[Hacker Rank] Interview Preparation Kit: Recursion: Davis' Staircase. Solved ✅.

Author: Gonzalo Diaz

docs/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci-recursive-staircase-solution-notes.md

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+# [Recursion: Davis' Staircase](https://www.hackerrank.com/challenges/ctci-recursive-staircase)
+
+Find the number of ways to get from the bottom of a staircase
+to the top if you can jump 1, 2, or 3 stairs at a time.
+
+- Difficulty:  `#medium`
+- Category: `#ProblemSolvingIntermediate`
+
+## Failed solution
+
+This solution correctly calculates the result. The problem is its performance,
+since due to the traversal of the recursion tree,
+it is eventually easy to reach repeated cases that are recalculated each time.
+
+```typescript
+def step_perms_compute(n: number): number
+    if (n == 0) {
+      return 0
+    }
+    if (n == 1) {
+      return 1
+    }
+    if (n == 2) {
+      return 2
+    }
+    if (n == 3) {
+      return 4
+    }
+
+    return
+        step_perms_compute(n - 3) +
+        step_perms_compute(n - 2) +
+        step_perms_compute(n - 1)
+```
+
+## Alternative solution
+
+The final solution introduces a simple caching mechanism,
+so that repeated cases are not recalculated.
+
+The trade-off is that the algorithm now requires
+more memory to run in less time.
+
+## Generalized solution
+
+In order to comply with some clean code best practices,
+I noticed that the step limit in the algorithm is a hard-coded number,
+so to comply with the "no magic numbers" rule,
+I was forced to find a more generalized solution.
+
+Then I found the following pattern:
+
+- First cases are:
+
+$$ \begin{matrix}
+    \text{stepPerms(0)} = 0 \\
+    \text{stepPerms(1)} = 1 \\
+    \text{stepPerms(2)} = 2 \\
+  \end{matrix}
+$$
+
+- Next step combinations above 2 and less than the step limit are:
+
+$$ \text{stepPerms(number of steps)} = 2^\text{number of steps} + 1 $$
+
+- When `number of steps` are above the limit, the pattern is
+the sum of latest `number of steps` previous calls of
+`stepPerms(x)` results as follows:
+
+$$ \displaystyle\sum_{
+  i=\text{number of steps} - \text{limit}}
+  ^\text{number of steps}
+  stepPerms(\text{number of steps} - i)
+$$

docs/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci-recursive-staircase.md

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+# [Recursion: Davis' Staircase](https://www.hackerrank.com/challenges/ctci-recursive-staircase)
+
+Find the number of ways to get from the bottom of a staircase
+to the top if you can jump 1, 2, or 3 stairs at a time.
+
+- Difficulty:  `#medium`
+- Category: `#ProblemSolvingIntermediate`
+
+Davis has a number of staircases in his house and he likes to
+climb each staircase `1`, `2`, or `3` steps at a time.
+Being a very precocious child, he wonders how many ways there
+are to reach the top of the staircase.
+
+Given the respective heights for each of the  staircases in his house,
+find and print the number of ways he can climb each staircase,
+module $10^10 + 7 $ on a new line.
+
+## Example
+
+`n = 5`
+
+The staircase has `5` steps. Davis can step on the following sequences of steps:
+
+```text
+1 1 1 1 1
+1 1 1 2
+1 1 2 1
+1 2 1 1
+2 1 1 1
+1 2 2
+2 2 1
+2 1 2
+1 1 3
+1 3 1
+3 1 1
+2 3
+3 2
+```
+
+There are `13` possible ways he can take these `5` steps and `13 modulo 10000000007`
+
+## Function Description
+
+Complete the stepPerms function using recursion in the editor below.
+
+stepPerms has the following parameter(s):
+
+- int n: the number of stairs in the staircase
+
+## Returns
+
+int: the number of ways Davis can climb the staircase, modulo 10000000007
+
+## Input Format
+
+The first line contains a single integer, `s`, the number of staircases in his house.
+Each of the following `s` lines contains a single integer,
+`n`, the height of staircase `i`.
+
+## Constraints
+
+- $ 1 \leq s \leq 5 $
+- $ 1 \leq n \leq 36 $
+
+## Subtasks
+
+- 1 \leq n \leq 20 for `50%` of the maximum score.
+
+## Sample Input
+
+```text
+STDIN   Function
+-----   --------
+3       s = 3 (number of staircases)
+1       first staircase n = 1
+3       second n = 3
+7       third n = 7
+```
+
+## Sample Output
+
+```text
+1
+4
+44
+```
+
+## Explanation
+
+Let's calculate the number of ways of climbing
+the first two of the Davis' `s = 3` staircases:
+
+1. The first staircase only has `n = 1` step,
+    so there is only one way for him to
+    climb it (i.e., by jumping `1` step). Thus, we print `1` on a new line.
+
+2. The second staircase has `n = 3` steps and he can climb it in any of the
+    four following ways:
+
+    1. 1 -> 1 -> 1
+    2. 1 -> 2
+    3. 2 -> 1
+    4. 3
+
+Thus, we print `4` on a new line.

src/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci_recursive_staircase.js

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+/**
+ * @link Problem definition [[docs/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci-recursive-staircase.md]]
+ * @see Solution Notes: [[docs/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci-recursive-staircase-solution-notes.md]]
+ */
+
+const TOP_LIMIT = 10 ** 10 + 7;
+const STEPS_LIMIT = 3;
+
+export function stepPermsComputWithCache(nSteps, cache, stepsLimit) {
+  if (nSteps >= 0 && nSteps <= 2) {
+    return nSteps;
+  }
+
+  const keys = new Set(Object.values(cache));
+  let result = 0;
+
+  for (let i = 1; i <= Math.min(stepsLimit, nSteps); i++) {
+    const searchKey = nSteps - i;
+    if (!keys.has(searchKey)) {
+      cache[searchKey] = stepPermsComputWithCache(searchKey, cache, stepsLimit);
+    }
+
+    result += cache[searchKey];
+  }
+
+  return result + (nSteps <= stepsLimit ? 1 : 0);
+}
+
+export function stepPerms(n) {
+  const initialCache = {};
+  return stepPermsComputWithCache(n, initialCache, STEPS_LIMIT) % TOP_LIMIT;
+}
+
+export default { stepPerms, stepPermsComputWithCache };

src/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci_recursive_staircase.test.js

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+import { describe, expect, it } from '@jest/globals';
+import { logger as console } from '../../../logger.js';
+
+import {
+  stepPerms,
+  stepPermsComputWithCache
+} from './ctci_recursive_staircase.js';
+import TEST_CASES from './ctci_recursive_staircase.testcases.json';
+import TEST_CASES_GENERALIZED from './ctci_recursive_staircase_generalized.testcases.json';
+
+describe('ctci_recursive_staircase', () => {
+  it('stepPerms test cases', () => {
+    expect.assertions(8);
+
+    TEST_CASES.forEach((testSet) => {
+      testSet?.tests.forEach((test) => {
+        const answer = stepPerms(test.input);
+
+        console.debug(`stepPerms(${test.input}) solution found: ${answer}`);
+
+        expect(answer).toStrictEqual(test.expected);
+      });
+    });
+  });
+
+  it('stepPermsComputWithCache test cases', () => {
+    expect.assertions(3);
+
+    TEST_CASES_GENERALIZED.forEach((testSet) => {
+      testSet?.tests.forEach((test) => {
+        const initialCache = {};
+        const answer = stepPermsComputWithCache(
+          test.input,
+          initialCache,
+          test.limit
+        );
+
+        console.debug(
+          `stepPermsComputWithCache(${test.input}, ${initialCache}, ${test.limit}) solution found: ${answer}`
+        );
+
+        expect(answer).toStrictEqual(test.expected);
+      });
+    });
+  });
+});

src/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci_recursive_staircase.testcases.json

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+[
+    {
+        "title": "Sample Test case 0",
+        "tests": [
+            {
+                "input": 1,
+                "expected": 1
+            },
+            {
+                "input": 3,
+                "expected": 4
+            },
+            {
+                "input": 7,
+                "expected": 44
+            }
+        ]
+    },
+    {
+        "title": "Sample Test case 9",
+        "tests": [
+            {
+                "input": 5,
+                "expected": 13
+            },
+            {
+                "input": 8,
+                "expected": 81
+            }
+        ]
+    },
+    {
+        "title": "Sample Test case 10",
+        "tests": [
+            {
+                "input": 15,
+                "expected": 5768
+            },
+            {
+                "input": 20,
+                "expected": 121415
+            },
+            {
+                "input": 27,
+                "expected": 8646064
+            }
+        ]
+    }
+]

src/hackerrank/interview_preparation_kit/recursion_and_backtracking/ctci_recursive_staircase_generalized.testcases.json

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+[
+  {
+    "title": "Own sample 1",
+    "tests": [
+      {
+        "input": 4,
+        "limit": 3,
+        "expected": 7
+      }
+    ]
+  },
+  {
+    "title": "Own sample 2",
+    "tests": [
+      {
+        "input": 5,
+        "limit": 4,
+        "expected": 15
+      }
+    ]
+  },
+  {
+    "title": "Own sample 3",
+    "tests": [
+      {
+        "input": 6,
+        "limit": 2,
+        "expected": 13
+      }
+    ]
+  }
+]