Climbing Stairs Problem in DSA Go - Execution Trace

Concept Flow - Climbing Stairs Problem

Start at step 0

↓

Can climb 1 or 2 steps

↓

Calculate ways to reach step n

↓

Use ways(n-1) + ways(n-2)

↓

Repeat until step n

↓

Return total ways to reach step n

We start at step 0 and can climb either 1 or 2 steps at a time. The total ways to reach step n is the sum of ways to reach steps n-1 and n-2, repeated until we reach step n.

Execution Sample

DSA Go

func climbStairs(n int) int {
    if n <= 2 {
        return n
    }
    a, b := 1, 2
    for i := 3; i <= n; i++ {
        a, b = b, a+b
    }
    return b
}

This code calculates the number of ways to climb n stairs by summing ways to reach the previous two steps.

Execution Table

Step	Operation	i	a (ways to i-2)	b (ways to i-1)	New b (ways to i)	Visual State
1	Initialize	-	1	2	-	a=1, b=2 (ways for steps 1 and 2)
2	Check n <= 2	-	-	-	-	If n=2, return 2 directly
3	Loop start	3	1	2	3	Calculate ways for step 3: 1+2=3
4	Loop iteration	4	2	3	5	Calculate ways for step 4: 2+3=5
5	Loop iteration	5	3	5	8	Calculate ways for step 5: 3+5=8
6	Loop iteration	6	5	8	13	Calculate ways for step 6: 5+8=13
7	Loop iteration	7	8	13	21	Calculate ways for step 7: 8+13=21
8	Loop iteration	8	13	21	34	Calculate ways for step 8: 13+21=34
9	Loop iteration	9	21	34	55	Calculate ways for step 9: 21+34=55
10	Loop iteration	10	34	55	89	Calculate ways for step 10: 34+55=89
11	Return result	-	-	-	89	Return total ways for step 10

💡 Loop ends when i exceeds n; final b holds total ways to climb n stairs

Variable Tracker

Variable	Start	After i=3	After i=4	After i=5	After i=6	After i=7	After i=8	After i=9	After i=10	Final
i	-	3	4	5	6	7	8	9	10	-
a	1	2	3	5	8	13	21	34	55	55
b	2	3	5	8	13	21	34	55	89	89

Key Moments - 3 Insights

Why do we start with a=1 and b=2 instead of 0 and 1?

Because ways to climb 1 step is 1 and ways to climb 2 steps is 2, as shown in execution_table row 1. Starting with these values sets the base for the loop.

Why does the loop start from i=3?

The first two steps (1 and 2) are base cases handled before the loop. The loop calculates ways for steps 3 to n, as seen from execution_table rows 3 onwards.

What does the variable 'b' represent after each iteration?

'b' holds the total number of ways to reach the current step i, as shown in the 'New b' column of execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at step 5 (i=5). What is the value of 'b' after this iteration?

A 8

B 3

C 5

D 13

💡 Hint

Check the 'New b' column for i=5 in the execution_table.

At which iteration does the loop calculate the number of ways to climb 7 steps?

A i=6

B i=7

C i=5

D i=8

💡 Hint

Look at the 'Step' and 'i' columns in the execution_table to find when i=7.

If we change the initial value of 'a' to 0, how would the total ways for step 3 change?

A It would become 3

B It would become 1

C It would become 2

D It would become 0

💡 Hint

Refer to the calculation in execution_table row 3 where new b = a + b.

Concept Snapshot

Climbing Stairs Problem:
- Ways to climb n stairs = ways(n-1) + ways(n-2)
- Base cases: ways(1)=1, ways(2)=2
- Use two variables to track previous two steps
- Loop from 3 to n, update ways
- Return ways for step n

Full Transcript

The Climbing Stairs Problem counts how many ways to reach the top by climbing 1 or 2 steps at a time. We start with base cases: 1 way to climb 1 step, 2 ways to climb 2 steps. Then, for each step from 3 to n, we calculate the total ways as the sum of ways to the previous two steps. Variables 'a' and 'b' track these counts. The loop updates these values until reaching step n, then returns the total ways stored in 'b'.

Practice

1. What does the Climbing Stairs problem ask you to find?
Given n steps, how many distinct ways can you climb to the top if you can take 1 or 2 steps at a time?

easy

A. The total number of distinct ways to reach the top

B. The minimum number of steps to reach the top

C. The maximum number of steps you can take at once

D. The number of steps you skip

Solution

1. Step 1: Understand the problem statement

   The problem asks for the count of different ways to climb to the top by taking 1 or 2 steps at a time.

2. Step 2: Identify what is being counted

   We count all possible sequences of 1-step and 2-step moves that sum to n steps.

3. Final Answer:

   The total number of distinct ways to reach the top -> Option A

4. Quick Check:

   Counting sequences of 1 or 2 steps = total ways [OK]

Hint: Count sequences of 1 or 2 steps summing to n [OK]

Common Mistakes:

• Confusing number of ways with minimum steps
• Thinking about maximum steps allowed
• Counting skipped steps instead of ways

2. Which Go function signature correctly represents a function to solve the Climbing Stairs problem for n steps?

easy

A. func climbStairs(n int) bool

B. func climbStairs(n string) string

C. func climbStairs() int

D. func climbStairs(n int) int

Solution

1. Step 1: Identify input and output types

   The input is the number of steps (integer), output is the count of ways (integer).

2. Step 2: Match function signature

   Only func climbStairs(n int) int matches input int and output int.

3. Final Answer:

   func climbStairs(n int) int -> Option D

4. Quick Check:

   Input int, output int = correct signature [OK]

Hint: Input and output both must be int for steps and ways [OK]

Common Mistakes:

• Using string instead of int for steps
• Missing input parameter
• Returning bool instead of int

3. What is the output of this Go code for n = 4?

func climbStairs(n int) int {
    if n <= 2 {
        return n
    }
    a, b := 1, 2
    for i := 3; i <= n; i++ {
        a, b = b, a+b
    }
    return b
}

fmt.Println(climbStairs(4))

medium

A. 7

B. 6

C. 5

D. 4

Solution

1. Step 1: Trace initial values and loop

   Start with a=1, b=2. Loop i=3 to 4:
   i=3: a=2, b=3
   i=4: a=3, b=5

2. Step 2: Return final value

   Return b=5 after loop ends.

3. Final Answer:

   5 -> Option C

4. Quick Check:

   Ways for 4 steps = 5 [OK]

Hint: Use Fibonacci-like addition for steps count [OK]

Common Mistakes:

• Returning a instead of b
• Off-by-one error in loop range
• Incorrect initial values

4. Find the error in this Go code for climbing stairs:

func climbStairs(n int) int {
    if n == 1 {
        return 1
    }
    a, b := 1, 2
    for i := 2; i <= n; i++ {
        a, b = b, a+b
    }
    return b
}

medium

A. Return statement should return a instead of b

B. Loop starts at wrong index causing extra iteration

C. Initial values of a and b are incorrect

D. Missing base case for n == 2

Solution

1. Step 1: Analyze loop start and iterations

   Loop starts at i=2 but initial values already cover steps 1 and 2, causing one extra iteration.

2. Step 2: Correct loop start

   Loop should start at i=3 to avoid counting extra ways.

3. Final Answer:

   Loop starts at wrong index causing extra iteration -> Option B

4. Quick Check:

   Loop index off by one causes wrong count [OK]

Hint: Start loop at 3 when initial values cover steps 1 and 2 [OK]

Common Mistakes:

• Starting loop at 2 instead of 3
• Changing initial values incorrectly
• Returning wrong variable

5. You want to find the number of ways to climb 10 steps but can take 1, 2, or 3 steps at a time. How would you modify the climbing stairs approach?

hard

A. Use dynamic programming with dp[i] = dp[i-1] + dp[i-2] + dp[i-3]

B. Use the same formula dp[i] = dp[i-1] + dp[i-2]

C. Only count ways taking 3 steps at a time

D. Multiply the number of ways for 1 and 2 steps

Solution

1. Step 1: Understand the new step options

   Now you can take 1, 2, or 3 steps at a time, so ways to reach step i depend on previous 3 steps.

2. Step 2: Formulate recurrence relation

   Number of ways dp[i] = dp[i-1] + dp[i-2] + dp[i-3] to include all possibilities.

3. Final Answer:

   Use dynamic programming with dp[i] = dp[i-1] + dp[i-2] + dp[i-3] -> Option A

4. Quick Check:

   Include all step sizes in sum for dp [OK]

Hint: Sum ways from last 3 steps for 1,2,3 step moves [OK]

Common Mistakes:

• Using old formula for only 1 or 2 steps
• Ignoring 3-step moves
• Multiplying counts instead of adding