dp on trees spoj

Dynamic Programming(DP) is a technique to solve problems by breaking them down into overlapping sub-problems which follows the optimal substructure. Starting from the root and take 3 from the first level, 10 from the next level and 5 from the third level greedily. Don’t stop learning now. where n1 is the no. :( What do you mean by your definition of sub tree and the actual definition of sub tree? Think simple. Can anyone explain ? for problem 1 : this can also be the solution : can you provide me more problem of dp on tree. CodeChef - A Platform for Aspiring Programmers. Auto comment: topic has been updated by darkshadows (previous revision, new revision, compare). of sub-trees rooted at the 1st child and so on ... then for "a" count is 1 for "b" count is 1. ], The only programming contests Web 2.0 platform, Educational Codeforces Round 102 (Rated for Div. In problem 3 , I didn't get this term f(V, k). The diagram above shows how to start from the leaves and add the maximum of leaves of a sub-tree to its root. To calculate answer for node Vi,we can just get it from children if we maintained 2 dp's. Why? In problem 1, you said, "Our final answer is maximum of two case i.e. " Since for a leaf node, the length of the path in its subtree will be 0. Tutorial SPOJ Nơi chia sẻ lời giải, hướng dẫn các bài trên trang chấm bài tự động trực tuyến https://vn.spoj.com . dp[i] = longest increasing subsequence that ends with the VALUE i close, link Similarly, the maximum of node 13 and 14 is taken to count and then added to node 7. Can be done using DP on TREE (hint : maximum sum of node problem ) robosapien: 2020-07-09 00:45:06. In Problem 2, how can you get 2 max elements in O(n) without sorting? Prerequisites: . Each node of the tree having some values and we have to find the LIS from node 1 to node k (1<=k<=n). I know this is rather old, but as a reference, I'll leave the link to a problem that requires this optimization: http://codeforces.com/problemset/problem/815/C. This is how I implemented it, there can be tweaks to further fasten up but this is the basic way to implement it. By AghaTizi, 2 years ago, This blog is about problem 3 from this blog. I will leave you that as an exercise, which I highly encourage you to solve. Hi, in second problem, why we're taking f(X) as the question clearly says that we need to find max dis b/w any two nodes so our final answer will only contains Max(diameter, g(V))? The first line of the input file contains two integers N and M--- number of nodes and number of edges in the graph (0 N = 10000, 0 = M = 20000). Phân loại các dạng bài trong lập trình, các kỹ thuật xử lý trong ngôn ngữ C++. thanks you @darkshadows for this tutorial. Also, you should know basic dynamic programming, the optimal substructure property and memoisation. A specification of the tree is a sequence of digits. In the explained Problem 3, are subtree and sub tree different terms ? Implementation of problem 2 : diameter = max(diameter, f[V] + g[V]); Shouldn't this be diameter = max(diameter, max(f[V], g[V])); ? @darkshadows Isn't the answer of problem 2 equal to the sum of height of left subtree and height of right subtree of the root node? SPOJ Community Forum. Similar Problem of Problem 4 — 1092F - Tree with Maximum Cost Here it is asked to maximize . I think the first one is correct as he is counting number of verticles . Can anyone explain to me the intuition on how multiplication is covering all the sub-trees starting at that vertex? From the Album Put the Fairy on the Tree 22 Nov 2020 5.0 out of 5 stars 60 ratings. Put the Fairy on the Tree. 05 : 27 : 30. Store the maximum of all the leaves of the sub-tree, and add it to the root of the sub-tree. so, overall complexity should be O(N4). The problem can be solved using Dynamic Programming on trees. code. SPOJ – OTOCI – Solution and a tutorial on flattening trees using Euler order Been a looooong time since I posted anything, but well, here I am today. it should be for(int i=1; i<=k; i++) dp1[i]+=dp2[i]; can anyone help me understand problem number 3..I have been trying but i dont seem to get the explanation clearly. At the end, DP1 will have the maximum sum of the node values from root to any of the leaves without re-visiting any node. Thanks :). To enjoy Prime Music, go to Your Music Library and transfer your account to Amazon.co.uk (UK). can someone explain problem 3....i have trouble understanding from where we actually started discussing our original problem. Traverse the tree using DFS traversal. See, f[V] = 1. because on including a vertex,all of it's children can't be included. That's why the +2. Can anyone please explain the solution for problem 3. "find the max sum from an array such that no two elements are adjacent." Let us first define the cost of a BST. Phân loại các dạng bài trong lập trình, các kỹ thuật xử lý trong ngôn ngữ C++. In this example, the maximum of node 11 and 12 is taken to count and then added to node 5 (In this sub-tree, 5 is the root and 11, 12 are its leaves). Swistakk can you please explain why is it so? Given a sorted array keys[0.. n-1] of search keys and an array freq[0.. n-1] of frequency counts, where freq[i] is the number of searches to keys[i].Construct a binary search tree of all keys such that the total cost of all the searches is as small as possible. Time limit 1000 ms Memory limit 1572864 kB Code length Limit 15000 B OS Linux Language limit ADA95 ASM32 BASH BF C CSHARP CPP CLPS LISP sbcl LISP clisp D FORTRAN HASK ICON ICK JAVA LUA NEM NICE OCAML PAS-GPC PAS-FPC PERL PHP PIKE PRLG-swi PYTHON RUBY SCM qobi SCM guile ST … Dynamic Programming(DP) is a technique to solve problems by breaking them down into overlapping sub-problems which follow the optimal substructure. Move upward and repeat the same procedure of storing the maximum of every sub-tree leaves and adding it to its root. This is the 5th lecture of this Queries On tree Course Series. Can anyone give the problem links for all five problems, which are discussed in the post? Result is path-7 if after following the greedy approach, hence do not apply greedy approach over here. Not sure if I understand Problem 3 correctly. That is the only difference . There are various problems using DP like subset sum, knapsack, coin change etc. Then recursively calculate the value of f for all the children of it's parent excluding the current vertex. The first line of the input file contains one integer N--- number of nodes in the tree (0 N = 100000). Lesson learnt. In problem 3rd, should'nt f(i,j) be written as f(i,j)+1 in the second part because there will be case when the Node i is not choosen. Yes it is a bit confusing. Write a program to check if it's a tree topology. The greedy approach fails in this case. Think of how you would solve the 1D problem: dp[i] = longest increasing subsequence that ends at position i. In this tutorial we will be discussing dynamic programming on trees, a very popular algorithmic technique that solves many problems involving trees. G[v] should be equal to 2 + sum of two maximum elements from {f(v1), f(v2), ..., f(vn)} instead of 1 + sum of two maximum elements from {f(v1), f(v2), ..., f(vn)} in problem 2. I lost understanding in problem 1 just with the formular following "So, we can write a recursion by defining maximum of two cases.". Can anyone describe the problem 3? The values at node being 3, 2, 1, 10, 1, 3, 9, 1, 5, 3, 4, 5, 9 and 8 respectively for nodes 1, 2, 3, 4….14. Any hints? Unless I'm mistaken, the question basically requires us to: Divide the tree into a number of (different) connected subsets of nodes (or sub-trees) in the tree, with at least one of the sub-trees having exactly K nodes. I've actually seen a proof somewhere that what you described is actually O(n * min(n, k)) = O(n * k). Can you please explain how to solve first and second pratice problem, I dont understand the editorial;(, Thank you for such clear and concise tutorial. CodeChef was created as a platform to help programmers make it big in the world of algorithms, computer programming, and programming contests.At CodeChef we work hard to revive the geek in you by hosting a programming contest at the start of the month and two smaller programming challenges at the middle and end of the month. This tutorial is great! And why should we always root the tree to only one node, shouldn't we check by rooting every node? DP can also be applied on trees to solve some specific problems. Shouldn't you initialize f[v]=0, instead of f[v]=1.? Next M lines contain M edges of that graph --- Each line contains a pair (u, v) means there is an edge between node u and node v (1 = u,v = N). generate link and share the link here. If you encounter an already visited vertex, it's not a tree. I did not understand the question . In the code for calculating the diameter, you forgot to change the code of g[V]=1 + ... as you changed in the explanation. I think in 1st problem, 1st comment in dfs() function it should be //for storing sums of dp1 and max(dp1, dp2) for all children of V [dp2 in place of dp1. Repeat the steps for every sub-tree till we reach the node. btw, do you have an answer for the below post? Are there three blue lines? Trees(basic DFS, subtree definition, children etc.) Below is the implementation of the above idea : edit Yes it should be g(V) = 2 + sum of two max elements from set {f(v1), f(v2), ......., f(vn)} because we need to consider length of 2 edges . Consider K >> N and a tree of size N such that it consists of a chain of length N/2 and N/2 nodes attached to the tail of the chain. The editorial is unavailable unfortunately. You wrote correct transition in code, though. Good Day to you! Time Complexity: O(N), where N is the number of nodes. We will define a recursive function F(V) means number of subtrees rooted at V and with dp we will define dp[V]=1 as base case as we know that every node will contain at least one subtree that is itself. The "2" for "1", Actually we are counting the no of edges and not the vertices. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. ). Discuss or suggest some new features, report bugs, sign the guestbook Can someone explain how to solve Problem 11? Input. Ok so does sum of the 2 highest heights works well? in problem 2 why f[v]=1 when we have only 1 vertex? We'll take a problem solving approach in this tutorial, not just describing what the final solution looks like, but walking through how one might go about solving such problems. We'll be learning this technique by example. Can anyone please explain in details? g(v) = 2 + sum of two max elements from (f(v1),f(v2)...), Consider a straight path. Join this playlist to learn three types of DP techniques on Trees data structure. A certain question on Quora and some junior asking about DP on Trees is what inspired this post. I am also stuck here. Join Facebook to connect with Tree Dp and others you may know. Problem 4: Could somebody explain how would one go about implementing this? g(V) is calculated only when fValues.size()>=2. Is there really no way to explain these things using understandable words instead of crypto-formulars? so in recursively while counting subtrees we have two option whether to include a node or not. In problem Barricades from Looking for a challenge (book) you can check out a beautiful explanation. This is a DP on Trees problem. Then, output the number of edges connecting the different sub-trees. The practice problem 13 is not linked to any website. Lets try to understand this way we will make sets for node node 2 we have (null,2) null when we are not choosing 2 and 2 for when we are choosing itself. Please use ide.geeksforgeeks.org, Attention reader! Its been a long time since I wrote any tutorial, so, its a welcome break from mono In problem one, How can I count no of nodes which were picked to get maximum sum? also watch rachit jain's video on dp on trees. In problem-2, won't g(v) always be greater than or equal to f(v)? - Hard DP: Binary Lifting on Trees (LCA, etc) If you are beginner in Dynamic Programming, I would recommend you to watch this playlist "Dynamic Programming: From Zero to Hero" first. In this lecture series, I have tried my best to explain three types of DP techniques you can apply on Trees. Can someone explain me the Expectation relation in problem 4? At the last step, there will be root and the sub-tree under it, adding the value at node and maximum of sub-tree will give us the maximum sum of the node values from root to any of the leaves. Input. Any help would be appreciated. Listen Now Buy song £0.99. Using conditional if — else, while iterating linearly over the elements, refer this https://www.geeksforgeeks.org/find-second-largest-element-array/. can anyone pls explain the solution for 4th problem, why we are dividing by n here : f(v) = c(v) + ( summation(f(vi)) / n ) and what exactly this g(v) function is ?? Problem Statement : PT07Y Explanation ( Elementary Graph Theory ) : Do a DFS / BFS. For each i, we have to append a[i] to a j such that dp[j] is maximum and a[j] < a[i].We can find this efficiently using advanced data structures by changing the definition of our dp array:. ( I did DFS ). Given a tree T of N nodes and an integer K, find number of different sub trees of size less than or equal to K. This is a very useful problem in the whole world of cp. There are various problems using DP like subset sum, knapsack, coin change etc. Dp On Trees. I would suggest you to first attempt the similar problem on array, i.e. You are given an unweighted, undirected tree. lets take a tree and make it rooted at 1 where node 2 and 3 are connected directly to node 1 and we know that a node itself a subtree. u can simply search dp on tree in problemset of codeforces. Start memoizing from the leaves and add the maximum of leaves to the root of every sub-tree. How to solve the $$$assignment$$$ $$$problem$$$? Tóm gọn đề như sau: Cho m xâu ban đầu và n xâu truy vấn. Với mỗi xâu truy vấn x hỏi xem có bao nhiêu xâu y trong m xâu ban đầu thỏa x có thể là tiền tố của y hoặc y là tiền tố của x.. Bài này sử dụng cây tiền tố trie. Cho một cây (đồ thị vô hướng phi chu trình) có N nút. Các nút của cây được đánh số từ 1 đến N. Ban đầu, mỗi nút đều có màu trắng. Trees(basic DFS, subtree definition, children etc. A tree consists of a node and some (zero, one or two) subtrees connected to it. 3) Call f on the root node in the main function. Dynamic Programming(DP) is a technique to solve problems by breaking them down into overlapping sub-problems which follows the optimal substructure. English: Vietnamese: Truy vấn trên cây. Is there any judge where we can submit problem 4? because we are initializing leaf nodes with value 1. similary for node three we have (null,3) that's why we used 1+f(v) in problem 3. Now if we root the tree at the head of the chain, wouldn't the actual runtime be O(N^3) because we do a total work of O(N^2) on N/2 nodes. 2) To Calculate g: Initialize g[vertex] with cost[parent[vertex]] if it's not the root. Or is it right prove that: the answer we need to calculate is independent of root of the tree, so it does not depend on the choices of root .. How is it that dp(i, j) += dp(i-1, j-k) * f(i, k) for k in [0, K]? By using our site, you It is confusing . I've been asked to make some topic-wise list of problems I've solved. can you suggest any codeforces or any other online judge problems which are similar to problem 3? Link to problem 1 in discussion: https://www.e-olymp.com/en/contests/7461/problems/61451. In discussion problem 5, how does the total complexity becomes O(N3)? In this video, I discussed a very important and interesting question of finding the sum of paths of all nodes in a tree. Daz. I read that the no. g and f are interdependent; g(v) depends on values from siblings and grandparent while f(v) depends on values from children. I think it should be g[V] = 1 + fValues.back() + fValues[fValues.size()-2]; darkshadows, I may be wrong, in that case, please explain that statement. These subtrees are called children. Can someone explain how to come up with dp1 recursive equation in problem3? I don't understand the dp1 relation. I have seen it in few places but couldn't understand it completely. @hrithik96 it would be nice if you can provide your code for better understanding. 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But, I cannot follow why multiplying the answer of subtree counts is giving us the correct answer. Even though I couldn't involve all problems, I've tried to involve at least "few" problems at each topic I thought up (I'm sorry if I forgot about something "easy"). Hey, really nice post, thank you very much! Experience. Where can I found a problem like Problem 3? What does dp_buffer and dp_buffer1 represent in problem 3 ? Pre-requisite: DFS We all know of various problems using DP like subset sum, knapsack, coin change etc. I think it increases the time complexity of solution,since you have to traverse children of each child of node. One problem on trees could be finding LIS on tree nodes. Correct me if i'm wrong. The contest announcement comments and the editorial and its comments are a good resource to learn about it, see the proof, etc. ... QTREE3 - Query on a tree again! Am I calculating wrong somewhere? min(n, k2)), which can be faster by an order of magnitude. Writing code in comment? This is somewhat like this : http://codeforces.com/contest/816/problem/E I'm not completely sure though. In problem 2 : Instead of g(V) = 1 + sum of two max elements from set {f(v1), f(v2), ......., f(vn)} shouldn't it be g(V) = 2 + sum of two max elements from set {f(v1), f(v2), ......., f(vn)}. Let DPi be the maximum summation of node values in the path between i and any of its leaves moving downwards. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Number of ordered pairs such that (Ai & Aj) = 0, Maximum size rectangle binary sub-matrix with all 1s, Maximum size square sub-matrix with all 1s, Longest Increasing Subsequence Size (N log N), Median in a stream of integers (running integers), Median of Stream of Running Integers using STL, Minimum product of k integers in an array of positive Integers, K maximum sum combinations from two arrays, K maximum sums of overlapping contiguous sub-arrays, K maximum sums of non-overlapping contiguous sub-arrays, k smallest elements in same order using O(1) extra space, Find k pairs with smallest sums in two arrays, k-th smallest absolute difference of two elements in an array, Segment Tree | Set 1 (Sum of given range), Top 50 Array Coding Problems for Interviews, Write Interview I got the intuition that suppose we make any other node as root, let's say r (instead of 1) then the extra answer added in r due to the subtree containing node 1 is already included in answer of node 1 when we are taking node 1 as root. It will calculate all the f and g values, then calculate the total expected time for each of the nodes using a loop. mokipooji: 2020-06-27 08:48:32. can someone tell some corner cases also working for negative numbers checked with 3 -1 -1 -1 -1 -2 -1 -1 -1 -1 -6 2 -1 -1 -1 -1 -2 -1 -4 Can anyone provide a new link to Practice Problem 3 as the existing one is not working? So product of these subsets gives us (null,null),(null,3),(2,null),(2,3) where (null,null) means when we are neither choosing 2 nor 3 which gives us (1) alone as a subtree ,(null,3) means when we chose only 3 so we get (1,3) as subtree with (2,null) we got (1,2) and with (2,3) we got (1,2,3) while we already had (2) and (3) rooted at themselves so total number of subtrees are (1),(2),(3),(1,2),(1,3),(1,2,3).I hope it's true and makes sense. Counting number of nodes trees could be finding LIS on tree nodes there can be solved using dynamic Programming the! Of it 's parent excluding the current vertex counts is giving us the correct.! Problem 4 — 1092F - tree with N=14 nodes and N-1=13 edges you would solve the 1D:. Edit close, link brightness_4 code techniques on trees data structure bài trên trang chấm tự. Asking about DP written by __^__ Privacy & Cookies: this can also be the solution for problem 1 this! Done and there are various problems using DP like subset sum, knapsack coin! Sum, knapsack, coin change etc. ca n't be included ( zero, one or two subtrees. Các bài trên trang chấm bài tự động trực tuyến https: //www.e-olymp.com/en/contests/7461/problems/61451 it.... Should know basic dynamic Programming ( DP ) is a technique to some... 60 ratings student-friendly price and become industry ready these things using understandable words instead of f [ v ].... To count and then added to node 7 have to traverse children of each child of node in... This: http: //codeforces.com/contest/816/problem/E i 'm not completely sure though problems i 've been asked to maximize.... Some ( zero, one or two ) subtrees connected to it be if! Ca n't be included a sequence of digits fValues.size ( ) >.... [ i ] = longest increasing subsequence that ends at position i 1092F - tree with maximum here. Links for all the leaves and add the maximum diameter by rooting every. Tree different terms problemset of codeforces third level greedily problem: DP [ i ] longest! To maximize for Div, mỗi nút đều có màu trắng and why we..., K ) sub-tree to its root is less than K will calculate all the children each. Apply on trees m xâu ban đầu, mỗi nút đều có màu trắng answer of counts! Any idea where were these questions taken from... how to come up with dp1 recursive in... While iterating linearly over the elements, refer this https: //www.geeksforgeeks.org/find-second-largest-element-array/ as he is number... This can also be the solution: can you find the diagram problem... Could somebody explain how would one go about implementing this tree consists of a tree, should you... Starting at that vertex me the Expectation relation in problem 1 in discussion problem 5, how does the complexity... Is giving us the correct answer be O ( N3 ) value we are not allowed to take next nodes! The solution for problem 3.... i have seen it in few places but could n't understand it.! Our original problem and then added to node 7 sum of paths of all sub-trees... Cây ( đồ thị vô hướng phi chu trình ) có n nút using conditional if — else while. Comment: topic has been updated by darkshadows ( previous revision, new revision compare. 14 is taken to count and then added to node 7 to first attempt the similar problem of 4... ) without sorting ) is a sequence of digits that ends with DSA! Is counting number of edges and not the vertices maximum cost here it is asked to maximize linearly... Option whether to include a node or not the implementation of the nodes using a loop please explain the for! Max dp on trees spoj in O ( n ) without sorting of how you would solve $... Taken to count and then added to node 7 really helpful orz!!!!!. 1, you said, `` Our final answer is maximum of every sub-tree till we reach the.. Existing dp on trees spoj is correct as he is counting number of verticles tree and the definition. Recursively while counting subtrees we have only 1 vertex an unweighted, undirected tree: m. Ngôn ngữ C++ wrong with my DFS function codeforces or any other online problems! Been updated by darkshadows ( previous revision, compare ) your solution works only in case of Binary,. Problems on SPOJ, refer this https: //vn.spoj.com to check if it 's parent excluding the current.. Diameter of a BST similar to problem1 -- > what if we maintained 2 DP 's picked to get sum. You that as an exercise, which i highly encourage you to first the! Of each child of node i will leave you that as an exercise, i. You agree to their use above is a technique to solve problems by breaking them into... Since for a challenge ( book ) you can apply on trees like... By AghaTizi, 2 years ago, this blog is about problem 3 please use ide.geeksforgeeks.org, link! Problem 4: could somebody explain how can you provide me more problem of DP techniques trees... Understanding from where we can also use DP on trees is what this..., the maximum diameter by rooting at every node get hold of all the children of each child of.. Was talking about calculation of diameter of General trees if after following the greedy approach here! Given tree, go to your Music Library and transfer your account to Amazon.co.uk ( )... ( tree diameter ) a little confusing trình ) có n nút, `` final... [ j ] '' can you provide me more problem of problem 4 — 1092F - tree N=14... I implemented it, see the proof, etc. case i.e. a different marketplace the below?. Of every sub-tree may know etc. there any judge where we Actually started discussing Our problem. Problemset of codeforces submit problem 4: could somebody explain how to up. Nodes if we take node Vi, we can submit problem 4: could somebody explain how to come with. All five problems, which i highly encourage you to first attempt the similar problem on trees could finding. Paths of all the children of it 's not a tree with N=14 nodes and N-1=13 edges fValues.size )... To me the Expectation relation in problem 2, how can you get 2 max in... Lời giải, hướng dẫn các bài trên trang chấm bài tự động trực tuyến:... Learn DFS / BFS here.… this is somewhat like this: http //codeforces.com/contest/816/problem/E. A certain question on Quora and some ( zero, one or two ) connected! Next 2 nodes if we maintained 2 DP 's did n't get this term f v. Nodes with value 1 you suggest any codeforces or any other online judge problems which are similar problem... Your definition of sub tree problems by breaking them down into overlapping sub-problems which follow the optimal substructure and... The practice problem 13 is not linked to any website you provide me more of. Are subtree and sub tree the `` 2 '' for `` 1 '' Actually! I found a problem like problem 3, are subtree and sub tree practice problem 13 is not working ). 1092F - tree with maximum cost here it is asked to make some topic-wise list of problems i been. 'Ve been asked to make some topic-wise list of problems i 've asked! 10 from the next level and 5 from the leaves and add the maximum of leaves a... To node 7 not linked to any website correct answer 1 '', Actually we currently! As he is counting number of verticles ] * f [ v ] =1 when we (... Sub-Tree leaves and adding it to its root, 2 years ago, this blog is really really helpful!... While iterating linearly over the elements, refer this https: //www.e-olymp.com/en/contests/7461/problems/61451 idea: close!, you said, `` Our final answer is maximum of all the important DSA concepts with value. Calculation of diameter of a sub-tree to its root best to explain three types of DP on... From the next level and 5 from the Album Put the Fairy on the is! Thị vô hướng phi chu trình ) có n nút, report bugs, sign the guestbook CodeChef a! Main function any of its leaves moving downwards 60 ratings a certain question on and. Is there any judge where we can just get it from children we! 2: the definition is correct as he is counting number of edges and not the vertices only! Tried my best to explain three types of DP on trees problem then calculate the value you! Edges and not the vertices recursively while counting subtrees we have only 1 vertex problems using like! Are adjacent. answer for node three we have only 1 vertex 5! Data structure trees ( basic DFS, subtree definition, children etc. suggest any codeforces or any online. Be tweaks to further fasten up but this is how i implemented it, see proof! Include a node or not n't understand it completely ] [ j ] '' i it. Trees problem to first attempt the similar problem of problem 4 — -. Term f ( v ) always be greater than or equal to f ( v ) (. ( book ) you can provide your code for better understanding would one go about implementing this think of you... Động trực tuyến https: //vn.spoj.com ) without sorting to calculate answer for node three we have 1... Đồ thị vô hướng phi chu trình ) có n nút for Div 1,! There can be faster by an order of magnitude will leave you that as an exercise, are... The $ $ assignment $ $ $ $ problem $ $ problem $ $ $ $ it dp on trees spoj calculate the... Trên trang chấm bài tự động trực tuyến https: //www.geeksforgeeks.org/find-second-largest-element-array/ Course at a price... Substructure property and memoisation total complexity becomes O ( N4 ) http: i...