Decode Ways
A medium Dynamic Programming interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Dynamic Programming
- Sheets
- 2
- Core for
- 9 roles
The problem
A message containing letters from A-Z can be encoded into digits using the mapping A=1, B=2, ..., Z=26. Given a string of digits, return the total number of ways to decode it.
Example 1
- Input
- s = "12"
- Output
- 2
- Why
- It could be decoded as "AB" (1,2) or "L" (12).
Example 2
- Input
- s = "226"
- Output
- 3
- Why
- It could be decoded as "BZ" (2,26), "VF" (22,6), or "BBF" (2,2,6).
Example 3
- Input
- s = "06"
- Output
- 0
- Why
- "06" cannot be mapped to "F" because there is no leading zero mapping; 06 is not valid.
Constraints
- 1 <= s.length <= 100
- s contains only digits and may contain leading zeroes
How to think about it
Updated 2026-09-29At any point in the string, a valid character comes from taking either the single current digit (if it is not '0') or the pair formed with the preceding digit (if between 10 and 26). This is Fibonacci in disguise, where steps are conditioned on digit validity.
Approaches, worst first
Backtracking exploration
time O(2^n) · space O(n)
From each index, branch into reading one digit or two digits if valid. Without caching, strings of repeating digits like '111111' branch exponentially.
Dynamic programming array
time O(n) · space O(n)
Let dp[i] be the number of ways to decode prefix s[0..i-1]. Set dp[0] = 1. If s[i-1] != '0', add dp[i-1]. If s[i-2..i-1] forms a number from 10 to 26, add dp[i-2].
Rolling variablesWrite this one
time O(n) · space O(1)
Since dp[i] only looks back two indices, compress the state into two integers representing ways(i-1) and ways(i-2). Update them in one forward pass.
Where people lose marks · 3
- A leading '0' at index 0 makes the entire string undecodable; it should immediately return 0.
- Inner zeros like '30' or '70' cannot stand alone and cannot form valid two-digit characters, causing the ways to drop to 0.
- Treating '05' as 5. Leading zeros are disallowed in multi-digit mappings; only '10' through '26' are valid two-digit combinations.
The theory behind it
Dynamic Programming — the ground this problem stands on. All Dynamic Programming problems
What Dynamic Programming is
Dynamic programming is a method for solving a complex problem by breaking it into overlapping subproblems, solving each subproblem only once, and remembering the answers in a lookup table. Instead of recalculating identical questions over and over, future steps look up previous answers directly. By assembling these saved pieces from the bottom up or storing them during recursion, a task that would take billions of steps finishes in a fraction of a second.
When to reach for it
Reach for dynamic programming when questions ask for the maximum profit, minimum cost, total number of distinct ways to achieve a goal, or whether a target can be formed. Signals include overlapping choices where making a choice now affects what choices remain later, but greedy picking fails to find the true global optimum. If drawing a recursive decision tree reveals the same subproblem states repeating across branches, dynamic programming is needed.
How the pattern works
Identify the state variables that uniquely describe a subproblem, such as an array index and remaining capacity. Write the base cases first, representing states whose answers are known without calculation. Next, write the recurrence relation that expresses the current state using previously solved states, taking the minimum, maximum, or sum among your options. Build the solution either top-down by caching recursive returns in a memo table, or bottom-up by filling an array in topological dependency order. When each state depends only on the previous row, compress storage down to a single array.
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