Patience Sorting

19 Jul, 2021
Patience sorting is a sorting algorithm based on card game Patience. In this sorting algorithm the rules of patience game is used to sort an list of elements based on their values.Rules of Patience Game:  
Cards with lower value can be placed over the card.
If there is no possible position for a card, then a new pile can be created.
Goal is to form as much as few piles possible.
Below is the visualization of the game as follows:  
As in the above visualization, It’s clear that cards are only placed when the value of them is less than the highest card of the pile. Otherwise, If there is no such pile then create a new one.Patience Sorting: There are generally two steps in the patience sorting that is creation of the piles and merging the piles. Below is the illustration of the steps:  
Initialize a 2D array to store the piles.
Traverse the given array and perform the following operations: Iterate over all the piles and check the top of the stack of each pile is less than the current element or not. IF found to be true, then push the current element to the top of the stack.Otherwise, create a new pile with the current element as the top of that stack.
Iterate over all the piles and check the top of the stack of each pile is less than the current element or not. IF found to be true, then push the current element to the top of the stack.Otherwise, create a new pile with the current element as the top of that stack.
Iterate over all the piles and check the top of the stack of each pile is less than the current element or not. IF found to be true, then push the current element to the top of the stack.
Otherwise, create a new pile with the current element as the top of that stack.
Merge the Piles:The idea is to perform a k-way merge of the p piles, each of which is internally sorted. Iterate over all the piles while count of elements in the pile is greater than or equal to 0 and find the minimum element from the top of each stack and push it into the sorted array.
Below is the visualization of the sorting steps:  
Below is the implementation of the above approach: 
C++
Javascript
// C++ program of the above approach #include <bits/stdc++.h>using namespace std; // Function to merge piles in a sorted ordervector<int> merge_piles(vector<vector<int> >& v){     // Store minimum element from    // the top of stack    vector<int> ans;     // In every iteration find the smallest element    // of top of pile and remove it from the piles    // and store into the final array    while (1) {         // Stores the smallest element        // of the top of the piles        int minu = INT_MAX;         // Stores index of the smallest element        // of the top of the piles        int index = -1;         // Calculate the smallest element        // of the top of the every stack        for (int i = 0; i < v.size(); i++) {             // If minu is greater than            // the top of the current stack            if (minu > v[i][v[i].size() - 1]) {                 // Update minu                minu = v[i][v[i].size() - 1];                 // Update index                index = i;            }        }         // Insert the smallest element        // of the top of the stack        ans.push_back(minu);         // Remove the top element from        // the current pile        v[index].pop_back();         // If current pile is empty        if (v[index].empty()) {             // Remove current pile            // from all piles            v.erase(v.begin() + index);        }         // If all the piles are empty        if (v.size() == 0)            break;    }    return ans;} // Function to sort the given array// using the patience sortingvector<int> patienceSorting(vector<int> arr){     // Store all the created piles    vector<vector<int> > piles;     // Traverse the array    for (int i = 0; i < arr.size(); i++) {         // If no piles are created        if (piles.empty()) {             // Initialize a new pile            vector<int> temp;             // Insert current element            // into the pile            temp.push_back(arr[i]);             // Insert current pile into            // all the piles            piles.push_back(temp);        }        else {             // Check if top element of each pile            // is less than or equal to            // current element or not            int flag = 1;             // Traverse all the piles            for (int j = 0; j < piles.size(); j++) {                 // Check if the element to be                // inserted is less than                // current pile's top                if (arr[i] < piles[j][piles[j].size() - 1]) {                    piles[j].push_back(arr[i]);                     // Update flag                    flag = 0;                    break;                }            }             // If flag is true            if (flag) {                 // Create a new pile                vector<int> temp;                 // Insert current element                // into temp                temp.push_back(arr[i]);                 // Insert current pile                // into all the piles                piles.push_back(temp);            }        }    }     // Store the sorted sequence    // of the given array    vector<int> ans;     // Sort the given array    ans = merge_piles(piles);     // Traverse the array, ans[]    for (int i = 0; i < ans.size(); i++)        cout << ans[i] << " ";     return ans;} // Driver Codeint main(){    vector<int> arr = { 6, 12, 2, 8, 3, 7 };     // Function Call    patienceSorting(arr);}
<script>// Javascript program of the above approach // Function to merge piles in a sorted orderfunction merge_piles(v) {     // Store minimum element from    // the top of stack    let ans = [];     // In every iteration find the smallest element    // of top of pile and remove it from the piles    // and store into the final array    while (1) {         // Stores the smallest element        // of the top of the piles        let minu = Number.MAX_SAFE_INTEGER;         // Stores index of the smallest element        // of the top of the piles        let index = -1;         // Calculate the smallest element        // of the top of the every stack        for (let i = 0; i < v.length; i++) {             // If minu is greater than            // the top of the current stack            if (minu > v[i][v[i].length - 1]) {                 // Update minu                minu = v[i][v[i].length - 1];                 // Update index                index = i;            }        }         // Insert the smallest element        // of the top of the stack        ans.push(minu);         // Remove the top element from        // the current pile        v[index].pop();         // If current pile is empty        if (v[index].length == 0) {             // Remove current pile            // from all piles            v.splice(index, 1);        }         // If all the piles are empty        if (v.length == 0)            break;    }    return ans;} // Function to sort the given array// using the patience sortingfunction patienceSorting(arr) {     // Store all the created piles    let piles = [];     // Traverse the array    for (let i = 0; i < arr.length; i++) {         // If no piles are created        if (piles.length == 0) {             // Initialize a new pile            let temp = [];             // Insert current element            // into the pile            temp.push(arr[i]);             // Insert current pile into            // all the piles            piles.push(temp);        }        else {             // Check if top element of each pile            // is less than or equal to            // current element or not            let flag = 1;             // Traverse all the piles            for (let j = 0; j < piles.length; j++) {                 // Check if the element to be                // inserted is less than                // current pile's top                if (arr[i] < piles[j][piles[j].length - 1]) {                    piles[j].push(arr[i]);                     // Update flag                    flag = 0;                    break;                }            }             // If flag is true            if (flag) {                 // Create a new pile                let temp = [];                 // Insert current element                // into temp                temp.push(arr[i]);                 // Insert current pile                // into all the piles                piles.push(temp);            }        }    }     // Store the sorted sequence    // of the given array    let ans = [];     // Sort the given array    ans = merge_piles(piles);     // Traverse the array, ans[]    for (let i = 0; i < ans.length; i++)        document.write(ans[i] + " ");     return ans;} // Driver Code let arr = [6, 12, 2, 8, 3, 7]; // Function CallpatienceSorting(arr); // This code is contributed by _saurabh_jaiswal.</script>
2 3 6 7 8 12
 
Time Complexity: O(N2) Auxiliary Space: O(N)
Note: The above approach can be optimized by merging of piles using priority_queue. Time Complexity: O(N * log(N)) Auxiliary Space: O(N)
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_saurabh_jaiswal
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