646 Maximum Length of Pair Chain

This problem does have a greedy solution, but I wanted to practice me some good ‘ol DP. But please feel free to go to leetcode solutions and understand the greedy solution too (it’s linear) - link.

Intuition

All possibilities of pair combinations can be checked as far as the condition p1[1] < p2[0] is met to make a chain. Then keep track of the maxchain length.

Note: remember that when we check all possibilities, it’s basically a decision tree. And how do we find the longest length (a.k.a depth) of any branch in a tree? By DFS-ing and increasing result by one.

    leftDepth = 1 + dfs(node.left)
    rightDepth = 1 + dfs(node.right)

Code

Python3

def findLongestChain(self, pairs: List[List[int]]) -> int:
    pairs.sort(key=lambda x: x[1])
 
    def dfs(arr, start: int, memo):
        if start in memo: return memo[start]
 
        if start >= len(arr):
            return
        
        maxChain = 1    # a single element is a valid chain <- so the minimum is 1
 
        for end in range(start+1, len(arr)):
            a, b = arr[start]
            c, d = arr[end]
            if b < c:
                maxChain = max(maxChain, 1 + dfs(arr, end, memo))
 
        memo[start] = maxChain
        
        return maxChain
    
    return dfs(pairs, 0, defaultdict(int))

Big O Analysis

• Runtime

  The runtime complexity here is $O(N^2)$ since although memoisation works, we are atmost visiting each element twice.

• Memory

  The memory usage is $O(N)$ since we use the collections.defaultdict object to store memoised values.

— A

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