GATE CS 2021 Set 2 Q.59 (CS Q.49): Functions, Recursion, Scope and Storage Classes

Consider the following ANSI C program.

#include <stdio.h>

int foo(int x, int y, int q)
{
    if ((x <= 0) && (y <= 0))
        return q;
    if (x <= 0)
        return foo(x, y - q, q);
    if (y <= 0)
        return foo(x - q, y, q);
    return foo(x, y - q, q) + foo(x - q, y, q);
}

int main()
{
    int r = foo(15, 15, 10);
    printf("%d", r);
    return 0;
}

The output of the program upon execution is ______.

Your answer

Official answer and solution

Official answer: 60
From: Official GATE 2021 answer key

Solution

Trace the recursive calls of foo(15, 15, 10) using dynamic programming/memoization of subproblem values.

The function is called with foo(15, 15, 10), where $q = 10$ .
Let $F(x, y)$ denote foo(x, y, 10). Note that $F(x, y)$ is symmetric with respect to $x$ and $y$ .

Base cases and transitions:

If both $x \le 0$ and $y \le 0$ : returns $q = 10$ .
If $x \le 0$ and $y > 0$ : returns $F(x, y - 10)$ .
If $x > 0$ and $y \le 0$ : returns $F(x - 10, y)$ .
If both $x > 0$ and $y > 0$ : returns $F(x, y - 10) + F(x - 10, y)$ .

Compute subproblems bottom-up:

For negative coordinates:
$F(-5, -5) = 10$
$F(5, -5) = F(5 - 10, -5) = F(-5, -5) = 10$
$F(15, -5) = F(15 - 10, -5) = F(5, -5) = 10$
By symmetry, $F(-5, 5) = 10$ and $F(-5, 15) = 10$ .

For $(5, 5)$ : $F(5, 5) = F(5, -5) + F(-5, 5) = 10 + 10 = 20$

For $(15, 5)$ : $F(15, 5) = F(15, -5) + F(5, 5) = 10 + 20 = 30$ By symmetry, $F(5, 15) = 30$ .

For $(15, 15)$ : $F(15, 15) = F(15, 5) + F(5, 15) = 30 + 30 = 60$

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 59 of 65, printed as CS Q.49
Section: Computer Science and Information Technology

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