GATE CS 2011 Q.25 (CS Q.25): Dynamic Programming

An algorithm to find the length of the longest monotonically increasing sequence of numbers in an array $A[0:n-1]$ is given below.

Let $L_i$ denote the length of the longest monotonically increasing sequence starting at index $i$ in the array.

Initialize $L_{n-1} = 1$ .

For all $i$ such that $0 \le i \le n-2$ :

L_i = \begin{cases} 1 + L_{i+1} & \text{if } A[i] < A[i+1] \\ 1 & \text{otherwise} \end{cases}

Finally the length of the longest monotonically increasing sequence is $\max(L_0, L_1, \dots, L_{n-1})$ .

Which of the following statements is TRUE?

The algorithm uses dynamic programming paradigm

The algorithm has a linear complexity and uses branch and bound paradigm

The algorithm has a non-linear polynomial complexity and uses branch and bound paradigm

The algorithm uses divide and conquer paradigm

Official answer and solution

Solution

Classify the given recurrence and memoization scheme among algorithm design paradigms.

The algorithm computes optimal solutions to subproblems $L_i$ (the length of the longest increasing sequence starting at index $i$ ) using the recurrence:

L_i = 1 + L_{i+1} \quad \text{if } A[i] < A[i+1], \quad \text{else } 1

The subproblems exhibit optimal substructure and overlapping subproblems: the solution for $L_i$ depends directly on the already-computed value of $L_{i+1}$ .

The algorithm evaluates subproblems in reverse order (from index $n-2$ down to 0) and stores the intermediate results in a table $L$ , which is the classic bottom-up dynamic programming paradigm.

Final answer

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