdc++.h> using namespace std;   struct MatrixNode {     int _val;     MatrixNode* _u; // pointer to node above     MatrixNode* _d; // pointer to node below     MatrixNode* _l; // pointer to node towards left     MatrixNode* _r; // pointer to node towards right       // Constructor for MatrixNode     MatrixNode( int val = 0,                 MatrixNode* u = nullptr,                 MatrixNode* d = nullptr,                 MatrixNode* l = nullptr,                 MatrixNode* r = nullptr )         {             _val = val;             _u = u;             _d = d;             _l = l;             _r = r;         } };   MatrixNode* BuildOrthogonalList(int matrix[][3], int r, int c) {     // an unordered_map to store the {value, pointers} pair     // for easy access while building the list     unordered_map<int, MatrixNode*> mp;       for(int i = 0; i < r; i++)     {         for(int j = 0; j < c; j++)         {             // create a newNode for each entry in the matrix             MatrixNode* newNode = new MatrixNode(matrix[i][j]);             // store the pointer of the new node             mp[(matrix[i][j])] = newNode;               // set the up and down pointing pointers correctly             if(i != 0)             {                 newNode->_u = mp[(matrix[i - 1][j])];                 mp[(matrix[i - 1][j])]->_d = newNode;             }               // similarly set the left and right pointing pointers             if(j != 0)             {                 newNode->_l = mp[(matrix[i][j - 1])];                 mp[(matrix[i][j - 1])]->_r = newNode;             }         }     }       // return the start of the list     return mp[(matrix[0][0])]; }   void PrintOrthogonalList(MatrixNode* head) {     MatrixNode* curRow; // will point to the begin of each row     MatrixNode* cur; // will traverse each row and print the element     for(curRow = head; curRow != nullptr; curRow = curRow->_d)     {         for(cur = curRow; cur != nullptr; cur = cur->_r)         {             cout << cur->_val << " ";         }         cout << endl;     } }   int main() {     int matrix[3][3] = {         {1, 2, 3},         {4, 5, 6},         {7, 8, 9}     };       MatrixNode* list = BuildOrthogonalList(matrix, 3, 3);     PrintOrthogonalList(list);       return 0; } |
Java
// Java code for the above approach import java.io.*; import java.util.*;   class MatrixNode {   int val;   MatrixNode u, d, l, r;     MatrixNode(int val) { this.val = val; } }   class GFG {     static MatrixNode buildOrthogonalList(int[][] matrix,                                         int r, int c)   {     // a map to store the {value, pointers} pair for     // easy access     Map<Integer, MatrixNode> mp = new HashMap<>();       for (int i = 0; i < r; i++) {       for (int j = 0; j < c; j++) {         // create a newNode for each entry in the         // matrix         MatrixNode newNode           = new MatrixNode(matrix[i][j]);         // store the pointer of the new node         mp.put(matrix[i][j], newNode);           // set the up and down pointing pointers         // correctly         if (i != 0) {           newNode.u = mp.get(matrix[i - 1][j]);           mp.get(matrix[i - 1][j]).d = newNode;         }           // similarly set the left and right pointing         // pointers         if (j != 0) {           newNode.l = mp.get(matrix[i][j - 1]);           mp.get(matrix[i][j - 1]).r = newNode;         }       }     }       // return the start of the list     return mp.get(matrix[0][0]);   }     public static void printOrthogonalList(MatrixNode head)   {     MatrixNode curRow       = head; // will point to the begin of each row     MatrixNode cur = null; // will traverse each row and     // print the element     while (curRow != null) {       cur = curRow;       while (cur != null) {         System.out.print(cur.val + " ");         cur = cur.r;       }       System.out.println();       curRow = curRow.d;     }   }     public static void main(String[] args)   {     int[][] matrix       = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };       MatrixNode list = buildOrthogonalList(matrix, 3, 3);     printOrthogonalList(list);   } }   // This code is contributed by lokeshmvs21. |
Python3
# Python code for the above approach class MatrixNode:     def __init__(self, val = 0, u = None, d = None, l = None, r = None):         self._val = val         self._u = u         self._d = d         self._l = l         self._r = r   def BuildOrthogonalList(matrix, r, c):     """     Builds an orthogonal list from a given matrix     """    # a dictionary to store the {value, pointers} pair     # for easy access while building the list     mp = {}       for i in range(r):         for j in range(c):             # create a newNode for each entry in the matrix             newNode = MatrixNode(matrix[i][j])             # store the pointer of the new node             mp[(matrix[i][j])] = newNode               # set the up and down pointing pointers correctly             if i != 0:                 newNode._u = mp[(matrix[i - 1][j])]                 mp[(matrix[i - 1][j])]._d = newNode               # similarly set the left and right pointing pointers             if j != 0:                 newNode._l = mp[(matrix[i][j - 1])]                 mp[(matrix[i][j - 1])]._r = newNode       # return the start of the list     return mp[(matrix[0][0])]   def PrintOrthogonalList(head):     """     Prints the given orthogonal list     """    curRow = head # will point to the begin of each row     cur = None # will traverse each row and print the element     while curRow:         cur = curRow         while cur:             print(cur._val, end=" ")             cur = cur._r         print()         curRow = curRow._d   matrix = [     [1, 2, 3],     [4, 5, 6],     [7, 8, 9] ]   list = BuildOrthogonalList(matrix, 3, 3) PrintOrthogonalList(list)   # This code is contributed by lokeshpotta20. |
C#
// C# code equivalent to the Java code using System; using System.Collections.Generic;   class MatrixNode {   public int val;   public MatrixNode u, d, l, r;     public MatrixNode(int val)   {     this.val = val;   } }   class Program {   static MatrixNode buildOrthogonalList(int[,] matrix, int r, int c)   {     // a dictionary to store the {value, pointers} pair for easy access     Dictionary<int, MatrixNode> mp = new Dictionary<int, MatrixNode>();       for (int i = 0; i < r; i++)     {       for (int j = 0; j < c; j++)       {         // create a newNode for each entry in the matrix         MatrixNode newNode = new MatrixNode(matrix[i, j]);         // store the pointer of the new node         mp[(matrix[i, j])] = newNode;           // set the up and down pointing pointers correctly         if (i != 0)         {           newNode.u = mp[(matrix[i - 1, j])];           mp[(matrix[i - 1, j])].d = newNode;         }           // similarly set the left and right pointing pointers         if (j != 0)         {           newNode.l = mp[(matrix[i, j - 1])];           mp[(matrix[i, j - 1])].r = newNode;         }       }     }       // return the start of the list     return mp[(matrix[0, 0])];   }     public static void printOrthogonalList(MatrixNode head)   {     MatrixNode curRow = head; // will point to the begin of each row     MatrixNode cur = null; // will traverse each row and print the element     while (curRow != null)     {       cur = curRow;       while (cur != null)       {         Console.Write(cur.val + " ");         cur = cur.r;       }       Console.WriteLine();       curRow = curRow.d;     }   }     static void Main(string[] args)   {     int[,] matrix = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };       MatrixNode list = buildOrthogonalList(matrix, 3, 3);     printOrthogonalList(list);   } } |
Javascript
// JavaScript code for the above approach   class MatrixNode {   constructor(val) {     this.val = val;     this.u = null;     this.d = null;     this.l = null;     this.r = null;   } }   function buildOrthogonalList(matrix, r, c) {   // a map to store the {value, pointers} pair of easy access   const mp = new Map();     for (let i = 0; i < r; i++) {     for (let j = 0; j < c; j++) {       // create a newNode for each entry in the matrix       const newNode = new MatrixNode(matrix[i][j]);               // store the pointer of the new Node       mp.set(matrix[i][j], newNode);         // set the up and down pointing pointers correctly       if (i !== 0) {         newNode.u = mp.get(matrix[i - 1][j]);         mp.get(matrix[i - 1][j]).d = newNode;       }         // similarly set the left and right pointing pointers       if (j !== 0) {         newNode.l = mp.get(matrix[i][j - 1]);         mp.get(matrix[i][j - 1]).r = newNode;       }     }   }     // return the start of the list   return mp.get(matrix[0][0]); }   function printOrthogonalList(head) {   let curRow = head; // will point to the begin of each row   let cur = null; // will traverse each row and print the element   while (curRow !== null) {     cur = curRow;     while (cur !== null) {       console.log(cur.val + " ");       cur = cur.r;     }     console.log("<br>");     curRow = curRow.d;   } }   const matrix = [   [1, 2, 3],   [4, 5, 6],   [7, 8, 9] ];   const list = buildOrthogonalList(matrix, 3, 3); printOrthogonalList(list);   // This code is contributed by sankar. |
Application:
The most common application of orthogonal linked list is in sparse matrix representation. In brief, a sparse matrix is a matrix in which most of its elements are zeroes (or any known constant). They appear often in scientific applications. Representing sparse matrices as a 2D array is a huge wastage of memory. Instead, sparse matrices are represented as an orthogonal linked list. We create a node only for non-zero elements in the matrix and in each node, we store the value, the row index and the column index along with the necessary pointers to other nodes. This saves a lot of performance overhead and is the most memory-efficient way to implement a sparse matrix.
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