@ -27,38 +27,54 @@ For each topic, study links are recommended to help you master the topic. There
## General Tips
## General Tips
- Input validation:
Always validate input first. Check for invalid/empty/negative/different type input. Never assume you are given the valid parameters. Alternatively, clarify with the interviewer whether you can assume valid input (usually yes), which can save you time from writing code that does input validation.
- Always validate input first. Check for invalid/empty/negative/different type input. Never assume you are given the valid parameters. Alternatively, clarify with the interviewer whether you can assume valid input (usually yes), which can save you time from writing code that does input validation.
- Are there any time/space complexity requirements/constraints?
Are there any time/space complexity requirements/constraints?
- Check corner cases:
- Check for off-by-one errors.
Check for off-by-one errors.
- In languages where there are no automatic type coercion, check that concatenation of values are of the same type: `int`/`str`/`list`.
- After finishing your code, use a few example inputs to test your solution.
In languages where there are no automatic type coercion, check that concatenation of values are of the same type: `int`/`str`/`list`.
- Is the algorithm meant to be run multiple times, for example in a web server? If yes, the input is likely to be preprocess-able to improve the efficiency in each call.
- Use a mix of functional and imperative programming paradigms:
After finishing your code, use a few example inputs to test your solution.
- Write pure functions as much as possible.
- Pure functions are easier to reason about and can help to reduce bugs in your implementation.
Is the algorithm meant to be run multiple times, for example in a web server? If yes, the input is likely to be preprocess-able to improve the efficiency in each call.
- Avoid mutating the parameters passed into your function especially if they are passed by reference unless you are sure of what you are doing.
- However, functional programming is usually expensive in terms of space complexity because of non-mutation and the repeated allocation of new objects. On the other hand, imperative code is faster because you operate on existing objects. Hence you will need to achieve a balance between accuracy vs efficiency, by using the right amount of functional and imperative code where appropriate.
Use a mix of functional and imperative programming paradigms:
- Avoid relying on and mutating global variables. Global variables introduce state.
- If you have to rely on global variables, make sure that you do not mutate it by accident.
- Write pure functions as much as possible.
- Generally, to improve the speed of a program, we can either choose a more appropriate data structure/algorithm or use more memory. It's a classic space/time tradeoff.
- Pure functions are easier to reason about and can help to reduce bugs in your implementation.
- Data structures are your weapons. Choosing the right weapon for the right battle is the key to victory. Be very familiar about the strengths of each data structure and the time complexities for its various operations.
- Avoid mutating the parameters passed into your function especially if they are passed by reference unless you are sure of what you are doing.
- Data structures can be augmented to achieve efficient time complexities across different operations. For example, a hash map can be used together with a doubly-linked list to achieve O(1) time complexity for both the `get` and `put` operation in an [LRU cache](https://leetcode.com/problems/lru-cache/).
- However, functional programming is usually expensive in terms of space complexity because of non-mutation and the repeated allocation of new objects. On the other hand, imperative code is faster because you operate on existing objects. Hence you will need to achieve a balance between accuracy vs efficiency, by using the right amount of functional and imperative code where appropriate.
- Hashmaps are probably the most commonly used data structure for algorithm questions. If you are stuck on a question, your last resort can be to enumerate through the possible data structures (thankfully there aren't that many of them) and consider whether each of them can be applied to the problem. This has worked for me sometimes.
- Avoid relying on and mutating global variables. Global variables introduce state.
- If you are cutting corners in your code, state that out loud to your interviewer and say what you would do in a non-interview setting (no time constraints). E.g., I would write a regex to parse this string rather than using `split()` which does not cover all cases.
- If you have to rely on global variables, make sure that you do not mutate it by accident.
Generally, to improve the speed of a program, we can either choose a more appropriate data structure/algorithm or use more memory. It's a classic space/time tradeoff.
Data structures are your weapons. Choosing the right weapon for the right battle is the key to victory. Be very familiar about the strengths of each data structure and the time complexities for its various operations.
Data structures can be augmented to achieve efficient time complexities across different operations. For example, a hash map can be used together with a doubly-linked list to achieve O(1) time complexity for both the `get` and `put` operation in an [LRU cache](https://leetcode.com/problems/lru-cache/).
Hashmaps are probably the most commonly used data structure for algorithm questions. If you are stuck on a question, your last resort can be to enumerate through the possible data structures (thankfully there aren't that many of them) and consider whether each of them can be applied to the problem. This has worked for me sometimes.
If you are cutting corners in your code, state that out loud to your interviewer and say what you would do in a non-interview setting (no time constraints). E.g., I would write a regex to parse this string rather than using `split()` which does not cover all cases.
## Sequence
## Sequence
#### Notes
#### Notes
- Arrays and strings are considered sequences (a string is a sequence of characters). There are tips relevant for dealing with both arrays and strings which will be covered here.
Arrays and strings are considered sequences (a string is a sequence of characters). There are tips relevant for dealing with both arrays and strings which will be covered here.
- Are there duplicate values in the sequence, would it affect the answer?
- Check for sequence out of bounds.
Are there duplicate values in the sequence, would it affect the answer?
- Be mindful about slicing or concatenating sequences in your code. Typically, slicing and concatenating sequences require O(n) time. Use start and end indices to demarcate a subarray/substring where possible.
- Sometimes you can traverse the sequence from the right rather than from the left.
Check for sequence out of bounds.
- Master the [sliding window technique](https://discuss.leetcode.com/topic/30941/here-is-a-10-line-template-that-can-solve-most-substring-problems) that applies to many substring/subarray problems.
- When you are given two sequences to process, it is common to have one index per sequence to traverse/compare the both of them. For example, we use the same approach to merge two sorted arrays.
Be mindful about slicing or concatenating sequences in your code. Typically, slicing and concatenating sequences require O(n) time. Use start and end indices to demarcate a subarray/substring where possible.
Sometimes you can traverse the sequence from the right rather than from the left.
Master the [sliding window technique](https://discuss.leetcode.com/topic/30941/here-is-a-10-line-template-that-can-solve-most-substring-problems) that applies to many substring/subarray problems.
When you are given two sequences to process, it is common to have one index per sequence to traverse/compare the both of them. For example, we use the same approach to merge two sorted arrays.
#### Corner Cases
#### Corner Cases
@ -70,10 +86,13 @@ For each topic, study links are recommended to help you master the topic. There
#### Notes
#### Notes
- Is the array sorted or partially sorted? If it is, some form of binary search should be possible. This also usually means that the interviewer is looking for a solution that is faster than O(n).
Is the array sorted or partially sorted? If it is, some form of binary search should be possible. This also usually means that the interviewer is looking for a solution that is faster than O(n).
- Can you sort the array? Sometimes sorting the array first may significantly simplify the problem. Make sure that the order of array elements do not need to be preserved before attempting a sort.
- For questions where summation or multiplication of a subarray is involved, pre-computation using hashing or a prefix/suffix sum/product might be useful.
Can you sort the array? Sometimes sorting the array first may significantly simplify the problem. Make sure that the order of array elements do not need to be preserved before attempting a sort.
- If you are given a sequence and the interviewer asks for O(1) space, it might be possible to use the array itself as a hash table. For example, if the array only has values from 1 to N, where N is the length of the array, negate the value at that index (minus one) to indicate presence of that number.
For questions where summation or multiplication of a subarray is involved, pre-computation using hashing or a prefix/suffix sum/product might be useful.
If you are given a sequence and the interviewer asks for O(1) space, it might be possible to use the array itself as a hash table. For example, if the array only has values from 1 to N, where N is the length of the array, negate the value at that index (minus one) to indicate presence of that number.
#### Practice Questions
#### Practice Questions
@ -96,7 +115,10 @@ For each topic, study links are recommended to help you master the topic. There
#### Notes
#### Notes
- Questions involving binary representations and bitwise operations are asked sometimes and you must be absolutely familiar with how to convert a number from decimal form into binary form (and vice versa) in your programming language of choice.
Questions involving binary representations and bitwise operations are asked sometimes and you must be absolutely familiar with how to convert a number from decimal form into binary form (and vice versa) in your chosen programming language.
Some helpful utility snippets:
- Test k<sup>th</sup> bit is set: `num & (1 << k) != 0`.
- Test k<sup>th</sup> bit is set: `num & (1 << k) != 0`.
- Set k<sup>th</sup> bit: `num |= (1 << k)`.
- Set k<sup>th</sup> bit: `num |= (1 << k)`.
- Turn off k<sup>th</sup> bit: `num &= ~(1 << k)`.
- Turn off k<sup>th</sup> bit: `num &= ~(1 << k)`.
@ -124,9 +146,9 @@ For each topic, study links are recommended to help you master the topic. There
#### Notes
#### Notes
- Usually used to solve optimization problems.
Dynamic Programming (DP) is usually used to solve optimization problems. The only way to get better at DP is to practice. It takes some amount of practice to be able to recognize that a problem can be solved by DP.
- The only way to get better at DP is to practice. It takes some amount of practice to be able to recognize that a problem can be solved by DP.
- Sometimes you do not need to store the whole DP table in memory, the last two values or the last two rows of the matrix will suffice.
Sometimes you do not need to store the whole DP table in memory, the last two values or the last two rows of the matrix will suffice.
#### Practice Questions
#### Practice Questions
@ -146,8 +168,9 @@ For each topic, study links are recommended to help you master the topic. There
#### Notes
#### Notes
- When comparing euclidean distance between two pairs of points, using dx<sup>2</sup> + dy<sup>2</sup> is sufficient. It is unnecessary to square root the value.
When comparing euclidean distance between two pairs of points, using dx<sup>2</sup> + dy<sup>2</sup> is sufficient. It is unnecessary to square root the value.
- To find out if two circles overlap, check that the distance between the two centers of the circles is less than the sum of their radii.
To find out if two circles overlap, check that the distance between the two centers of the circles is less than the sum of their radii.
## Graph
## Graph
@ -159,17 +182,22 @@ For each topic, study links are recommended to help you master the topic. There
#### Notes
#### Notes
- Be familiar with the various graph representations, graph search algorithms and their time and space complexities.
Be familiar with the various graph representations, graph search algorithms and their time and space complexities.
- You can be given a list of edges and tasked to build your own graph from the edges to perform a traversal on. The common graph representations are:
You can be given a list of edges and tasked to build your own graph from the edges to perform a traversal on. The common graph representations are:
- Adjacency matrix.
- Adjacency matrix.
- Adjacency list.
- Adjacency list.
- Hashmap of hashmaps.
- Hashmap of hashmaps.
- Some questions look like they are trees but they are actually graphs. In that case you will have to handle cycles and keep a set of visited nodes when traversing.
- Graph search algorithms:
Some inputs look like they are trees but they are actually graphs. In that case you will have to handle cycles and keep a set of visited nodes when traversing.
- In coding interviews, graphs are commonly represented as 2-D matrices where cells are the nodes and each cell can traverse to its adjacent cells (up/down/left/right). Hence it is important that you be familiar with traversing a 2-D matrix. When recursively traversing the matrix, always ensure that your next position is within the boundary of the matrix. More tips for doing depth-first searches on a matrix can be found [here](https://discuss.leetcode.com/topic/66065/python-dfs-bests-85-tips-for-all-dfs-in-matrix-question/). A simple template for doing depth-first searches on a matrix goes like this:
In coding interviews, graphs are commonly represented as 2-D matrices where cells are the nodes and each cell can traverse to its adjacent cells (up/down/left/right). Hence it is important that you be familiar with traversing a 2-D matrix. When recursively traversing the matrix, always ensure that your next position is within the boundary of the matrix. More tips for doing depth-first searches on a matrix can be found [here](https://discuss.leetcode.com/topic/66065/python-dfs-bests-85-tips-for-all-dfs-in-matrix-question/). A simple template for doing depth-first searches on a matrix goes like this:
```py
```py
def traverse(matrix):
def traverse(matrix):
@ -214,12 +242,17 @@ def traverse(matrix):
#### Notes
#### Notes
- Interval questions are questions where you are given an array of two-element arrays (an interval) and the two values represent a start and an end value. Interval questions are considered part of the array family but they involve some common techniques hence they are extracted out to this special section of their own.
Interval questions are questions where you are given an array of two-element arrays (an interval) and the two values represent a start and an end value. Interval questions are considered part of the array family but they involve some common techniques hence they are extracted out to this special section of their own.
- An example interval array: `[[1, 2], [4, 7]]`.
- Interval questions can be tricky to those who have not tried them before because of the sheer number of cases to consider when they overlap.
An example interval array: `[[1, 2], [4, 7]]`.
- Do clarify with the interviewer whether `[1, 2]` and `[2, 3]` are considered overlapping intervals as it affects how you will write your equality checks.
- A common routine for interval questions is to sort the array of intervals by each interval's starting value.
Interval questions can be tricky to those who have not tried them before because of the sheer number of cases to consider when they overlap.
- Be familiar with writing code to check if two intervals overlap and merging two overlapping intervals:
Do clarify with the interviewer whether `[1, 2]` and `[2, 3]` are considered overlapping intervals as it affects how you will write your equality checks.
A common routine for interval questions is to sort the array of intervals by each interval's starting value.
Be familiar with writing code to check if two intervals overlap and merging two overlapping intervals:
- Like arrays, linked lists are used to represent sequential data. The benefit of linked lists is that insertion and deletion from anywhere in the list is O(1) whereas in arrays the following elements will have to be shifted.
Like arrays, linked lists are used to represent sequential data. The benefit of linked lists is that insertion and deletion from anywhere in the list is O(1) whereas in arrays the following elements will have to be shifted.
- Adding a dummy node at the head and/or tail might help to handle many edge cases where operations have to be performed at the head or the tail. The presence of dummy nodes essentially ensures that operations will never have be done on the head or the tail, thereby removing a lot of headache in writing conditional checks to dealing with null pointers. Be sure to remember to remove them at the end of the operation.
- Sometimes linked lists problem can be solved without additional storage. Try to borrow ideas from reverse a linked list problem.
Adding a dummy node at the head and/or tail might help to handle many edge cases where operations have to be performed at the head or the tail. The presence of dummy nodes essentially ensures that operations will never have be done on the head or the tail, thereby removing a lot of headache in writing conditional checks to dealing with null pointers. Be sure to remember to remove them at the end of the operation.
- For deletion in linked lists, you can either modify the node values or change the node pointers. You might need to keep a reference to the previous element.
- For partitioning linked lists, create two separate linked lists and join them back together.
Sometimes linked lists problem can be solved without additional storage. Try to borrow ideas from reverse a linked list problem.
- Linked lists problems share similarity with array problems, think about how you would do it for an array and try to apply it to a linked list.
- Two pointer approaches are also common for linked lists. For example:
For deletion in linked lists, you can either modify the node values or change the node pointers. You might need to keep a reference to the previous element.
For partitioning linked lists, create two separate linked lists and join them back together.
Linked lists problems share similarity with array problems, think about how you would do it for an array and try to apply it to a linked list.
Two pointer approaches are also common for linked lists. For example:
- Getting the k<sup>th</sup> from last node - Have two pointers, where one is k nodes ahead of the other. When the node ahead reaches the end, the other node is k nodes behind.
- Getting the k<sup>th</sup> from last node - Have two pointers, where one is k nodes ahead of the other. When the node ahead reaches the end, the other node is k nodes behind.
- Detecting cycles - Have two pointers, where one pointer increments twice as much as the other, if the two pointers meet, means that there is a cycle.
- Detecting cycles - Have two pointers, where one pointer increments twice as much as the other, if the two pointers meet, means that there is a cycle.
- Getting the middle node - Have two pointers, where one pointer increments twice as much as the other. When the faster node reaches the end of the list, the slower node will be at the middle.
- Getting the middle node - Have two pointers, where one pointer increments twice as much as the other. When the faster node reaches the end of the list, the slower node will be at the middle.
- Be familiar with the following routines because many linked list questions make use of one or more of these routines in the solution:
Be familiar with the following routines because many linked list questions make use of one or more of these routines in the solution:
- Counting the number of nodes in the linked list.
- Counting the number of nodes in the linked list.
- Reversing a linked list in-place.
- Reversing a linked list in-place.
- Finding the middle node of the linked list using fast/slow pointers.
- Finding the middle node of the linked list using fast/slow pointers.
- If code involves division or modulo, remember to check for division or modulo by 0 case.
#### Notes
- When a question involves "a multiple of a number", perhaps modulo might be useful.
- Check for and handle overflow/underflow if you are using a typed language like Java and C++. At the very least, mention that overflow/underflow is possible and ask whether you need to handle it.
If code involves division or modulo, remember to check for division or modulo by 0 case.
- Consider negative numbers and floating point numbers. This may sound obvious, but under interview pressure, many obvious cases go unnoticed.
- If the question asks to implement an operator such as power, squareroot or division and want it to be faster than O(n), binary search is usually the approach to go.
When a question involves "a multiple of a number", perhaps modulo might be useful.
- Some common formulas:
- Sum of 1 to N = (n+1) * n/2
Check for and handle overflow/underflow if you are using a typed language like Java and C++. At the very least, mention that overflow/underflow is possible and ask whether you need to handle it.
- Sum of GP = 2<sup>0</sup> + 2<sup>1</sup> + 2<sup>2</sup> + 2<sup>3</sup> + ... 2<sup>n</sup> = 2<sup>n+1</sup> - 1
- Permutations of N = N! / (N-K)!
Consider negative numbers and floating point numbers. This may sound obvious, but under interview pressure, many obvious cases go unnoticed.
- Combinations of N = N! / (K! * (N-K)!)
If the question asks to implement an operator such as power, squareroot or division and want it to be faster than O(n), binary search is usually the approach to go.
#### Some common formulas:
- Sum of 1 to N = (n+1) * n/2
- Sum of GP = 2<sup>0</sup> + 2<sup>1</sup> + 2<sup>2</sup> + 2<sup>3</sup> + ... 2<sup>n</sup> = 2<sup>n+1</sup> - 1
- A matrix is a 2-dimensional array. Questions involving matrices are usually related to dynamic programming or graph traversal.
#### Notes
- For questions involving traversal or dynamic programming, you almost always want to make a copy of the matrix with the same dimensions that is initialized to empty values to store the visited state or dynamic programming table. Be familiar with such a routine:
A matrix is a 2-dimensional array. Questions involving matrices are usually related to dynamic programming or graph traversal.
For questions involving traversal or dynamic programming, you almost always want to make a copy of the matrix with the same dimensions that is initialized to empty values to store the visited state or dynamic programming table. Be familiar with such a routine:
```py
```py
rows, cols = len(matrix), len(matrix[0])
rows, cols = len(matrix), len(matrix[0])
copy = [[0 for _ in range(cols)] for _ in range(rows)]
copy = [[0 for _ in range(cols)] for _ in range(rows)]
```
```
- Many grid-based games can be modeled as a matrix, such as Tic-Tac-Toe, Sudoku, Crossword, Connect 4, Battleship, etc. It is not uncommon to be asked to verify the winning condition of the game. For games like Tic-Tac-Toe, Connect 4 and Crosswords, where verification has to be done vertically and horizontally, one trick is to write code to verify the matrix for the horizontal cells, transpose the matrix and reuse the logic for horizontal verification to verify originally vertical cells (which are now horizontal).
Many grid-based games can be modeled as a matrix, such as Tic-Tac-Toe, Sudoku, Crossword, Connect 4, Battleship, etc. It is not uncommon to be asked to verify the winning condition of the game. For games like Tic-Tac-Toe, Connect 4 and Crosswords, where verification has to be done vertically and horizontally, one trick is to write code to verify the matrix for the horizontal cells, transpose the matrix and reuse the logic for horizontal verification to verify originally vertical cells (which are now horizontal).
- Remember to always define a base case so that your recursion will end.
#### Notes
- Useful for permutation, generating all combinations and tree-based questions.
- Be familiar with how to generate all permutations of a sequence as well as how to handle duplicates.
Recursion is useful for permutation, because it generates all combinations and tree-based questions. You should know how to generate all permutations of a sequence as well as how to handle duplicates.
- Recursion implicitly uses a stack. Hence all recursive approaches can be rewritten iteratively using a stack.
- Beware of cases where the recursion level goes too deep and causes a stack overflow (the default limit in Python is 1000).
Remember to always define a base case so that your recursion will end.
- Recursion will never be O(1) space complexity because a stack is involved unless there is [tail-call optimization](https://stackoverflow.com/questions/310974/what-is-tail-call-optimization). Do find out if your chosen language supports tail-call optimization.
Recursion implicitly uses a stack. Hence all recursive approaches can be rewritten iteratively using a stack. Beware of cases where the recursion level goes too deep and causes a stack overflow (the default limit in Python is 1000). You may get bonus points for pointing this out to the interviewer. Recursion will never be O(1) space complexity because a stack is involved, unless there is [tail-call optimization](https://stackoverflow.com/questions/310974/what-is-tail-call-optimization) (TCO). Find out if your chosen language supports TCO.
- Please read the above tips on [Sequence](#sequence) because they apply to strings too.
#### Notes
- Ask about input character set and case sensitivity. Usually the characters are limited to lower case Latin characters, i.e. a-z.
- When you need to compare strings where the order isn't important (like anagram), you may consider using a hash map as a counter. If your language has a built-in `Counter` class like Python, ask to use that instead.
Please read the above tips on sequence. They apply to strings too.
- If you need to keep a counter of characters, a common mistake to make is to say that the space complexity required for the counter is O(n). The space required for a counter is O(1) not O(n), because the upper bound is the range of characters which is usually a fixed constant of 26 when the input set is just lower case Latin characters.
- Common data structures for looking up strings efficiently:
Ask about input character set and case sensitivity. Usually the characters are limited to lowercase Latin characters, for example a to z.
When you need to compare strings where the order isn’t important (like anagram), you may consider using a HashMap as a counter. If your language has a built-in Counter class like Python, ask to use that instead.
- Common string algorithms:
- [Rabin Karp](https://en.wikipedia.org/wiki/Rabin%E2%80%93Karp_algorithm) for efficient searching of substring using a rolling hash.
If you need to keep a counter of characters, a common mistake is to say that the space complexity required for the counter is O(n). The space required for a counter is O(1) not O(n). This is because the upper bound is the range of characters, which is usually a fixed constant of 26. The input set is just lowercase Latin characters.
- [KMP](https://en.wikipedia.org/wiki/Knuth%E2%80%93Morris%E2%80%93Pratt_algorithm) for efficient searching of substring.
Common data structures for looking up strings efficiently are
- [Rabin Karp](https://en.wikipedia.org/wiki/Rabin%E2%80%93Karp_algorithm) for efficient searching of substring using a rolling hash.
- [KMP](https://en.wikipedia.org/wiki/Knuth%E2%80%93Morris%E2%80%93Pratt_algorithm) for efficient searching of substring.
#### Corner Cases
#### Corner Cases
@ -368,26 +430,33 @@ for c in set(word):
mask |= (1 << (ord(c) - ord('a')))
mask |= (1 << (ord(c) - ord('a')))
```
```
- To determine if two strings have common characters, perform `&` on the two bitmasks and if the result is non-zero, the two strings do have common characters: `mask_a & mask_b > 0`.
To determine if two strings have common characters, perform & on the two bitmasks. If the result is non-zero, `mask_a & mask_b > 0`, then the two strings have common characters.
### Anagram
### Anagram
- An anagram is direct word switch or word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once. In interviews, usually we are only bothered with words without spaces in them.
An anagram is word switch or word play. It is the result of re-arranging the letters of a word or phrase to produce a new word or phrase, while using all the original letters only once. In interviews, usually we are only bothered with words without spaces in them.
- To determine if two strings are anagrams, there are a few plausible approaches.
- Sorting both strings should produce the same resulting string. This takes O(nlgn) time and O(lgn) space.
To determine if two strings are anagrams, there are a few plausible approaches:
- If we map each character to a prime number and we multiply each mapped number together, anagrams should have the same multiple (prime factor decomposition). This takes O(n) time and O(1) space.
- Frequency counting of characters will help to determine if two strings are anagrams. This also takes O(n) time and O(1) space.
- Sorting both strings should produce the same resulting string. This takes O(nlgn) time and O(lgn) space.
- If we map each character to a prime number and we multiply each mapped number together, anagrams should have the same multiple (prime factor decomposition). This takes O(n) time and O(1) space.
- Frequency counting of characters will help to determine if two strings are anagrams. This also takes O(n) time and O(1) space.
### Palindrome
### Palindrome
- A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward, such as madam or racecar.
A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward, such as *madam* or *racecar*.
- Ways to determine if a string is a palindrome:
- Reverse the string and it should be equal to itself.
Here are ways to determine if a string is a palindrome:
- Have two pointers at the start and end of the string, move inwards till they meet. At any point in time the characters at both pointers should match.
- The order of characters within the string matters, so hash maps are usually not helpful.
- Reverse the string and it should be equal to itself.
- When a question is about counting the number of palindromes, a common trick is to have two pointer that move outwards, away from the middle. Note that palindromes can be even/odd length, and that for each middle pivot position, you would need to check twice, once including the character, and once without.
- Have two pointers at the start and end of the string. Move the pointers inward till they meet. At any point in time, the characters at both pointers should match.
- For substrings, you can terminate early once there is no match.
- For subsequences, use dynamic programming as there are overlapping subproblems. Check out [this question](https://leetcode.com/problems/longest-palindromic-subsequence/).
The order of characters within the string matters, so HashMaps are usually not helpful.
When a question is about counting the number of palindromes, a common trick is to have two pointers that move outward, away from the middle. Note that palindromes can be even or odd length. For each middle pivot position, you need to check it twice: Once that includes the character and once without the character.
- For substrings, you can terminate early once there is no match.
- For subsequences, use dynamic programming as there are overlapping subproblems. Check out [this question](https://leetcode.com/problems/longest-palindromic-subsequence/).
#### Practice Questions
#### Practice Questions
@ -410,13 +479,20 @@ for c in set(word):
#### Notes
#### Notes
- A tree is an undirected, connected, acyclic graph.
A tree is an undirected and connected acyclic graph.
- Recursion is a common approach for trees. When you notice the subtree problem can be used to solve the whole problem, you should try recursion.
- When using recursion, always remember to check for the base case, usually where the node is `null`.
Recursion is a common approach for trees. When you notice that the subtree problem can be used to solve the entire problem, try using recursion.
- When you are asked to traverse a tree by level, use depth-first search.
- Sometimes it is possible that your recursive function needs to return two values.
When using recursion, always remember to check for the base case, usually where the node is `null`.
- If the question involves summation of nodes along the way, be sure to check whether nodes can be negative.
- You should be very familiar with writing pre-order, in-order and post-order traversal recursively. As an extension, challenge yourself by writing them iteratively. Sometimes interviewers do ask candidates for the iterative approach, especially if the candidate finishes writing the recursive approach too fast.
When you are asked to traverse a tree by level, use depth first search.
Sometimes it is possible that your recursive function needs to return two values.
If the question involves summation of nodes along the way, be sure to check whether nodes can be negative.
You should be very familiar with writing pre-order, in-order, and post-order traversal recursively. As an extension, challenge yourself by writing them iteratively. Sometimes interviewers ask candidates for the iterative approach, especially if the candidate finishes writing the recursive approach too quickly.
#### Corner Cases
#### Corner Cases
@ -427,13 +503,15 @@ for c in set(word):
### Binary Tree
### Binary Tree
- In-order traversal of a binary tree is insufficient to uniquely serialize a tree. Pre-order or post-order traversal is also required.
In-order traversal of a binary tree is insufficient to uniquely serialize a tree. Pre-order or post-order traversal is also required.
### Binary Search Tree
### Binary Search Tree (BST)
- In-order traversal of a BST will give you all elements in order.
In-order traversal of a BST will give you all elements in order.
- Be very familiar with the properties of a BST and validating that a binary tree is a BST. This comes up more often than expected.
- When a question involves a BST, the interviewer is usually looking for a solution which runs faster than O(n).
Be very familiar with the properties of a BST and validating that a binary tree is a BST. This comes up more often than expected.
When a question involves a BST, the interviewer is usually looking for a solution which runs faster than O(n).
#### Practice Questions
#### Practice Questions
@ -454,13 +532,15 @@ for c in set(word):
#### Study Links
#### Study Links
- [Trying to Understand Tries](https://medium.com/basecs/trying-to-understand-tries-3ec6bede0014)
- [Trying to Understand Tries](https://medium.com/basecs/trying-to-understand-tries-3ec6bede0014)
- [Implement Trie (Prefix Tree)]
#### Notes
#### Notes
- Tries are special trees (prefix trees) that make searching and storing strings more efficient. Tries have many practical applications such as for searching and autocomplete. It will be helpful to know these common applications so that you can easily identify when a problem can be solved efficiently using a trie.
Tries are special trees (prefix trees) that make searching and storing strings more efficient. Tries have many practical applications, such as conducting searches and providing autocomplete. It is helpful to know these common applications so that you can easily identify when a problem can be efficiently solved using a trie.
- Sometimes preprocessing a dictionary of words (given in a list) into a trie will improve the efficiency when searching for a word of length k among n words. Searching becomes O(k) instead of O(n).
- LeetCode has written a [very comprehensive article](https://leetcode.com/articles/implement-trie-prefix-tree/) on tries which you are highly encouraged to read.
Sometimes preprocessing a dictionary of words (given in a list) into a trie, will improve the efficiency of searching for a word of length k, among n words. Searching becomes O(k) instead of O(n).
- Be familiar with implementing a `Trie` class and its `add`, `remove` and `search` methods from scratch.
Be familiar with implementing, from scratch, a `Trie` class and its `add`, `remove` and `search` methods.
#### Practice Questions
#### Practice Questions
@ -476,8 +556,9 @@ for c in set(word):
#### Notes
#### Notes
- If you see a top/lowest K being mentioned in the question, it is usually a signal that a heap can be used to solve the problem.
If you see a top or lowest *k* being mentioned in the question, it is usually a signal that a heap can be used to solve the problem, such as in [Top K Frequent Elements](https://leetcode.com/problems/top-k-frequent-elements/).
- If you require the top K elements use a Min Heap of size K. Iterate through each element, pushing it into the heap. Whenever the heap size exceeds K, remove the minimum element, that will guarantee that you have the K largest elements.
If you require the top *k* elements use a Min Heap of size *k*. Iterate through each element, pushing it into the heap. Whenever the heap size exceeds *k*, remove the minimum element, that will guarantee that you have the *k* largest elements.
@ -19,6 +19,8 @@ If you have been out of college for a while, it will be highly advisable to revi
This [interviews repository](https://github.com/kdn251/interviews) by Kevin Naughton Jr. served as a quick refresher for me.
This [interviews repository](https://github.com/kdn251/interviews) by Kevin Naughton Jr. served as a quick refresher for me.
The Medium publication [basecs](https://medium.com/basecs) by [Vaidehi Joshi](https://medium.com/@vaidehijoshi) is also a great and light-hearted resource to recap on the various data structures and algorithms.
### Mastery through Practice
### Mastery through Practice
Next, gain familiarity and mastery of the algorithms and data structures in your chosen programming language:
Next, gain familiarity and mastery of the algorithms and data structures in your chosen programming language:
Tay Yang Shun
7 years ago