Jums ir doti n skaitļu pāri. Katrā pārī pirmais skaitlis vienmēr ir mazāks par otro skaitli. Pāris (c d) var sekot citam pārim (a b), ja b< c. Chain of pairs can be formed in this fashion. Find the longest chain which can be formed from a given set of pairs. Piemēri:
Input: (5 24) (39 60) (15 28) (27 40) (50 90) Output: (5 24) (27 40) (50 90) Input: (11 20) {10 40) (45 60) (39 40) Output: (11 20) (39 40) (45 60) In iepriekšējā ziņa, ko esam apsprieduši par maksimālā garuma pāru ķēdes problēmu. Tomēr ziņojumā bija ietverts tikai kods, kas saistīts ar maksimālā izmēra ķēdes garuma noteikšanu, bet ne uz maksimālā izmēra ķēdes uzbūvi. Šajā ierakstā mēs apspriedīsim, kā izveidot maksimālo garuma pāru ķēdi. Ideja ir vispirms sakārtot dotos pārus to pirmā elementa pieaugošā secībā. Ļaujiet arr[0..n-1] būt pāru ievades masīvam pēc šķirošanas. Mēs definējam vektoru L tā, lai L[i] pats par sevi ir vektors, kas saglabā arr[0..i] pāru maksimālā garuma ķēdi, kas beidzas ar arr[i]. Tāpēc indeksam i L[i] var rekursīvi uzrakstīt kā -
L[0] = {arr[0]} L[i] = {Max(L[j])} + arr[i] where j < i and arr[j].b < arr[i].a = arr[i] if there is no such j Piemēram, (5 24) (39 60) (15 28) (27 40) (50 90)
L[0]: (5 24) L[1]: (5 24) (39 60) L[2]: (15 28) L[3]: (5 24) (27 40) L[4]: (5 24) (27 40) (50 90)
Lūdzu, ņemiet vērā, ka pāru šķirošana tiek veikta, jo mums ir jāatrod maksimālais pāru garums, un pasūtīšanai šeit nav nozīmes. Ja mēs nešķirosim, mēs iegūsim pārus augošā secībā, bet tie nebūs maksimāli iespējamie pāri. Zemāk ir iepriekš minētās idejas īstenošana -
C++/* Dynamic Programming solution to construct Maximum Length Chain of Pairs */ #include using namespace std; struct Pair { int a; int b; }; // comparator function for sort function int compare(Pair x Pair y) { return x.a < y.a; } // Function to construct Maximum Length Chain // of Pairs void maxChainLength(vector<Pair> arr) { // Sort by start time sort(arr.begin() arr.end() compare); // L[i] stores maximum length of chain of // arr[0..i] that ends with arr[i]. vector<vector<Pair> > L(arr.size()); // L[0] is equal to arr[0] L[0].push_back(arr[0]); // start from index 1 for (int i = 1; i < arr.size(); i++) { // for every j less than i for (int j = 0; j < i; j++) { // L[i] = {Max(L[j])} + arr[i] // where j < i and arr[j].b < arr[i].a if ((arr[j].b < arr[i].a) && (L[j].size() > L[i].size())) L[i] = L[j]; } L[i].push_back(arr[i]); } // print max length vector vector<Pair> maxChain; for (vector<Pair> x : L) if (x.size() > maxChain.size()) maxChain = x; for (Pair pair : maxChain) cout << '(' << pair.a << ' ' << pair.b << ') '; } // Driver Function int main() { Pair a[] = {{5 29} {39 40} {15 28} {27 40} {50 90}}; int n = sizeof(a)/sizeof(a[0]); vector<Pair> arr(a a + n); maxChainLength(arr); return 0; }
Java // Java program to implement the approach import java.util.ArrayList; import java.util.Collections; import java.util.List; // User Defined Pair Class class Pair { int a; int b; } class GFG { // Custom comparison function public static int compare(Pair x Pair y) { return x.a - (y.a); } public static void maxChainLength(List<Pair> arr) { // Sort by start time Collections.sort(arr Main::compare); // L[i] stores maximum length of chain of // arr[0..i] that ends with arr[i]. List<List<Pair>> L = new ArrayList<>(); // L[0] is equal to arr[0] List<Pair> l0 = new ArrayList<>(); l0.add(arr.get(0)); L.add(l0); for (int i = 0; i < arr.size() - 1; i++) { L.add(new ArrayList<>()); } // start from index 1 for (int i = 1; i < arr.size(); i++) { // for every j less than i for (int j = 0; j < i; j++) { // L[i] = {Max(L[j])} + arr[i] // where j < i and arr[j].b < arr[i].a if (arr.get(j).b < arr.get(i).a && L.get(j).size() > L.get(i).size()) L.set(i L.get(j)); } L.get(i).add(arr.get(i)); } // print max length vector List<Pair> maxChain = new ArrayList<>(); for (List<Pair> x : L) if (x.size() > maxChain.size()) maxChain = x; for (Pair pair : maxChain) System.out.println('(' + pair.a + ' ' + pair.b + ') '); } // Driver Code public static void main(String[] args) { Pair[] a = {new Pair() {{a = 5; b = 29;}} new Pair() {{a = 39; b = 40;}} new Pair() {{a = 15; b = 28;}} new Pair() {{a = 27; b = 40;}} new Pair() {{a = 50; b = 90;}}}; int n = a.length; List<Pair> arr = new ArrayList<>(); for (Pair anA : a) { arr.add(anA); } // Function call maxChainLength(arr); } } // This code is contributed by phasing17
Python3 # Dynamic Programming solution to construct # Maximum Length Chain of Pairs class Pair: def __init__(self a b): self.a = a self.b = b def __lt__(self other): return self.a < other.a def maxChainLength(arr): # Function to construct # Maximum Length Chain of Pairs # Sort by start time arr.sort() # L[i] stores maximum length of chain of # arr[0..i] that ends with arr[i]. L = [[] for x in range(len(arr))] # L[0] is equal to arr[0] L[0].append(arr[0]) # start from index 1 for i in range(1 len(arr)): # for every j less than i for j in range(i): # L[i] = {Max(L[j])} + arr[i] # where j < i and arr[j].b < arr[i].a if (arr[j].b < arr[i].a and len(L[j]) > len(L[i])): L[i] = L[j] L[i].append(arr[i]) # print max length vector maxChain = [] for x in L: if len(x) > len(maxChain): maxChain = x for pair in maxChain: print('({a}{b})'.format(a = pair.a b = pair.b) end = ' ') print() # Driver Code if __name__ == '__main__': arr = [Pair(5 29) Pair(39 40) Pair(15 28) Pair(27 40) Pair(50 90)] n = len(arr) maxChainLength(arr) # This code is contributed # by vibhu4agarwal
C# using System; using System.Collections.Generic; public class Pair { public int a; public int b; } public class Program { public static int Compare(Pair x Pair y) { return x.a - (y.a); } public static void MaxChainLength(List<Pair> arr) { // Sort by start time arr.Sort(Compare); // L[i] stores maximum length of chain of // arr[0..i] that ends with arr[i]. List<List<Pair>> L = new List<List<Pair>>(); // L[0] is equal to arr[0] L.Add(new List<Pair> { arr[0] }); for (int i = 0; i < arr.Count - 1; i++) L.Add(new List<Pair>()); // start from index 1 for (int i = 1; i < arr.Count; i++) { // for every j less than i for (int j = 0; j < i; j++) { // L[i] = {Max(L[j])} + arr[i] // where j < i and arr[j].b < arr[i].a if (arr[j].b < arr[i].a && L[j].Count > L[i].Count) L[i] = L[j]; } L[i].Add(arr[i]); } // print max length vector List<Pair> maxChain = new List<Pair>(); foreach (List<Pair> x in L) if (x.Count > maxChain.Count) maxChain = x; foreach (Pair pair in maxChain) Console.WriteLine('(' + pair.a + ' ' + pair.b + ') '); } public static void Main() { Pair[] a = { new Pair() { a = 5 b = 29 } new Pair() { a = 39 b = 40 } new Pair() { a = 15 b = 28 } new Pair() { a = 27 b = 40 } new Pair() { a = 50 b = 90 } }; int n = a.Length; List<Pair> arr = new List<Pair>(a); MaxChainLength(arr); } }
JavaScript <script> // Dynamic Programming solution to construct // Maximum Length Chain of Pairs class Pair{ constructor(a b){ this.a = a this.b = b } } function maxChainLength(arr){ // Function to construct // Maximum Length Chain of Pairs // Sort by start time arr.sort((cd) => c.a - d.a) // L[i] stores maximum length of chain of // arr[0..i] that ends with arr[i]. let L = new Array(arr.length).fill(0).map(()=>new Array()) // L[0] is equal to arr[0] L[0].push(arr[0]) // start from index 1 for (let i=1;i<arr.length;i++){ // for every j less than i for(let j=0;j<i;j++){ // L[i] = {Max(L[j])} + arr[i] // where j < i and arr[j].b < arr[i].a if (arr[j].b < arr[i].a && L[j].length > L[i].length) L[i] = L[j] } L[i].push(arr[i]) } // print max length vector let maxChain = [] for(let x of L){ if(x.length > maxChain.length) maxChain = x } for(let pair of maxChain) document.write(`(${pair.a} ${pair.b}) `) document.write('') } // driver code let arr = [new Pair(5 29) new Pair(39 40) new Pair(15 28) new Pair(27 40) new Pair(50 90)] let n = arr.length maxChainLength(arr) /// This code is contributed by shinjanpatra </script>
Izvade:
virkne un apakšvirkne
(5 29) (39 40) (50 90)
Laika sarežģītība Dinamiskās programmēšanas risinājums ir O(n2) kur n ir pāru skaits. Palīgtelpa Programma izmanto O (n2).