Orthogonal Linked List

Last Updated : 24 Mar, 2023

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.

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.
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.

Suggest improvement

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